Post by: hamdani yusuf on 02/11/2021 03:16:16
Post by: chiralSPO on 02/11/2021 03:39:07
Post by: hamdani yusuf on 02/11/2021 05:36:23
https://twitter.com/pickover/status/1454996588661719041
Let's see which one is the most correct.
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We learned it's an undefined value
As 0 to the power 0 looks like 0/0
But for Euler, he considered that 0^0 =1
(in his book "Introduction to Analysis of the Infinite")
There is no consensus among mathematicians about that..
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but doesn't the limit from the negative approach negative 1?
Quote
No. It approaches a complex number. The real part approaches 1. The imaginary part approaches an infinitesimally small negative number.
Post by: hamdani yusuf on 02/11/2021 05:40:08
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Any base raised to zero always equals to one. Also, zero raised to any exponent is zero. But by taking limits closer to zero let's say 0.00000001 base to the power of 0.00000001 is equal to 0.999999 which is closer to one.
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But since no value given current information, ie the expression does not determine original limit, so should be indeterminate and leave it undefined?
Correct me if I'm wrong.
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In short, from the perspective of mathematics, this is undefined.
Post by: hamdani yusuf on 02/11/2021 05:41:24
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I'd show him my phone ad hope he doesn't notice the question at the bottom -
(https://pbs.twimg.com/media/FDHo_xYXEAUIqB2?format=jpg&name=medium)
Post by: hamdani yusuf on 02/11/2021 05:51:03
And here's Wikipedia's answer.
Quote
https://en.wikipedia.org/wiki/Exponentiation#Zero_exponent
By definition, any nonzero number raised to the 0 power is 1:[16][1]
b^0 = 1.
This definition is the only possible that allows extending the formula
b^(m+n) = b^m . b^n
to zero exponents. It may be used in every algebraic structure with a multiplication that has an identity.
Intuitionally, b^0 may be interpreted as the empty product of copies of b. So, the equality b^0 = 1 is a special case of the general convention for the empty product.
The case of 0^0 is more complicated. In contexts where only integer powers are considered, the value 1 is generally assigned to 0^0 but, otherwise, the choice of whether to assign it a value and what value to assign may depend on context.
For more details, see Zero to the power of zero.
https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero
Post by: hamdani yusuf on 02/11/2021 05:58:50
Quote
0⁰ x must be equal to x. As it means that you multiplied 0 to x zero times. That is you never multiplied 0 to x. Hence x stay as it is. In that case 0⁰ has to be 1 for sure.
Post by: chiralSPO on 02/11/2021 06:03:12
y = xx is well-defined for all x > 0. The limit as you approach from positive side is clearly 1. Sure, the limit as you approach from the negative side, it's not well defined (in the reals), but neither are any of the points for x < 0, so I'm not sure that's a problem.
y = 0x is (boringly) well-defined as 0 for all x > 0. The limit as you approach from positive side is clearly not 1, but again, because the limit as you approach from the negative side is not well defined (in the reals), it doesn't seem that odd to me to have a jump discontinuity for one point before all hell breaks loose.
y = x0 is clearly (and boringly) 1 across all reals.
I was taught to think of y = xa as y = 1 × xa, interpreted as 1 times x, a times. So if a = 0, you're multiplying 1 by x zero times, and it doesn't matter what x is.
Post by: chiralSPO on 02/11/2021 06:05:12
Here's an unconventional approach.I'm not sure how unconventional that is. As I say in my previous post (which crossed with yours), this is how I was taught to think of this in school (8th grade, I think?)Quote0⁰ x must be equal to x. As it means that you multiplied 0 to x zero times. That is you never multiplied 0 to x. Hence x stay as it is. In that case 0⁰ has to be 1 for sure.
Post by: Petrochemicals on 02/11/2021 07:32:02
Post by: hamdani yusuf on 02/11/2021 09:40:20
Garbage.Someone's trash is someone else's treasure.
Post by: hamdani yusuf on 02/11/2021 09:44:22
y = xx is well-defined for all x < 0.
What does it mean? Doesn't it mean all negative x?
Post by: evan_au on 02/11/2021 10:08:49
Post by: hamdani yusuf on 02/11/2021 11:00:06
I'm not sure how unconventional that is. As I say in my previous post (which crossed with yours), this is how I was taught to think of this in school (8th grade, I think?)Perhaps just my ignorance. Wikipedia article doesn't explain it as obviously.
Many comments used limit to see the pattern, just like in Evan's video.
The video also appear in the Twitter conversation. Here's a response.
Quote
Great video, but it doesn't in any way show that 0^0=1, the real answer is undefined. That is about the limit of X^X in R+ as X approaches zero. In fact with with that kind of reasoning you might even argue that the answer is 0 because the same limit on R- is -1 (so 0 on average)
But Google's calculator has different answer.
Quote
(-0.0001)^(-0.0001) = 1.00092141 - 0.000314448745 i
Post by: Petrochemicals on 02/11/2021 12:21:28
4^0.5=2Garbage.Someone's trash is someone else's treasure.
4x(4x0.5)=8
(4x0.5)x(4x0.5)=4
4x(1÷0.5)=8
4^(1÷2)=2
4^1/2=2
Square root 4=2
Square root( 4^1) =2
4^3/4=2.828etc
Quad root (4^3)=2.828etc.
So basically you are looking for a root if the number drops below the whole number. But asking for a root of zero is nonsense for the same mathematical reasons that there are no positive square numbers.
But a zero root of a number is one?
Post by: chiralSPO on 02/11/2021 12:31:17
Sorry, I accidentally put the wrong sign (I meant all positive values). I have edited my post to correct the error. Thanks for pointing it out! :-)y = xx is well-defined for all x < 0.
What does it mean? Doesn't it mean all negative x?
Post by: chiralSPO on 02/11/2021 12:56:32
...ok.... so?4^0.5=2Garbage.Someone's trash is someone else's treasure.
4x(4x0.5)=8
(4x0.5)x(4x0.5)=4
4x(1÷0.5)=8
4^(1÷2)=2
4^1/2=2
Square root 4=2
Square root( 4^1) =2
4^3/4=2.828etc
Quad root (4^3)=2.828etc.
So basically you are looking for a root if the number drops below the whole number.
If you want to think of it that way, it still kinda works, but that is not the easiest way to think of it. Instead, think of exponents as multiple multiplication (just like multiplication can be thought of as multiple addition)
7 x 3 = 0 + 3 + 3 + 3 + 3 + 3 + 3 + 3 = 21
2.7 x 3 = 0 + 3 + 3 + 0.7 x 3 = 6 + 2.1 = 8.1
0 x 3 = 0 + (zero additions of three) = 0
0 x (any number) = 0 + (zero of that number) = 0
0 x 0 = 0
if we then go to exponents:
37 = 1 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 2187
32.7 = 1 x 3 x 3 x 30.7 = 9 x 2.16 = 19.42
30 = 1 x (zero multiplications by three) = 1
any number 0 = 1 x (zero multiplications by that number) = 1
03 = 1 x 0 x 0 x 0 = 0
0(any number) = 1 x 0 (that number of times) = 0
0 0 = 1 x (zero multiplications of zero) = 1
But asking for a root of zero is nonsense for the same mathematical reasons that there are no positive square numbers.
But a zero root of a number is one?
Presumably, you meant to say that there are no negative square numbers? (and that is easily fixed with imaginary numbers, but let's stick with real numbers for now).
Post by: Zer0 on 02/11/2021 14:06:29
0 * 0 = 0.
0 / 0 = 1.
0 ^ 0 = Undefined.
0 - 0 = Nothing.
Ps - Dyscalculia!
👻
Post by: chiralSPO on 02/11/2021 14:20:36
0 + 0 = 0.
0 * 0 = 0.
0 / 0 = 1.
0 ^ 0 = Undefined.
0 - 0 = Nothing.
Ps - Dyscalculia!
👻
I would switch the third and fourth: 0/0 = undefined and 00 = 1
Post by: hamdani yusuf on 02/11/2021 15:13:17
Let x be a number very close to 0.0 + 0 = 0.
0 * 0 = 0.
0 / 0 = 1.
0 ^ 0 = Undefined.
0 - 0 = Nothing.
Ps - Dyscalculia!
👻
I would switch the third and fourth: 0/0 = undefined and 00 = 1
let c be a finite non-zero number.
xc/x = c
x^xc = 1
thus
0/0 = undefined
0^0 = 1
Post by: Eternal Student on 02/11/2021 17:49:40
The original question was:
What's 0 to the power of 0?This is actually a very good question and it's something that isn't easily resolved.
It also turns out that whenever something like 00 is encountered in science, it has nearly always arisen as a Limt of xx as x→0+ (approaching 0 from the right). This limit is defined and does equal 1. As a consequence of this it has become an un-official convention that 00 = 1 and regrettably many calculators, like the one you were using, will show you that result.
Too often we start from a false assumption. It is easy to imagine that just because we can write some mathematical term down it must have some numerical value. For example, I have never calculated the value of 10067 + 765409 but I might assume it is some Real number. We also frequently assume that when there are patterns to follow we must be able to extend those patterns. For example, whatever my answer to that sum might be it should be bigger than the first number, 10067.
For exponentiation it's actually much safer if we start by assuming nothing at all. Do not assume ab defines any function from (a,b) → ℜ and don't even assume that the Real numbers exist. Instead start from more basic assumptions (axioms). If we do build up the Real Numbers and develop enough real Analysis to construct the exponential series then we will see that 00 was never defined and indeed it cannot be defined in any consistent way as a Real number.
I could just spit out some chapters from a textbook on Real Analysis or Complex Analysis that talk about the exponential series but I can't do that any better than the textbooks. Instead let's put out some minor problems to consider, which might help to identify just how complicated it is to raise numbers to an exponent:
1. What is 41/2 ? Why?
2. What is (-4)1/2 ? Is there no real solution?
4. By the rules of indicies we have (x2)1/2 = x for all x. This seems reasonable but what happens to the LHS and RHS when you let x = -3 ? Don't we obtain +3 = -3 ?
5. 3π cannot be written as the integer root of any integer power. Specifically 3π ≠ (a√3)b for any inetgers a,b. So what is the value of 3π? Could it be a negative number? If that's too easy consider (-3)π .
Best Wishes.
PS, yes I know question 3 was missing.
Post by: Petrochemicals on 02/11/2021 18:31:26
Presumably, you meant to say that there are no negative square numbers? (and that is easily fixed with imaginary numbers, but let's stick with real numbers for now).yes. I imagined something different.
Post by: hamdani yusuf on 02/11/2021 22:23:16
What is 41/2 ? Why?y=41/2
y=2
or
y=-2
because 22=4
and (-2)2=4
Post by: hamdani yusuf on 02/11/2021 22:28:56
What is (-4)1/2 ? Is there no real solution?(-1 . 4)1/2 = (-1)1/2 . (4)1/2
= 2i
or
= -2i
No real solution.
Post by: hamdani yusuf on 02/11/2021 22:35:37
but what happens to the LHS and RHS when you let x = -3 ? Don't we obtain +3 = -3 ?No.
I found some calculators treat exponentiation of negative value differently. Some treat the negative sign as part of the base, some others treat it as sign of the result. To avoid ambiguity, we can use brackets.
Post by: hamdani yusuf on 02/11/2021 23:01:09
So what is the value of 3π? Could it be a negative number?A fractional power produces more than one answers. Rational power has finite number of answers. Irrational power has infinite number of answers.
3π has infinite number of answers in complex plane.
In polar complex plane, the magnitude is around 31.544..., but any angle can be the argument.
Post by: hamdani yusuf on 02/11/2021 23:09:16
If that's too easy consider (-3)π .(-3)π = (-1)π . (3)π
Since any angle can be the answer, then it's the same as previous question.
Post by: Eternal Student on 03/11/2021 00:53:20
You're doing well there @hamdani yusuf .
You've already got enough to see that real numbers raised to some exponenet aren't always equal to a real number.
Sometimes, you can't find any real number that would be suitable. Sometimes you can find many suitable Real numbers. It's quite natural to extend the scope of the problem to consider complex numbers but then you can sometimes find an infinite set of numbers as a solution.
You already have enough evidence to recognise that 00 was never required to be a unique Real number. It's not even required to be a unique Complex number. You've also presented enough examples to see that it cannot be defined as a real number in any consistent way. If it has to be anything, it could be a bicycle. It's just a collection of mathematical symbols we can write down but it's not representative of any numerical value. ∀6 12 is another set of symbols that doesn't equal or represent any numerical value.
Technically, x1/n is defined to be the positive root wherever there was a choice. So that 91/2 is +3 and nothing else. This is done because it maintains exponentiation as a well defined function for as long as possible. Anyway, using that it would mean that your answer to q. 4 is wrong.
If you were given the expression (x2)1/2 = x and then set x= -3 then you would have to deduce that +3 = -3.
Obviously we don't really want anything that silly, so the only possibile resolutions are that we give up on considering ab as a well defined function for all a,b ∈ Z OR ELSE accept that the given equation (x2)1/2 = x was not valid for all x∈ℜ.
Mathematics has taken the second option, the rules of manipulating indicies that we were taught in school do not hold for all real numbers as a base for the exponentiation. A good teacher might have brought that to the attention of their students but it wouldn't matter much anyway: As human beings we want to follow patterns and we want to extend these patterns wherever we can, so we would have ignored any warning.
Thus (xm)1/n = xm/n is only a true statement for some values of x. The original question 4 that I presented was a little misleading. The most appropriate response should be "Exponentiation cannot follow this rule even though it seems like it should (because otherwise +3 = -3)".
I needed this to be considered because there are so many misconceptions and false proofs based on using "rules of indicies" even though these rules do not and cannot hold in the system of mathematics we commonly use.
Best Wishes.
Post by: hamdani yusuf on 03/11/2021 02:08:52
If you were given the expression (x2)1/2 = x and then set x= -3 then you would have to deduce that +3 = -3.No.
sin(0)=sin(π) doesn't imply 0=π
x2=y2 doesn't imply x=y
(x2)1/2=(y2)1/2 doesn't imply x=y
Post by: Eternal Student on 03/11/2021 09:36:49
If you were given the expression (x2)1/2 = x and then set x= -3 then you would have to deduce that +3 = -3.I don't know what to say in response to this. I can see that you don't want to accept it and of course you are free to do as you wish.
No.
Perhaps we should go back to one of the earlier posts you wrote.
Quote from: Eternal Student on Yesterday at 17:49:40You can't have y= +2 and also y= -2 because equals (=) is a transitive relation: If y = a and also y= b then a=b. So y= +2 and also y= -2 forces the relationship +2 = -2.
What is 41/2 ? Why?
- - - - - - -
Hamdani replied:
y=41/2
y=2
or
y= -2
because 22=4
and (-2)2=4
As a consequence, if you decide that 41/2 is equal to a real number then it can only be taken as one number.
In order to keep exponentiation as a well defined function for as long as possible (or if you want to phrase it another way - in order to maintain the consistency of Mathematics so that = is transitive), you are forced to assign only one number to 41/2 or else you must not assign any number to it at all.
By convention we take 41/2 = positive root of 4.
This is slightly different to the example Sin(0) = Sin(π). You can assign more than one value in the domain of a function to the same value in the Range. However, you cannot assign one value in the domain to more than one value in the Range. ("Many to One" is OK but "One to Many" is not).
Best Wishes.
Post by: Zer0 on 03/11/2021 19:17:13
😊
🙏
Just wanted to show you folks a lil something...
[ Invalid Attachment ]
[ Invalid Attachment ]
Ps - I use a Poco Android Mobile(model) by MI(company).
& I find it Strange that a built in default Chinese designed Calculator App is in clear defiance of the all mighty & powerful GooGle.
🤔
(The damm thing won't even Uninstall, & no I'm Not into Rooting so save that advice for someone else)
Post by: Petrochemicals on 03/11/2021 20:34:56
Thank You for the Corrections.Appalling grammer, it should be cannot, not "can't". You also cannot divide by 1 or anything beneath
Just wanted to show you folks a lil something...Screenshot_2021-11-04-00-41-12-800_com.miui.calculator.jpg (116.43 kB . 1080x2156 - viewed 3113 times)
Ps - I use a Poco Android Mobile(model) by MI(company).
& I find it Strange that a built in default Chinese designed Calculator App is in clear defiance of the all mighty & powerful GooGle.
(The damm thing won't even Uninstall, & no I'm Not into Rooting so save that advice for someone else)
Post by: Bored chemist on 03/11/2021 20:58:43
Appalling gramme, it should be cannot, not "can't". You also cannot divide by 1 or anything beneathWrong on so many levels.
I can divide by a half perfectly well.
Five divided by half is ten.
(five divided in half is rather less than ten).
I can divide by one very easily indeed.
x/1=x
There's nothing wrong with "can't" as a word.
And it looks like you can't spell grammar- though it may be that you can't abbreviate "programme"
Presumably, you consider these statements of fact to be "repetitive antagonism"
Post by: hamdani yusuf on 03/11/2021 22:02:10
For fractional power, we get a principal root and one or more other roots. But the selection of principal root is just a convention. It's neither more nor less true than the other roots.
Here's another example related to the original question here.
https://www.wolframalpha.com/input/?i=%28-0.01%29%5E-0.01
Post by: hamdani yusuf on 03/11/2021 22:05:00
This is slightly different to the example Sin(0) = Sin(π). You can assign more than one value in the domain of a function to the same value in the Range. However, you cannot assign one value in the domain to more than one value in the Range. ("Many to One" is OK but "One to Many" is not).Try arcsin(x) = 0
Post by: Eternal Student on 04/11/2021 02:28:36
But the selection of principal root is just a convention.Yes and no. It is a convention but it is a widely used and widely recognised one. I think it would be hard to find any credible source in any country of the world that didn't define a fractional power to be the positive or principal root. So that 41/2 = +2 and never -2.
There's another good reason for making this decision. It turns out that we're only going to get the positive root if we use the exponetial series.
The exponential series, Exp(x) has all the properties we want for ex.
Exp(x) =
I mentioned earlier that if you build up the Real numbers from first principles and develop some Real Anlysis then this series is found to be convergent for all x∈ℜ. We can take the definition further and apply it in the Complex plane and then it turns out the Exp(x) function is just fantastic (it has an infinite radius of convergence, it's holomorphic throughout C, it's just ideal and very well behaved).
The key point is that it has the properties we want for exponentiation, in particular Exp(x) . Exp (y) = Exp (x+y). From which all the other usual properties of exponents will follow Exp(-x) = 1/Exp(x) and for rational numbers a/b we will have Exp(a/b) = (b√e)a
It is therefore possible to define exponentiation with this series as your prototype. So it's not necessary to imagine that ab means that you start with the base a multiplied by itself a few times, take a suitable root and sometimes invert the final answer. Instead we have an analytical definition for ab it is the number you obtain as the limit of this series Exp ( b. Ln a) and it will be well defined whenever Ln(a) exists.
(The advantage of making the limit of this series your definition of ab is that it remains consistent with algebriac definitions for rational exponents but also allows you to evaluate irrrational exponents and, if you extend into complex analaysis, arbitrary complex exponents).
Anyway, we get two results from this identification. Firstly, Exp(1/2) = sum of terms all of which are positive, so we get a positive answer. This means that Exp(1/2) is finding the positive square root of e and not the negative root. So if we're going to use this analytical definition for exponentiation then 41/2 = +2 and not -2.
The second item of interest is that, despite all the advantages of defining exponentiation based on this series, it's surprisingly uselss for evaluating 0 raised to a power and we have to fall back to algebraic definitions for this. (This post is already long, so I'm stopping here).
You also mentioned this:
Try arcsin(x) = 0ArcSin is only a well defined function when you restrict the domain of the parent Sine function.
(https://www.intmath.com/analytic-trigonometry/svg/svgphp-inverse-trigo-functions-7-s6.svg)
Best Wishes.
Post by: hamdani yusuf on 04/11/2021 11:08:59
ArcSin is only a well defined function when you restrict the domain of the parent Sine function.The original question doesn't restrict the path to the answer to be a function. There's no restriction either for the solution to be in real number.
Post by: hamdani yusuf on 04/11/2021 11:12:25
it will be well defined whenever Ln(a) exists.When does Ln(a) not exist?
Post by: hamdani yusuf on 04/11/2021 11:24:14
https://www.wolframalpha.com/input/?i=%28-0.3%29%5E-0.3
Position in the complex plane
(https://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=83466.0;attach=32472)
I'm not sure why that particular point is selected as the principal root.
Plot of all roots in the complex plane
(https://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=83466.0;attach=32474)
Post by: Eternal Student on 04/11/2021 14:49:46
I'm losing track of what was being discussed here.
This bit I can do:
When does Ln(a) not exist?When a = 0.
Best Wishes.
Post by: Zer0 on 04/11/2021 19:08:53
Appalling gramme, it should be cannot, not "can't". You also cannot divide by 1 or anything beneathWrong on so many levels.
I can divide by a half perfectly well.
Five divided by half is ten.
(five divided in half is rather less than ten).
I can divide by one very easily indeed.
x/1=x
There's nothing wrong with "can't" as a word.
And it looks like you can't spell grammar- though it may be that you can't abbreviate "programme"
Presumably, you consider these statements of fact to be "repetitive antagonism"
[ Invalid Attachment ]
Umm...unfortunately, the designer is Chinese.
Maybe it provides correct grammatical sentence construction if the language is changed/switched to " Mandarin ".
(I ain't interested in figuring that out, i consider it as a default in-built Calculator Application...We all can agree courteously that it ain't a Wren & Martin.)
Ps - If anyone wishes to write in to Xiaomi or call their 1-800 Number, Please go ahead.
🙏
I feel my Life, and of Others around me gets Alot more easy n comfortable, provided i can just
" Let Go of Things, & Move On ".
(I shall stick to it)
😇
Post by: Bored chemist on 04/11/2021 19:12:49
Umm...unfortunately, the designer is Chinese.Why is that unfortunate?
Post by: Zer0 on 04/11/2021 19:38:34
Umm...unfortunately, the designer is Chinese.Why is that unfortunate?
At the time when i purchased this android mobile phone...
China was making leaps & bounds progress in almost all sectors.
👍
I viewed China as a Pioneer & Visionary.
& no doubt The device beats all other competitors in design, coverage, computational power & is economically priced.
But then, Unfortunately, things changed pretty quickly pretty drastically internationally.
(Covid19, TikTok Spying, Chinese aggression in South China Sea, Neighborhood disputes, Hong Kong, Taiwan, Afghanistan, CCP moving towards (complete)Totalitarianism.)
I have lost my trust & faith.
👎
There are quite a few in-built applications that are uninstallable.
I cannot root the device, orelse warranty snaps, & of course the pandora's jar opens.
I check complete details before updating os & security patches.
Ps - In short, Now i truly understand, what it means & how it feels like, to be sleeping with the enemy.
🤞
(Sure, i can divorce it & remarry, but to carry on flirting with disasters is in my lame genes.)
😉
Post by: Petrochemicals on 04/11/2021 22:58:51
You asked why I have my tag line, BC knows I do not respond but snipes anywhoo, that is why I have my tag line.Appalling gramme, it should be cannot, not "can't". You also cannot divide by 1 or anything beneathWrong on so many levels.
I can divide by a half perfectly well.
Five divided by half is ten.
(five divided in half is rather less than ten).
I can divide by one very easily indeed.
x/1=x
There's nothing wrong with "can't" as a word.
And it looks like you can't spell grammar- though it may be that you can't abbreviate "programme"
Presumably, you consider these statements of fact to be "repetitive antagonism"
[ Invalid Attachment ]
Umm...unfortunately, the designer is Chinese.
Maybe it provides correct grammatical sentence construction if the language is changed/switched to " Mandarin ".
(I ain't interested in figuring that out, i consider it as a default in-built Calculator Application...We all can agree courteously that it ain't a Wren & Martin.)
Ps - If anyone wishes to write in to Xiaomi or call their 1-800 Number, Please go ahead.
🙏
I feel my Life, and of Others around me gets Alot more easy n comfortable, provided i can just
" Let Go of Things, & Move On ".
(I shall stick to it)
😇
Post by: Bored chemist on 04/11/2021 23:06:25
BC knows I do not respond but snipes anywhoo,You mean "BC still points it out when I says stupid things".
And we know what the tag line is for- it's to warn others that they also might suffer the terrible loss; you might put them on your ignore list too.
Do I look bothered?
Post by: hamdani yusuf on 05/11/2021 13:22:14
Hi.What's the difference between something that doesn't exist and something that's not defined in math?
I'm losing track of what was being discussed here.
This bit I can do:When does Ln(a) not exist?When a = 0.
Best Wishes.
Imaginary number was once thought not to exist.
Post by: Eternal Student on 05/11/2021 15:38:33
What's the difference between something that doesn't exist and something that's not defined in math?One is a personality and the other is a mermaid?
(I know that's one a bit old. I don't get out much). Anyway.... What did I mean when I said Ln(a) doesn't exist?
In this context I did mean that it isn't defined.
Best Wishes.
Post by: Petrochemicals on 05/11/2021 22:23:48
Did he snipe?
Post by: Bored chemist on 06/11/2021 00:32:09
No; he just stated the obvious.Did he snipe?
Post by: hamdani yusuf on 06/11/2021 02:00:09
Quote
This had previously been the subject of some debate in the mathematical community until Donald Knuth set things straight in 1992, so it’s understandable that some confusion lingers, but the modern convention is to define 00=1 , for good reason.
https://www.quora.com/How-can-we-prove-0-0-is-equal-to-one
Quote
A definition cannot be “proven” right or wrong. That doesn’t even make sense.
Quote
It might appear that the “theorems” x0=1 and 0x=0 contradict each other in a way that forces us to leave 00 undefined. But 0x=0 cannot actually be a theorem, even if we ignore the 00 case: everyone agrees 0x is not defined for any negative x . It can’t be, because that would break the theorem that bx⋅b−x=1 .
Quote
So there are good reasons for wanting to define 00=1 , but does it break anything the way defining 0/0 would? In fact it doesn’t. All the usual laws of exponentiation remain valid with this definition.
That’s what we mean when we say that the choice to define 00=1 is the right one.
Post by: Eternal Student on 06/11/2021 12:52:08
This had previously been the subject of some debate in the mathematical community until Donald Knuth set things straight in 1992, so it’s understandable that some confusion lingers, but the modern convention is to define 00=1 , for good reason.I don't think there's a thing like the Oxford English Dictionary for Mathematics, somewhere that the very latest terminology and conventions are recorded and will be recognised as the definitive reference. So Donald will have a bit of trouble getting everyone to adopt that convention.
I Googled for "official mathematical conventions" and didn't find anything useful. I did find an extremely long and detailed booklet produced by ETS (Education Testing Service) which seems to be used as the standard for many examinations in the United States.
https://www.ets.org/s/gre/accessible/GRE_Math_Conventions_18_point.pdf.
On page 9 of 31 they write: Because all numbers are assumed to be real, some expressions
are not defined. Here are three examples: .........
Example 3: 00 is not defined.
This was produced in 2010, so about 18 years after Donald had apparently set the precedent that 00 will always be 1.
Anyway, defining 00 = 1 is something you can do and it obviously helps to keep things like power series written in a convenient form (just like the convention 0! = 1 ). It's certainly been an un-official convention for a long time. Just be careful if you write it in an examination where they aren't using that convention and be extremely careful if you ever obtain an expression like 0x ,or xy and need to examine the limiting behaviour as x→0 and/or y→0.
Best Wishes.
Post by: Zer0 on 06/11/2021 19:15:02
Did he snipe?
No; he just stated the obvious.Did he snipe?
ROFL!
🤣
I find myself rather precariously positioned between you two.
😔
I'd Request you both now to humbly forgive Me & kindly let me move on.
🙏
Throughout life, i have been corrected and put inline on various occasions by various Individuals.
I'm a bit Sensitive, and have sometimes disagreed with the ways in which Knowledge was imparted to me.
But I'm also a bit Sensible, hence mostly been appreciative that Knowledge was passed over to me.
An avid sweets trader was making the rounds. Going from village to village, selling his products.
🍭
As afternoon arrived, he decided to rest under the shades of a banyan tree.
🌳
As he was quite exhausted, soon he fell asleep, dreaming on the sweet profits he was generating.
😴
Only to be abruptly woken up by a Sage. Who seemed to be a venting in anger for reasons unknown.
😠
The Sage went on a rampage, thrashing the Trader left right & centre. Mostly targeting the trader's stomach.
🤛🤜
The Trader, a pious individual, had never before seen or known his molester, & kept on asking for forgiveness & begging for the beating to Stop!
🙏
As the Trader fell to the ground, the Sage trampled upon his stomach.
Immediately the Trader started vomiting, & lo & behold, A lil Snake sneaked out from the Trader's mouth.
😲
The Sage stopped, sat besides the Trader in a consolatory manner & explained...
As he was passing by, he saw the Trader resting with his mouth open.
Right about then, a lil snakelet made it's way right through inside.
Sage further explained, there was no time to wake up the Trader in an orderly manner, & no way to explain to the Trader what had just happened, considering the trader might have had a heart failure by fear, just knowing a snake had crawled inside of him.
Hence the Sage had to be decisive and act in time with brutality.
🐍
When the Trader Realised all that had happened...he fell onto the Sage's feet.
Thou pretty sensitive and in pain from the beatings, but alot more sensible to foresee what he had gained in return for a lil bit of bashing...his Life.
😇
All Sages have different ways of dealing with the Snake of Ignorance within Us.
✌️
& All Traders are not as appreciative. Humans just follow Logic, but are controlled by Emotions.
Ps - Wish to Apologize to the OP for hijacking it & taking it to a place where it clearly does Not belong.
I'm Sorry Yusuf.
I shall stop now.
🙏
Post by: hamdani yusuf on 07/11/2021 12:00:28
On page 9 of 31 they write: Because all numbers are assumed to be real, some expressionsFollowing the premise above,
are not defined. Here are three examples: .........
Example 3: 00 is not defined.
√(-1) would be undefined. Further implications are:
Euler's identity doesn't exist.
Wave equations don't exist.
Argument from authority has no weight in scientific debates.
Post by: Bored chemist on 07/11/2021 18:18:56
When will it matter?
Post by: Eternal Student on 07/11/2021 18:48:44
Argument from authority has no weight in scientific debates.By argument, I hope you mean discussion. I'm fine with you defining 00 =1 if you want to.
I'm genuinely surprised that there isn't an official list of mathematical conventions and symbols.... but there just doesn't seem to be. There is one for house building regulations, words in the English Language, Electrical engineers have various recognised industry standards etc. However, Maths doesn't seem to have anything like this.
Best Wishes.
Post by: hamdani yusuf on 08/11/2021 01:57:50
"What's 0^0 ?"When the answer is necessary to solve our problem.
When will it matter?
There were time when negative numbers didn't matter. There were also time when imaginary numbers didn't matter.
Post by: hamdani yusuf on 08/11/2021 02:31:20
By argument, I hope you mean discussion. I'm fine with you defining 00 =1 if you want to.
The word discussion is more suitable to replace the word debate. The argument from authority itself is a widely known logical fallacy.
https://en.wikipedia.org/wiki/Argument_from_authority
In this discussion, one side think that 00 is defined. The other side think that it's undefined. We try to convince the other side that our thought is better than the other. Many mathematical proofs are in the form of reductio ad absurdum, i.e. following the implications of an assumption to the point of absurdity/contradiction, to conclude that the assumption is false, hence we choose its opposite.
Arguing that our position is better because it's supported by an authoritative body can be useful in cases where each sides of the discussion have inadequate capacities to gather and process necessary data to produce correct and reasonable conclusions. Laymen discussion on complex problems like climate change or how to handle a pandemic are often cited as examples. Statements from the authorities can save our time and energy.
But if the goal is to find the fundamental truth, we need to process the raw data ourselves, and determine why a position is better than the other without relying on what the authority said.
Post by: Eternal Student on 08/11/2021 03:56:15
The word discussion is more suitable to replace the word debate.OK.
Perhaps I wasn't making myself clear. I'm not arguing with you in the sense of deliberately being unpleasant, or I certainly wasn't intending to do that anyway. It's always been a discussion as far I'm concerned.
As regards quoting authority, the most important point is that it isn't in my power to change the conventions. There doesn't even seem to be a place or document where current conventions, notation and terminology for Mathematics is recorded and provides a standard for the world. We could write a letter of complaint but I don't know where to send it.
If it's important to you then you have my support. It's a useful convention to have 00 = 1 it makes a fair level of sense and it certainly makes it easier to write out power series like
P(x) =
in one neat formula (which is commonly done) instead of writing something like this....P(x) = ao +
Or resorting to the use of two lines and splitting the definition of P(x) like this....
P(x) =
- - - - - - - - - - - - - - - -
But if the goal is to find the fundamental truth, we need to process the raw data ourselves, and determine why a position is better than the other without relying on what the authority said.Yes, seems like good advice.
You're actually doing amazingly well working through all the complications of defining what exponentiation should mean.
How (and why) will we define 0 raised to an irrational power, like 0π ? I honestly don't know. Most people would just take a rational sequence {xn} tending to Pi and look at the limit of 0xn - but what would you do and why?
Best Wishes.
Post by: Bored chemist on 08/11/2021 08:55:49
If the answer to 0^0 will solve our problem then the solution to the problem will tell us the answer."What's 0^0 ?"When the answer is necessary to solve our problem.
When will it matter?
There were time when negative numbers didn't matter. There were also time when imaginary numbers didn't matter.
However, in the same way that there's no useful problem that involves calculating 1/0- because the answer is undefined- there can be no useful problem that involves calculating 0^0.
Post by: hamdani yusuf on 08/11/2021 09:54:26
However, in the same way that there's no useful problem that involves calculating 1/0- because the answer is undefined- there can be no useful problem that involves calculating 0^0.
Quote
https://www.quora.com/What-is-0-0-the-zeroth-power-of-zero-1
And so we now assign 0^0 the value that’s useful, which is 1 . Why is that useful? Because it lets us manipulate exponentials without adding special cases.
Post by: Petrochemicals on 08/11/2021 14:08:01
If you can devide by zero
Post by: hamdani yusuf on 08/11/2021 14:51:01
If you can devide by zero
Let x be a number very close to 0.
let c be a finite non-zero number.
xc/x = c
x^xc = 1
thus
0/0 = undefined
0^0 = 1
Post by: Eternal Student on 08/11/2021 17:21:51
I don't really know what this was:
Let x be a number very close to 0.Why does that last line hold? I'm guessing you meant x^(xc) but perhaps it was (x^x).c Anyway x^(xc) is only approximately 1. Take x = 0.1 and c=10, Then x^(xc) = 0.1
let c be a finite non-zero number.
xc/x = c
x^xc = 1
Is this an argument based on continuity as x → 0 or something?
Could I also push this question again, please:
How (and why) will we determine 0 raised to an irrational power, like 0π or 0e or 0√2 ?
Can we prove that 0e = 0 from basic principles?
Perhaps we don't need to determine the value of these things, what do you think?
I'm not asking to be hostile. I'm just trying to see if you think that continuity is important. It doesn't have to be.
Best Wishes.
Post by: Bored chemist on 08/11/2021 17:25:10
Perhaps we don't need to determine the value of these things, what do you think?My best guess is that there will be no physical system where any measurable parameter is "predicted" to be 0 ^ pi or anything like that.
However, if there was, we could measure the parameter and find the result of the calculation.
Post by: Zer0 on 08/11/2021 18:04:23
Is 0 by default positive(+) ?
Or is it simply Neutral?
Could then there be a scenario in which one(person) could have say like a +0 & -0 occasions?
I've never ever heard of + or - zeroes.
& I suppose it's common sense that 0 is Neutral.
But if We ever had such an imaginary concept of +0 & -0...
Then what would come between them?
Singularity?
Ps - Please continue with your Positive discussions and do not go off track.
🙂👍
(Mathematics seems like a Universal Language, it is perhaps the No 1 largest & most spoken dialect on this planet)
Post by: Eternal Student on 08/11/2021 23:14:48
My best guess is that there will be no physical system where any measurable parameter is "predicted" to be 0 ^ pi or anything like that.Sounds sensible.
However, if there was, we could measure the parameter and find the result of the calculation.
I'm not ignoring your reply. It's just that the trhread might lose it's way if I discuss too much.
It seems that you appreciate that there can be a convention like 0! =1 without there being any fundamental requirement for 0! to be defined following from the basic axioms of mathematics. It is just a convention not some deep truth about 0! following from the axioms.
Same thing for 00 or even 0π. Exactly as you stated, we don't expect it to arise in any physically real situation. Limits are a different kettle of fish. Limits can appear, and 00 as a type of limiting form has all sorts of values depending on the exact situation.
- - - - - - - -
If you (Hamdani Yusuf) are still considering exponentiation. Then you may still like to cosider the question -
How can we determine 0π...... is continuity important in anything etc. (You don't have to. I'm not in charge).
- - - - - - -
Is 0 by default positive(+) ?For real numbers, the definition for "x being positive" is that "x is greater than 0".
Or is it simply Neutral?
Similarly "x is negative" means "x is less than 0". These are what we call strict inequalities, if x > 0 then x must be a bit bigger than 0, it cannot be exactly the same as 0.
Now, 0 is not greater than 0. So it is non-positive.
Similarly 0 is not less than 0. So it is non-negative.
There are several ways to describe 0. You could also say it is un-signed, although most of the time we don't bother, just say that it is 0 and we will all know it's not positive or neagtive. The only important thing to note is that when people say "this is positive" it isn't exactly the same as saying "this is non-negative". Non-negative numbers inlcude all the positive numbers PLUS the number 0.
Anyway, for ordinary mathematics +0 is identical to -0 and they are just plain old 0.
But if We ever had such an imaginary concept of +0 & -0...Only people with a cold heart?
Then what would come between them?
I can't really say. There are structures like the Hyperreal numbers where positive and neagtive non-zero infinitessimals exists. However, what lies between is not that interesting, there are just more infinitessimals including 0. I'm sure there are more interesting structures.
Best Wishes.
Post by: hamdani yusuf on 09/11/2021 01:35:09
Why does that last line hold? I'm guessing you meant x^(xc) but perhaps it was (x^x).c Anyway x^(xc) is only approximately 1. Take x = 0.1 and c=10, Then x^(xc) = 0.1I meant x^(cx)
Is this an argument based on continuity as x → 0 or something?
Try a smaller number for x, say 10-100. The equation applies for both positive and negative value of x, provided that its absolute value is much smaller than absolute value of c. Hence there's no reason to exclude only when x is exactly 0.
Post by: hamdani yusuf on 09/11/2021 02:03:01
Could I also push this question again, please:We need to distinguish between exact 0 and approximate 0.
How (and why) will we determine 0 raised to an irrational power, like 0π or 0e or 0√2 ?
Can we prove that 0e = 0 from basic principles?
Perhaps we don't need to determine the value of these things, what do you think?
dx in calculus is approximately 0. It doesn't work when it's exactly 0.
Fractional power is easier to understand when it's in polar format, because we can analyze the magnitude and phase angle separately.
For exact 0, the angle has no meaning. For approximate 0, when the power is irrational, any angle is elligible, including 0. For simplicity, it's usually chosen as principal value.
Post by: hamdani yusuf on 09/11/2021 02:06:57
Is 0 by default positive(+) ?Neutral
Or is it simply Neutral?
Post by: hamdani yusuf on 09/11/2021 02:10:28
is continuity important in anythingIt simplifies things, which improves efficiency. Efficiency is a universal instrumental goal.
Post by: Petrochemicals on 09/11/2021 04:13:17
I'm nOt sure what is supposed to meanIf you can devide by zeroLet x be a number very close to 0.
let c be a finite non-zero number.
xc/x = c
x^xc = 1
thus
0/0 = undefined
0^0 = 1
Post by: hamdani yusuf on 09/11/2021 05:59:39
Can you multiply by zero, Yx0 is essentially a non acting sum.Yes. The result is 0.
Post by: Petrochemicals on 09/11/2021 07:59:35
Is it? In what situation do you actually get such a sum? I know formula can render this sum under a condition, but the sum is discounted. Saying Yx0 is essentially a contradiction, 0xY is not a sum you would see as it would be discounted prior. 273 x 0 is essentially getting half way through a sum and then stopping, leaving it open. Is it undefined?Can you multiply by zero, Yx0 is essentially a non acting sum.Yes. The result is 0.
Post by: hamdani yusuf on 09/11/2021 10:37:05
Yes. It is.Is it? In what situation do you actually get such a sum? I know formula can render this sum under a condition, but the sum is discounted. Saying Yx0 is essentially a contradiction, 0xY is not a sum you would see as it would be discounted prior. 273 x 0 is essentially getting half way through a sum and then stopping, leaving it open. Is it undefined?Can you multiply by zero, Yx0 is essentially a non acting sum.Yes. The result is 0.
It's not half way. It's like planning to add 273 but cancel it.
Post by: Eternal Student on 09/11/2021 14:07:33
I meant x^(cx)Thanks, Hamdani. I can see this. So this is using some idea of continuity, the equation holds as you take a limit as x → 0.
Try a smaller number for x, say 10-100. The equation applies for both positive and negative value of x, provided that its absolute value is much smaller than absolute value of c. Hence there's no reason to exclude only when x is exactly 0.
We need to distinguish between exact 0 and approximate 0.But that's not what we are doing when we use limits or continuity ideas to suggest a value for 00. We are trying to make the behaviour at x=0 as much like the behaviour for other values of x as we possibly can. Specifically, we are keeping the value of the expression within some small difference of the value of the expression when x is just off 0 and we are confident that we can make this error as small as we want provided we take x close enough to 0.
If you really believed that the value at x=0 is significantly distinguished from values we obtain when x is approximately 0 but not 0, then continuity is certainly not what you're going to have. No matter how close we get to x=0 we should be prepared to accept that the value of the expression could be very different when x is precisely 0.
(From Eternal student) ....is continuity important in anything?OK. So continuity is of some importance to you and it was used in your argument x^(xc) = 1 when x→0.
Hamdani Yusuf replied: It simplifies things, which improves efficiency. Efficiency is a universal instrumental goal.
In which case, why are you so willing to lose continuity for 0x as x→0 ?
0^1 = 0, 0^0.1 = 0, 0^0.01 = 0, ....., 0^(any small positive rational) = 0, ......
If there's any hope for continuity then 0^0 = 0.
Don't get me wrong, there's no reason why continuity of these expressions should be important but we do need to ask the following question: Is it reasonable to imply that continuity is important in some situations and some expressions like the x^(xc) argument but then choose to ignore continuity in some other expressions like 0x ?
0x isn't an amazingly complicated expression or some example I have deliberately cooked up to get another limit. It just does look like one of the sensible expressions we might consider when we're trying to determine 00. What reason would there be for specifically ignoring that expression and prioritising continuity in an expression like x^(xc)?
There are plenty of more unusual expressions that we can "cook-up" (deliberately construct) to show other limits. Do we need some criteria to decide which expressions are more important than others?
Anyway, arguing for a value of 00 based on limits and continuity of apparently well defined expressions involving only exponents that are already well defined (so that will be rational exponents for example) doesn't usually get very far. It just shows that there doesn't have to be a unique value AND/OR that attempting to maintain continuity was not a suitable assumption or approach to tackle the problem. It isn't possible to maintain continuity for every simple expression involving exponents by assigning a single unique value for 00.
You do seem to have moved on from just using simple ideas based on continuity to something using complex number - but this post is already too long, so I'll stop here for a moment.
Best Wishes.
Post by: Eternal Student on 09/11/2021 16:08:48
I can't find much else going on in the forum today, So I'll continue with Hamdani's ideas concerning complex numbers and stuff...
dx in calculus is approximately 0. It doesn't work when it's exactly 0.I'm not sure what you meant by this or if you intend to develop the idea. The differential dx is not a number of any kind (unless we're using something like non-standard analysis and Hyperreal numbers). For standard analysis, the differential is just shorthand for expressing something about limits.
I think your main point was made in the previous line: We need to distinguish between exactly 0 and approximately 0. We can probably all agree that calculus does provide some precedent for the usefullness of doing this.
Fractional power is easier to understand when it's in polar format, because we can analyze the magnitude and phase angle separately.OK. So you're saying that we can't have exponentiation meaningfully defined just in the Real numbers. It only makes sense in the Complex plane. This is a bit controversial but it's ok, let's go with it.
For exact 0, the angle has no meaning. For approximate 0, when the power is irrational, any angle is elligible, including 0. For simplicity, it's usually chosen as principal value.
This helps with determining 0π since it has modulus 0 and no one cares about the argument since that won't actually change the location or value of the complex number anyway, it will be at the origin on an argand diagram so it's just plain old 0. That's great, we at least have a unique definition for 0π but we do seem to have paid a high price for this....
You mentioned in an earlier post that 3π is not uniquely defined as a Complex number. It is an infinite set of values, all with modulus approx. 31.5 but arbitray argument. So we've lost our ability to determine a unique value for most real numbers raised to an irrational exponent.
The best we can hope for is that a convention is applied for 3π so that an argument of 0 is assumed - but there is no fundamental value that is more truthful or follows more naturally from the axioms of mathematics. Is that right?
There's also another minor question I have to ask: How did you determine the modulus of 3π anyway? It's approximately 31.5 but why?
Best Wishes.
Post by: Zer0 on 09/11/2021 18:53:25
-1 ➕ +1 = ?
Ps - 🎃
Post by: hamdani yusuf on 10/11/2021 02:24:10
We are trying to make the behaviour at x=0 as much like the behaviour for other values of x as we possibly can. Specifically, we are keeping the value of the expression within some small difference of the value of the expression when x is just off 0 and we are confident that we can make this error as small as we want provided we take x close enough to 0.A picture speaks a thousand words.
(https://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=83466.0;attach=32480)
Post by: hamdani yusuf on 10/11/2021 02:29:08
If you really believed that the value at x=0 is significantly distinguished from values we obtain when x is approximately 0 but not 0, then continuity is certainly not what you're going to have. No matter how close we get to x=0 we should be prepared to accept that the value of the expression could be very different when x is precisely 0.Yes, in cases where positive x has different value than negative x, like 1/x.
(https://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=83466.0;attach=32482;image)
Post by: hamdani yusuf on 10/11/2021 02:34:22
Don't get me wrong, there's no reason why continuity of these expressions should be important but we do need to ask the following question: Is it reasonable to imply that continuity is important in some situations and some expressions like the x^(xc) argument but then choose to ignore continuity in some other expressions like 0x ?0^x discontinues at x exactly 0.
0x isn't an amazingly complicated expression or some example I have deliberately cooked up to get another limit. It just does look like one of the sensible expressions we might consider when we're trying to determine 00. What reason would there be for specifically ignoring that expression and prioritising continuity in an expression like x^(xc)?
(https://www.thenakedscientists.com/forum/index.php?action=dlattach;topic=83466.0;attach=32484;image)
Post by: hamdani yusuf on 10/11/2021 02:50:26
I'm not sure what you meant by this or if you intend to develop the idea. The differential dx is not a number of any kind (unless we're using something like non-standard analysis and Hyperreal numbers). For standard analysis, the differential is just shorthand for expressing something about limits.For example, in equation y=sin(2x). The gradient at the origin is dy/dx =2 when dx is close to 0, for both positive and negative values of dx. But if dx is exactly 0, dy is also exactly 0, which makes dy/dx undefined.
Post by: hamdani yusuf on 10/11/2021 02:54:04
OK. So you're saying that we can't have exponentiation meaningfully defined just in the Real numbers. It only makes sense in the Complex plane. This is a bit controversial but it's ok, let's go with it.Let's make it fractional exponentiation can't be meaningfully defined just in the Real numbers.
Post by: hamdani yusuf on 10/11/2021 03:30:29
You mentioned in an earlier post that 3^π is not uniquely defined as a Complex number. It is an infinite set of values, all with modulus approx. 31.5 but arbitray argument. So we've lost our ability to determine a unique value for most real numbers raised to an irrational exponent.Right. The reason is consistency with definition for root of unity.
The best we can hope for is that a convention is applied for 3^π so that an argument of 0 is assumed - but there is no fundamental value that is more truthful or follows more naturally from the axioms of mathematics. Is that right?
Quote
https://en.wikipedia.org/wiki/Root_of_unity
n mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power n.
(https://upload.wikimedia.org/wikipedia/commons/thumb/4/40/One5Root.svg/330px-One5Root.svg.png)
The 5th roots of unity (blue points) in the complex plane
PS I wonder why superscript formats keeps disappearing whenever I quote previous posts. Is it a bug, or is it a feature?
Post by: Eternal Student on 10/11/2021 04:47:36
You've been busy, Hamdani, well done. Looking through your posts it looks like you are actually trying to find good reasons to disregard some expressions and prioritise others. I'm quite impressed with the effort, most people would have given up on the idea already.
It looks like you've decided that a good reason for disregarding an expression is that it doesn't have a limit from the right and the left (and/or these aren't the same). That's actually quite reasonable.
If you're still interested in using that criteria, you should carefully check that there aren't any expressions that could still be a problem.
I might offer this example as a starting point:
[That doesn't always display well, it should be: Limit as x goes to 0 of (10 raised to -1/x) all raised to x ]
It has the general limiting form 00 which is what you wanted and it should have a well defined limit from the left and the right. You can either determine the limit or else plot it with some software or one of the web-sites offering a graph plotter online.
Best Wishes.
Post by: hamdani yusuf on 10/11/2021 13:29:54
In your example, the x cancel each other. So, no matter what the value of x is, the result is the same constant.
It's no longer an exponential function.
lim x->0 (10^(-1/x))^x
= lim x->0 (10^(-x/x))
= lim x->0 (10^(-1))
= 0.1
lim x->99 (10^(-1/x))^x
= 0.1
So, if we want to use limit to prove something, we need to make sure that the value of the limit affects the result. This is a criterion we often take for granted, and only realize when it's being violated.
Post by: Eternal Student on 11/11/2021 19:42:05
@hamdani yusuf .
1. Well done for getting the right limit. That thing tends to 1/10 or 0.1
2. The second part of your post isn't especially clear. Try explaining what you mean and what you would do another way and see if it still meakes sense. Then write it down.
Here's an example, which tries to generalise what you have done: Suppose a person wants to determine the height of the tower of London. They try various things and read various books until finally they find a statement like this "All towers are 100 metres high". Now you might think that they have their answer but as they consider it a bit more they realise that the height of towers never changes no matter where you are in the world or which tower it is. So they decide that this was information they really should NOT use. Since all towers are 100 metres high it just can't be giving any useful information about my tower in London.
This is the line of reasoning you are proposing for not using that expression (10^-1/x) ^ x. You seem to be saying that because the limit doesn't change when x→anywhere, it spoils the truth or usefulness of the information for indicating what happens as x→0.
Don't get me wrong.... if you are determined then you will eventually find some reasons for disregarding some expressions and only considering limits that arise from other expressions. However, you need to be a bit guarded against human nature. Ideally the criteria should be sensible and something that could reasonably have been established BEFORE examining all the possible situations. AFTER examining all the possibilties, you can certainly find or convince yourself of reasons for only using those expressions. For example, you can decide to use only those expressions for which the limit will evaluate to +1 (it's certainly going to get the result you wanted).
Mainstream pure mathematics, especially Real Analysis, went a different way to you. It's not better, it's just different. It was possible to define exponentiation based on the exponetial series. It's actually a very similar idea if you pull it all apart. In mainstream mathematics they do define AB to be the limit of something involving only whole number exponents. It's the limit of a suitable sum of terms involving only whole number exponents.
Best Wishes.
Post by: hamdani yusuf on 11/11/2021 20:45:13
It has the general limiting form 0^0 which is what you wanted and it should have a well defined limit from the left and the right.Except that it doesn't.
lim 10^(-1/0) is not 0. It's undefined.
So, your example represents undefined^0 instead of 0^0 as required. May be next time you can find a better example.
Post by: Eternal Student on 12/11/2021 01:25:04
lim 10^(-1/0) is not 0. It's undefined.You can put modulus signs around x, to have |x| instead of x if you're worried about left and right limits. I probably did miss that out - sorry, it was late and I was rushing and this LaTex equation format takes me ages to get working. By the time I've spell-checked and read through stuff I rarely get a post finished in less than an hour as it is.
If, for some reason you don't like a modulus, replace with an x2 term to get something like this:
and in the next step you can either continue to raise this whole thing to x (which gives you a differing two sided limit, not a limit of +1) or else raise all of the expression to the power of x2 to get back to something like the original idea.
May be next time you can find a better example.Sadly, I've got some other work to do and I'll probably be leaving this thread (and entire forum) alone for a while. It's been nice talking to you and I'm glad you've got some interest in mathematics.
Best Wishes and bye for now.
Post by: Petrochemicals on 23/11/2021 00:20:43
if we then go to exponents:To the power of 0 is 1 due to the answer = 1 portion of.
37 = 1 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 2187
32.7 = 1 x 3 x 3 x 30.7 = 9 x 2.16 = 19.42
30 = 1 x (zero multiplications by three) = 1
This is similar to as division where the answer is one portion of, the answer is equated to a singular.
20÷10equated to 1 is 10 singular portions of 20, if this is apparent? It may seem like I am just talking drivel? 20÷0.5 if 0.5 is equated to a singular the process is to find how many portions of 0.5 are in 1 or how many portions of 10 are in 1.
Vi's a vi, 2 portions of 0.5 in 1 meaning when 10 is the dividend the answer is 2 portions of the dividend or 20.
If you follow a similar method for multiplication 20x 10, 20 is a singular portion, vi's a vi 10 singular portions of 20 is 200.
So
3^2 = 3 x 3 = three singular portions that are 3 each= 9
3^1 = 3x1 = one singular portion that is 3 =3
3^0 =? = nil singular portions that are 3 each=? 0?
Post by: Bored chemist on 23/11/2021 08:49:33
Word salad.if we then go to exponents:To the power of 0 is 1 due to the answer = 1 portion of.
37 = 1 x 3 x 3 x 3 x 3 x 3 x 3 x 3 = 2187
32.7 = 1 x 3 x 3 x 30.7 = 9 x 2.16 = 19.42
30 = 1 x (zero multiplications by three) = 1
This is similar to as division where the answer is one portion of, the answer is equated to a singular.
20÷10equated to 1 is 10 singular portions of 20, if this is apparent? It may seem like I am just talking drivel? 20÷0.5 if 0.5 is equated to a singular the process is to find how many portions of 0.5 are in 1 or how many portions of 10 are in 1.
Vi's a vi, 2 portions of 0.5 in 1 meaning when 10 is the dividend the answer is 2 portions of the dividend or 20.
If you follow a similar method for multiplication 20x 10, 20 is a singular portion, vi's a vi 10 singular portions of 20 is 200.
So
3^2 = 3 x 3 = three singular portions that are 3 each= 9
3^1 = 3x1 = one singular portion that is 3 =3
3^0 =? = nil singular portions that are 3 each=? 0?
Post by: hamdani yusuf on 27/11/2021 22:59:25
The others like Eternal Student said it's undefined. Some counter examples using limits have been proposed to support it.
The dispute may come from the fact that two zeros in the question give different answers. 0^x=0 for positive x
0^x=undefined for negative x
x^0=1 for both positive and negative x
the question is when x is neither positive nor negative.
Let's campare them with other operations, *, /, + and -.
0*x=x*0=0
for positive and negative x
x/0=undefined
0/x=0
0+x=x+0=x
x-0=x
0-x=-x
For multiplication, addition, and subtraction, there's no dispute that the result is 0 when x=0. It's different for division and exponentiation.
Post by: Petrochemicals on 28/11/2021 01:16:36
There in lies the answer, given multiplication and division are the reverse of each other.
For multiplication, addition, and subtraction, there's no dispute that the result is 0 when x=0. It's different for division and exponentiation.
Post by: hamdani yusuf on 28/11/2021 10:55:29
What's the reverse of exponentiation?There in lies the answer, given multiplication and division are the reverse of each other.
For multiplication, addition, and subtraction, there's no dispute that the result is 0 when x=0. It's different for division and exponentiation.
Post by: Origin on 28/11/2021 13:46:55
What's the reverse of exponentiation?The inverse is the logarithm, just as division is the inverse of multiplication.
Why is this thread still going? You received your answer in the first page.
Post by: hamdani yusuf on 29/11/2021 04:13:09
Do you stop questioning things after you get first answer? Don't you want to hear some alternatives which might be a better answer, or provide a deeper understanding?What's the reverse of exponentiation?The inverse is the logarithm, just as division is the inverse of multiplication.
Why is this thread still going? You received your answer in the first page.
What's your response to Eternal Student's answer?
Post by: Petrochemicals on 29/11/2021 07:26:51
What is the process first of all?What's the reverse of exponentiation?There in lies the answer, given multiplication and division are the reverse of each other.
For multiplication, addition, and subtraction, there's no dispute that the result is 0 when x=0. It's different for division and exponentiation.
Post by: Petrochemicals on 28/02/2022 16:21:02
(2x2)+2+3 =9 =3x3
(3x3)+3+4=16=4x4
And so on
0x0=(-1x-1) +0+ (-1)=0
Whether you can consider -1 to be prior to zero is another matter.
Post by: Eternal Student on 28/02/2022 16:48:42
Looks ok @Petrochemicals .
I'm not sure how you intend to connect 0 x 0 with 0^0 = 00 which was the original question in this thread but that's probably not too important.
Best Wishes.
Post by: Petrochemicals on 28/02/2022 21:14:18
Hi.Short of fiddling in your brain I am powerless to do anything more.
Looks ok @Petrochemicals .
I'm not sure how you intend to connect 0 x 0 with 0^0 = 00 which was the original question in this thread but that's probably not too important.
Best Wishes.
Post by: hamdani yusuf on 02/03/2022 07:09:17
log0(x)=0
log0(x)=ln(x)/ln(0)
ln(x)=0
x=1
Quote
https://en.wikipedia.org/wiki/Logarithm#Definition
The logarithm of a positive real number x with respect to base b[nb 1] is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the unique real number y such that (https://wikimedia.org/api/rest_v1/media/math/render/svg/7b51b8dcca3e64714e2f3aba12bb8c68812f77cc)
The logarithm is denoted "logb x" (pronounced as "the logarithm of x to base b", "the base-b logarithm of x", or most commonly "the log, base b, of x").
An equivalent and more succinct definition is that the function logb is the inverse function to the function (https://wikimedia.org/api/rest_v1/media/math/render/svg/873987f9618fe2c30ce4e72cbd1a967ff759c1d4).
Post by: Eternal Student on 02/03/2022 16:12:44
How do you get line 2?
You let x = 0^0 so x isn't known to be a unique well defined real number. You are assuming it is.
If x = a battleship then Log0(x) = ?
This isn't intended to be a criticism. What you've done is quite common and sensible. It's human nature to start by assuming if 0^0 is a number then ask what number can it be? It's just that we will have already imposed some conditions on 0^0 just by tacitly assuming it is a real number.
A more thorough development of 0^0 would start by describing what you think 0^0 should be just in terms of some numbers which you accept or assume to exist (perhaps the integers or the standard set of Reals if you wish) and elementary operations defined on that set of numbers (so that could be binary operations like + and x). You have somewhat jumped over the first obstacle when you assume 0^0 is a real number.
However, if you do assume 0^0 is a real number then methods such as yours do start to indicate what number it must be. There's a few minor issues in your reasoning, like line 3 where you seem to have divided by Ln(0). Ln(0) is not a real number, it's -∞ in the extended reals but not defined at all in the reals. Perhaps you were only intending to identify 0^0 as an element of the extended reals.
Best Wishes.
Post by: hamdani yusuf on 03/03/2022 22:18:44
if x=0^0, then
log0(x)=0
log0(x)=ln(x)/ln(0)
ln(x)=0
x=1
Let's say that ln(0) is unknown. It can even be a complex number, for the sake of the argument.
log0(x)=0 by definition.
it can only happen if ln(x)=0, which means x=1.
Post by: Eternal Student on 05/03/2022 21:54:05
log0(x)=0 by definition.You should ask yourself what or who's definition of a Logarithm you are using.
Here's one example of a definition from Wikipedia:
Definition
The logarithm of a positive real number x with respect to base b is the exponent by which b must be raised to yield x.
So that definition only applies when both of the following things apply:
i) x is a real number.
ii) x > 0.
In my opinion the Wikipedia definition isn't actually perfect either, for example it makes very little sense if the base b = 0 - but that's a separate issue.
Best Wishes.
Post by: hamdani yusuf on 06/03/2022 04:07:18
Hi.In my example, x=1, which is both real and greater than 0.log0(x)=0 by definition.You should ask yourself what or who's definition of a Logarithm you are using.
Here's one example of a definition from Wikipedia:
Definition
The logarithm of a positive real number x with respect to base b is the exponent by which b must be raised to yield x.
So that definition only applies when both of the following things apply:
i) x is a real number.
ii) x > 0.
In my opinion the Wikipedia definition isn't actually perfect either, for example it makes very little sense if the base b = 0 - but that's a separate issue.
Best Wishes.
Post by: Eternal Student on 06/03/2022 12:13:07
I'm not sure what you want me to say here, @hamdani yusuf .
I've already said this
This isn't intended to be a criticism. What you've done is quite common and sensible.
I'm always quite happy when anyone is showing some interest in Mathematics and more than willing to spend some time in discussion like I've done here.
It's regrettable if I need to disagree with something or suggest a minor problem exists. However, this is where you need to recognise that a good student CAN learn from problems and only the weaker students (like myself, I'm quite a poor student) need to have their actions constantly re-inforced. There is a lot of what you've done that is good but hopefully you're a better student than I am. Hopefully, you can consider what might have been a problem.
The issue is that, presumably, you did not know x=1 until after the last line in the work.
So in line 2 you were making an assumption about the properties of x, you were assuming x is real and x>0 .
To phrase this another way, by performing line 2 in your proof you have forced the conditions:
(i) x is real
and (ii) x>0.
This is unless you are using some other definition of the Logarithm function (i.e. not a definition like that found in Wikipedia).
Best Wishes.
Post by: Eternal Student on 06/03/2022 15:22:42
This is something I used to do in school and it took me ages to work out where the problems are:
Suppose you have an equation x2 = x and you're asked to solve it.
It's reasonable to divide by x on both sides, you obtain:
x = 1 which is perfectly good, it's a totally acceptable solution.
The thing is the mathematics has accepted the tacit assumption you made when you divided by x. It has accepted that x could not have been 0 and it does then lead you to a perfectly valid solution. The only solution now possible is that x=1.
You've done much the same sort of thing when you perform line 2 in your proof. You have tacitly excluded possibilities like x = 0 or even that x is not a real number. The mathematics doesn't stop to question if that exclusion is reasonable, it just does as it is was told to do. You can proceed to perform valid mathematical operations after this but it is, of course, very likely to lead you to a solution where x is real and x ≠ 0.
What you have got is a proof that 00 =1 under certain assumptions:
Assumption 1) That 00 is a real number.
Assumption 2) That 00 > 0.
You also made this comment:
Let's say that ln(0) is unknown. It can even be a complex number, for the sake of the argument.Ln(0) isn't well defined. By the sound of it, you wish to force the condition that Ln(0) is a complex number. This is similar to the earlier situation. The mathematics will only find a solution given the conditions that were forced whether that is directly stated as an assumption or else tacitly assumed by the mathematical operations you perform.
Best Wishes.
Post by: hamdani yusuf on 06/03/2022 22:29:07
What you have got is a proof that 00 =1 under certain assumptions:I didn't make those assumptions. The answer was obtained as conclusion from the available definitions in mathematical operations.
Assumption 1) That 00 is a real number.
Assumption 2) That 00 > 0.
I'm open to alternative solutions if strong supports can be provided.
What do we get if we assume the contrary of those statements?
Post by: Eternal Student on 07/03/2022 00:55:47
The answer was obtained as conclusion from the available definitions in mathematical operations.
What definition of a Logarithm are you using?
Using your definition can you evaluate the following:
1. Log0(10)
2. Log10 (0)
Most definitions for Logb(x) fail to give you a well defined function for some input numbers b and/or x.
Best Wishes.
Post by: Eternal Student on 08/03/2022 19:42:56
There is now a considerable amount of discussion about 00 presented on Wikipedia.
https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero
Interestingly, using Google with the phrase "What is 0 to the power of 0" seems to bring up a slightly different set of results than I think it did when this thread was new.
The workings of Google and Wikipedia are a bit mysterious but it is supposed to be influenced by current internet activity and especially interactive sites. So it's possible that some interest in 00 shown in this forum has had a small role in influencing the recent changes in Google results and the Wiki entry. (Ok... I'm probably just unrealistic. More realistically, time spent on the forum is just time spent).
Best Wishes.
Post by: hamdani yusuf on 08/03/2022 22:28:35
Reverse of exponential.
What definition of a Logarithm are you using?
Using your definition can you evaluate the following:
1. Log0(10)
2. Log10 (0)
Most definitions for Logb(x) fail to give you a well defined function for some input numbers b and/or x.
Best Wishes.
1. Log(10)/Log(0) = 1/(-~) =-0
2. -~
How do you define "undefined" in mathematical sense?
Is following expression defined?
0 . 0
0 / 0
1/0
~
-~
√1
√(-1)
π
e^π
e^(πi)
arcsine(0)
arcsine(2)
Post by: DudleyC on 08/03/2022 23:48:45
I saw a fantastic explanation of this in Commercial link removed (http://Spam) some time ago,
Post by: Eternal Student on 09/03/2022 00:53:54
Reverse of exponential.That's a personal definition. I guess I did ask what definition you are using but I meant can you site a definition from a text-book or online reference.
Using your definition there are at least two issues:
1. Take b= 0 as the base for the logarithm (or the exponentiation). 0^1 = 0 but also 0^2 = 0 and 0^ 3 = 1 etc.
So which value do you map the value 0 back to when you try to invert that function? Is Log0 (0) = 1 , 2 or 3 ? To say that another way, exponentiation is sometimes a many-to-one function so that no inverse function exists.
What I'm trying to say is that it isn't as simple as saying "Logarithms are just the reverse of exponentiation", sometimes you can't find a reverse procedure.
2. Exponentiation isn't a trivial operation either. It certainly doesn't follow easily just from the basic field operations of (R, +, x). Algebra will allow a definition for ab where a and b are rational numbers but you need some results from real analysis to ascribe a meaning to ab where a and b could be irrational numbers.
- - - - - - - - -
You (@hamdani yusuf ) gave this as an evaluation of Log0 (10)
Log(10)/Log(0) = 1/(-~) =-0So, you're saying that Log0 (10) = 0 if I've understood that correctly.
So, by your definition of a Logarithm we have that 0 0 = 10.
How do you reconcile that with the proposal you made that 00 = 1 ?
- - - - - - - - - -
How do you define "undefined" in mathematical sense?
It depends on the situation. A function, f, is well defined if it can be expressed as subset, F, of A X B where A = Domain(f) and B = Range(f) such that (a,b) ∈ F and (a,c) ∈ F implies b=c.
To paraphrase that, a function is a rule that maps things in a domain to things in the range. It might map many things in the domain to one thing in the range but it cannot map one thing in the domain to many things in the range. It can be many-to-one but not one-to-many.
This probably isn't what you were asking, so I'm going to guess the general nature of the thing you were asking:
1. If we can write down some symbols and some numbers where there is clear mathematical meaning for some sets of numbers, does it follow that there is always a well defined mathematical meaning where we just change the numbers but keep the same symbols in the same places?
Answer: No.
2. If there is some pattern that seems to hold over a good range of values, can we just assume that the pattern does hold and push the numbers to the very limit?
Answer: No, not always.
- - - - - - - - - - - - - - - - - - - -
About the set of example expressions you (Hamdani) gave and asked if they were defined:
It depends on the context. For example ∞ is not a real number, so we would say that it is not defined in the Reals. However, you can use the symbol to mean something else, something that does have some meaning in a different mathematical structure. For example, in the extended Reals, there is an element denoted by the symbol ∞.
The symbol doesn't mean the same thing as it did in the Reals. In the Reals there was no such thing and the axioms for the incorporation of ∞ into the basic algebraic operations of + and x on other elements (finite numbers) in the extended reals do not force any equivalent results to hold in the reals.
- - - - - - - - - - - - - - - - - - - - - - - - -
Any number to the zeroth power is equal to 1 by definition.....(here's my website)....That looks like advertising. The best thing about it is that you've clearly shown the website got something wrong and probably isn't worth the subscription fee. Anyway, on the off chance that you were genuinely trying to join the discussion - hello and welcome.
Best Wishes.
Post by: hamdani yusuf on 09/03/2022 07:23:23
That's a personal definition. I guess I did ask what definition you are using but I meant can you site a definition from a text-book or online reference.
An equivalent and more succinct definition is that the function logb is the inverse function to the function (https://wikimedia.org/api/rest_v1/media/math/render/svg/873987f9618fe2c30ce4e72cbd1a967ff759c1d4).
https://en.wikipedia.org/wiki/Logarithm#Definition
Post by: Eternal Student on 09/03/2022 12:30:09
An equivalent and more succinct definition is that the function logb is the inverse function to the function .It might have been accidental but this editing has totally corrupted the original meaning. It's like quoting "Yes...I agree" from an extract that read "Yes that is interesting. However I can't agree with your conclusion".
1. It's the last line in a block of text where the first lines were more specific and excluded certain values.
2. There was a hyperlink on the word "inverse function" which has been lost. That link discusses the inability to find an inverse functions in some cases.
- - - - - - - - - - - - - - - -
You also haven't addressed this point:
So, you're saying that Log0 (10) = 0 if I've understood that correctly.
So, by your definition of a Logarithm we have that 0 0 = 10.
How do you reconcile that with the proposal you made that 00 = 1 ?
- - - - - - - - - - - - - - - - -
NOTE: I am not trying to be rude here. It just takes me hours to keep writing long posts. In the same style as your own posts, it's more useful to keep the points short and repeat anything that was important that was missed.
If you want to have 00 =1 then that's fine by me. I was just assuming you wanted some discussion.
Best Wishes.
Post by: hamdani yusuf on 09/03/2022 13:14:40
1. Take b= 0 as the base for the logarithm (or the exponentiation). 0^1 = 0 but also 0^2 = 0 and 0^ 3 = 1 etc.The thread title is about finding a value for a mathematical expression, regardless if it's a function or not.
So which value do you map the value 0 back to when you try to invert that function? Is Log0 (0) = 1 , 2 or 3 ? To say that another way, exponentiation is sometimes a many-to-one function so that no inverse function exists.
What I'm trying to say is that it isn't as simple as saying "Logarithms are just the reverse of exponentiation", sometimes you can't find a reverse procedure.
√1=1 but also √1=-1
I don't think it's a reason to say that √1 is undefined.
Post by: hamdani yusuf on 09/03/2022 13:18:02
It might have been accidental but this editing has totally corrupted the original meaning. It's like quoting "Yes...I agree" from an extract that read "Yes that is interesting. However I can't agree with your conclusion".It just shows that the definition I used here is not just my personal opinion like what you said.
1. It's the last line in a block of text where the first lines were more specific and excluded certain values.
2. There was a hyperlink on the word "inverse function" which has been lost. That link discusses the inability to find an inverse functions in some cases.
Post by: hamdani yusuf on 09/03/2022 13:25:04
NOTE: I am not trying to be rude here. It just takes me hours to keep writing long posts. In the same style as your own posts, it's more useful to keep the points short and repeat anything that was important that was missed.I think it's better to keep a post focused on one or a few points, which would make it easier to address in a response. Besides, sometimes I have to open the site from a mobile device, which makes long posts harder to read or write. Sometimes the page restarted unexpectedly, which made my writing lost. When it happens, sometimes I just don't bother to rewrite it.
If you want to have 00 =1 then that's fine by me. I was just assuming you wanted some discussion.
Post by: hamdani yusuf on 09/03/2022 14:36:32
So, you're saying that Log0 (10) = 0 if I've understood that correctly.Here are some other results.
So, by your definition of a Logarithm we have that 0^0 = 10.
How do you reconcile that with the proposal you made that 0^0 = 1 ?
Log0 (100) = 0
Log0 (1000) = 0
Log0 (-1) = 0
Log0 (-10) = 0
Apparently, using logarithm is not a correct way to answer the question in the OP.
Simple negative exponent is better for this task, as offered by Mascheroni.
(https://wikimedia.org/api/rest_v1/media/math/render/svg/95a871955aba3ddb741357a9ae1a827355aca7d1)
Post by: Eternal Student on 09/03/2022 21:41:41
as offered by Mascheroni.I hadn't actually heard of him (The internet says Lorenzo Mascheroni was an Italian mathematician).
I'm not sure how he got from
to = 1. It looks suspiciously like some repeated division by (a-a) was done. i.d.k. maybe he had something else in mind.
Best Wishes.
Post by: chris on 14/03/2022 13:11:31
What's 0 to the power of 0?
0^0 = Owl, no?
Post by: alancalverd on 14/03/2022 13:40:02
Post by: Eternal Student on 14/03/2022 23:58:51
Looks sensible, @alancalverd . There might be one symbol out of place but it's not serious.
∀n∩ℜ∀n∈ℜ
Best Wishes.
Post by: alancalverd on 15/03/2022 08:24:32
Post by: Colin2B on 15/03/2022 08:36:32
0^0 = Owl, no?Ah, yes. I see it now said piglet
Post by: hamdani yusuf on 16/03/2022 04:17:23
Hi.The equation is found in Euler's book, which attributes it to Mascheroni.as offered by Mascheroni.I hadn't actually heard of him (The internet says Lorenzo Mascheroni was an Italian mathematician).
I'm not sure how he got from
i.d.k. maybe he had something else in mind.
Best Wishes.
Quote
In 1752, Euler in Introductio in analysin infinitorum wrote that a0 = 1[15] and explicitly mentioned that 00 = 1.[16] An annotation attributed[17] to Mascheroni in a 1787 edition of Euler's book Institutiones calculi differentialis[18] offered the "justification"
https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero#As_a_value
Post by: hamdani yusuf on 16/03/2022 04:57:04
∀n∩ℜ, logn0 = -∞ so 00 is undefined by this route.Is logn0 well defined?
How do you define "undefined" in mathematical sense?
Post by: syhprum on 16/03/2022 17:56:13
Post by: alancalverd on 16/03/2022 22:26:32
Is logn0 well defined?Yes.Of all the infinities, the real continuum infinity is perhaps the best characterised and the easiest to work with.
Post by: Eternal Student on 16/03/2022 23:58:24
Is logn0 well defined?No, not in the Reals. There's no Real number which is assigned to Logn (0).
How do you define "undefined" in mathematical sense?It depends on the context. In it's simplest usage "undefined" just means "this thing is not defined". In the context of this thread I think the phrase "undefined" has most commonly been used to suggest that no real value is assigned to the expression. However, I'm sure some people have used it in a different context or to convey some other meaning. It may be best if you identitfy the specific sections where the phrase was used.
Best Wishes.
Post by: hamdani yusuf on 17/03/2022 03:53:50
Hi.In your definition, √(-1) is undefined. It makes i, as well as Euler's identity, eπi=-1, meaningless.Is logn0 well defined?No, not in the Reals. There's no Real number which is assigned to Logn (0).How do you define "undefined" in mathematical sense?It depends on the context. In it's simplest usage "undefined" just means "this thing is not defined". In the context of this thread I think the phrase "undefined" has most commonly been used to suggest that no real value is assigned to the expression. However, I'm sure some people have used it in a different context or to convey some other meaning. It may be best if you identitfy the specific sections where the phrase was used.
Best Wishes.
Post by: alancalverd on 17/03/2022 11:26:48
I bow to ES's statement: since you can't assign a numerical value to −∞ it isn't a member of ℜ. But it isn't imaginary either!
Post by: hamdani yusuf on 17/03/2022 13:57:30
Entirely true, since √-1 is not a real number and is therefore not defined within the set of reals. But Euler uses complex numbers, which include an imaginary part, a multiple of i.So, i is defined, even though it's not a real number? It implies that being a real number is not a necessity for something to be mathematically defined.
I bow to ES's statement: since you can't assign a numerical value to −∞ it isn't a member of ℜ. But it isn't imaginary either!
Can you assign a numerical value to π?
What makes −∞ different?
Post by: alancalverd on 17/03/2022 14:35:20
You can assign a real number that is infinintesimally less than π, and one that is greater, so you can define π as "that which lies between..." . You can't assign a real number that is infinitesimally greater or less than -∞, so as ES says, it isn't defined on the real number line.
Quote
a real number is a value of a continuous quantity that can represent a distance along a line
i is completely defined as √-1 but it doesn't lie on the line that passes through 0, 1, - 2.995, π2.7 , etc. so it isn't a real number.
Post by: Eternal Student on 18/03/2022 02:40:27
So, i is defined, even though it's not a real number?As alancalverd said, that is entirely true.
It's not to be given too much importance. The meaning of a word like "unconstitutional" is defined in some system (we might call that system the English language, for example) but it is not defined in the real number system.
Which leads us to the next comment. There are a whole load of things that are defined but aren't real numbers. There's even a whole load of things that are "mathematically defined" but aren't real numbers.
It implies that being a real number is not a necessity for something to be mathematically defined.Yes, entirely true. There are many axiom systems, or many different systems of mathematics. It is quite possible for something to be "mathematically defined" but not be a real number. The set of all rotations in 3-dimensional space is a structure called a group under the group operation • which is just the composition of two rotations. It's a perfectly good mathematical structure, all the objects in it are mathematically defined etc. - you can do things with that structure that we would all consider as being mathematics. However, none of the objects in that structure are real numbers.
I should think it is possible to make everything perfectly well defined in SOME system but that system isn't always useful. (Actually this is debatable - if the system is built from standard set theory then there are problems defining things like the set of all sets which do not contain themselves - but let's just assume that in general you can make everything well defined in some system of mathematics). Let's just bring this back to the main topic of the thread: There may be some system in which the expression 00 is defined, if that's what you wanted to know. However, it's not the conventional real number system and where the symbol 0 and the notation for exponentiation ab is used to produce the expression 00. In the real number system, the expression 00 is deliberately left "undefined": It is a meaningless expression, it's just a collection of symbols put together. There is a convention that 00 =1 which is very useful for things like writing out power series in the form
because you don't have to keep writing the constant term a0 separately from the sigma notation when x could be 0. (But that's just a convention for ease of notation).Why is it deliberately left undefined? See earlier discussions. There are problems maintaining all the properties we may want from the exponential function (like continuity) and/or all the usual rules of manipulating indices in an algebraic expression.
Can you just add the definition 00 =1 ? Sure you can, just declare it and the job is done. It doesn't make the aforementioned problems go away but there's no law that says you can't extend the definition of ab like that and no special squad of mathematicians, let's call them the continuity police, that will come knocking on your door to punish you for that decision.
However, since the problems haven't actually just gone away, you're going to have to be very careful when working with expressions like 00 and not assume all the usual rules of algebraic manipulation and/or analytical conditions like continuity apply. So you're still almost always going to have to treat the appearance of an expression like 00 as a special or separate case from the appearance of most other exponential expressions like ab. You gained nothing useful just by adding the definition 00 =1. What we would really want is that all the properties of exponents we've come to expect for ab continue to hold when a and b are both 0 and that does seem to be impossible (it produces inconsistencies) in the Real Number system. You can't have all the usual properties holding you have to pick and choose just a few. I agree that setting 00 = 1 probably allows the greatest amount of these properties to hold but the important thing is that it's not all of them.
Best Wishes.
Post by: hamdani yusuf on 18/03/2022 10:46:21
I agree that setting 00 = 1 probably allows the greatest amount of these properties to hold but the important thing is that it's not all of them.OK, deal.
I see the reluctance from accepting to define a value of a mathematical operation is related to stability of the result when the operands are perturbed. Here are my findings for mathematical operations with defined results:
0 * 0 is extremely stable. The operands can be perturbed heavily without changing the result (e.g. by multiplying them with some constants). Adding one of the operands with a finite constant doesn't change the result. Adding both operands with a finite constant does change the result.
1 * 0 is very stable. The first operand can be perturbed heavily without changing the result. Multiplying the second operand with a finite constant doesn't change the result, but adding it with a finite constant does change the result.
1 * 1 is less stable. Perturbing one of the operands with a finite constant does change the result. But multiplying them with a constant infinitesimally close to 1, or adding them with a constant infinitesimally close to 0 doesn't change the result.
Post by: alancalverd on 18/03/2022 16:04:31
1 * 1 is less stable. Perturbing one of the operands with a finite constant does change the result. But multiplying them with a constant infinitesimally close to 1, or adding them with a constant infinitesimally close to 0 doesn't change the result.
Oh yes it does!
(1 * Δ)2 = Δ2 ≠ 1
(1+Δ) 2 = 1 + 2Δ + Δ2 ≠ 1.
Post by: Eternal Student on 18/03/2022 22:00:40
I see the reluctance from accepting to define a value of a mathematical operation is related to stability of the result when the operands are perturbed.I don't really know what you mean by "stability". It sounds like you're considering continuity when you consider infinitessimal perturbations.
Best Wishes.
Post by: alancalverd on 18/03/2022 23:00:55
Post by: hamdani yusuf on 20/03/2022 05:38:14
Ok. They don't change the result much.1 * 1 is less stable. Perturbing one of the operands with a finite constant does change the result. But multiplying them with a constant infinitesimally close to 1, or adding them with a constant infinitesimally close to 0 doesn't change the result.
Oh yes it does!
(1 * Δ)2 = Δ2 ≠ 1
(1+Δ) 2 = 1 + 2Δ + Δ2 ≠ 1.
Post by: hamdani yusuf on 20/03/2022 05:40:38
Hi.Yeah it does.I see the reluctance from accepting to define a value of a mathematical operation is related to stability of the result when the operands are perturbed.I don't really know what you mean by "stability". It sounds like you're considering continuity when you consider infinitessimal perturbations.
Best Wishes.
0^0 is more stable than 1*1 when the perturbation is in form of multiplication with the operands.
The first operand is stable against addition, but the second operand is not.
Post by: hamdani yusuf on 20/03/2022 05:44:16
Acording to Mathmetica which I consider the last word on maths 0^0 is inditerminateWhat's the consideration behind its answer? Argument from authority is not a strong one in a scientific discussion.
Post by: alancalverd on 20/03/2022 15:50:32
Ok. They don't change the result much.Squeeze the trigger slowly and you'll appreciate the enormous difference between "not much" and "unstable".
Post by: hamdani yusuf on 20/03/2022 22:10:23
Ok. They don't change the result much.1 * 1 is less stable. Perturbing one of the operands with a finite constant does change the result. But multiplying them with a constant infinitesimally close to 1, or adding them with a constant infinitesimally close to 0 doesn't change the result.
Oh yes it does!
(1 * Δ)2 = Δ2 ≠ 1
(1+Δ) 2 = 1 + 2Δ + Δ2 ≠ 1.
Squeeze the trigger slowly and you'll appreciate the enormous difference between "not much" and "unstable".How can 1*1 unstable?
Post by: alancalverd on 20/03/2022 23:34:02
Post by: hamdani yusuf on 21/03/2022 01:42:46
It isn't. Only you suggested it was.I said 1*1 is less stable than 0*0.
Post by: Eternal Student on 21/03/2022 03:02:52
This kind of "stability" idea just isn't worried about too much, to the best of my knowledge. I think that's why you (Hamdani) aren't getting many good replies about the issue. The term "stability" is usually applied to other things in mathematics and I'm not sure that you (Hamdani) need to develop a new set of ideas for the stability of basic field operations like addition and multiplication. There is already a simple idea called continuity which seems to offer all we need.
However, if you want to develop ideas for this sort of stability, then I don't suppose there's any law against it.
Here's something to think about if you are developing some ideas for stability of basic field operations like + and x on different choices of input numbers:
You mentioned that 0 x 0 was very stable but that depends on how you measure stability. Surely you'd want to consider a proportional or percentage change? A change of +/- 1 unit in an expected answer of 100 units is nothing to worry about but a change of +/- 1 unit in an answer of 0.01 units is a ten-thousand percent error. Anyway, using this notion of proportional difference, 0 x 0 and also 0 + 0 isn't stable, it's terrifyingly unstable. Perturb the input numbers even a tiny amount and you generate a proportional difference that is unbounded. No other choice of input numbers is that bad.
Best Wishes.
Post by: alancalverd on 21/03/2022 11:37:43
Post by: hamdani yusuf on 21/03/2022 11:38:44
You mentioned that 0 x 0 was very stable but that depends on how you measure stability.
0 * 0 is extremely stable. The operands can be perturbed heavily without changing the result (e.g. by multiplying them with some constants). Adding one of the operands with a finite constant doesn't change the result. Adding both operands with a finite constant does change the result.Perturbation by multiplication already implies ratio. Percentage change simply converts perturbation from addition to multiplication.
Anyway, using this notion of proportional difference, 0 x 0 and also 0 + 0 isn't stable, it's terrifyingly unstable. Perturb the input numbers even a tiny amount and you generate a proportional difference that is unbounded. No other choice of input numbers is that bad.Can you show how?
Post by: hamdani yusuf on 21/03/2022 12:15:08
A simple example of a trigger instability is the equation y = 1/x. As x passes through zero its value switches between +∞ and -∞.Yes. I understand if that instability makes people think that 1/0 is undefined.
https://www.wolframalpha.com/input?i=1%2F0
Wolframalpha says that 1/0 is complex infinity instead of undefined.
1/0=-1/0=i/0=-i/0=complex infinity. In Riemann's sphere, they are located on the same point.
Quote
There are seven indeterminate forms involving 0, 1, and infinity:
https://mathworld.wolfram.com/Indeterminate.html
(https://mathworld.wolfram.com/images/equations/Indeterminate/NumberedEquation1.svg)
If complex infinity is allowed as well, then six additional indeterminate forms result:
(https://mathworld.wolfram.com/images/equations/Indeterminate/NumberedEquation2.svg)
Post by: Eternal Student on 21/03/2022 13:21:43
Anyway, using this notion of proportional difference, 0 x 0 and also 0 + 0 isn't stable, it's terrifyingly unstable. Perturb the input numbers even a tiny amount and you generate a proportional difference that is unbounded. No other choice of input numbers is that bad.
Can you show how?
0 + 0 = 0 = the expected answer for 0 + 0.
Perturb the input values slightly and consider δ + ε where δ, ε are arbitrarily small but positive.
Then δ + ε ≠ 0, instead δ + ε = something small but positive. The percentage change from the expected answer, 0, is then undefined:
(something positive) / 0 x 100%
- the division by 0 is a problem.
Best Wishes.
Post by: hamdani yusuf on 22/03/2022 12:19:27
0 + 0 = 0 = the expected answer for 0 + 0.
Perturb the input values slightly and consider δ + ε where δ, ε are arbitrarily small but positive.
Then δ + ε ≠ 0, instead δ + ε = something small but positive. The percentage change from the expected answer, 0, is then undefined:
(something positive) / 0 x 100%
- the division by 0 is a problem.
Best Wishes.
Perturbation by multiplication already implies ratio. Percentage change simply converts perturbation from addition to multiplication.
Post by: Eternal Student on 22/03/2022 12:43:06
Perturbation by multiplication already implies ratio. Percentage change simply converts perturbation from addition to multiplication.? I've no idea what you neant there.
Best Wishes.
Post by: hamdani yusuf on 10/06/2023 03:49:23
Quote
0^0 is highly debated in the mathematical community. Why is 0^0 undefined? Does 0^0=0? Does 0^0=1? In this video I'll address the 0^0 meaning and give you a solid 0^0 proof.
SPOILER ALERT: 0^0=1 proof can be found just using the definition of how we define exponents.
Post by: Bored chemist on 10/06/2023 13:42:56
Most of the time it wouldn't matter.
But, if I wanted to cover my 3 by4 metre room, I wouldn't know what area of carpet to buy.
On the other hand, I could get some carpet and cut it to fit the room.
And then I could weigh it and also weigh a square metre of carpet and if I divided one weight be the other I would get 12 which is the product that I somehow couldn't calculate.
I can't (reliably) calculate 0^0.
But if I ever find myself in a situation where I need to know what 0^0 is, I can work it out from the context.
If the value of 0^0 matters to some real-life calculation, you can use reality to calculate 0^0.
If it never occurs in any real-world calculation then it never really matters what the value is, and you can set it equal to the number of angels who can dance on the head of a pin.
Post by: hamdani yusuf on 10/06/2023 23:51:16
Post by: hamdani yusuf on 29/03/2024 12:32:24
Quote
TimelineShortly, it depends on context.
---
0:00 Opening Quote from Donald Knuth
0:18 The Integer Exponentiation Context
2:36 The Discrete Math Argument
5:57 Examples from Polynomials
7:55 The Abstract Algebra Argument
14:00 Indeterminate Form and The History of 0^0: Team Euler vs. Team Cauchy
17:00 Handling of 0^0 in Programming and Softwares
18:12 Ending words