[hamdani yusuf (OP)]

What's 0 to the power of 0?

[chiralSPO]

1

[hamdani yusuf (OP)]

I brought this up when I saw a Twitter feed, and the replies showed various answers.
https://twitter.com/pickover/status/1454996588661719041
Let's see which one is the most correct.

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..

but doesn't the limit from the negative approach negative 1?

No. It approaches a complex number. The real part approaches 1. The imaginary part approaches an infinitesimally small negative number.

[hamdani yusuf (OP)]

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.

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.

In short, from the perspective of mathematics, this is undefined.

[hamdani yusuf (OP)]

I'd show him my phone ad hope he doesn't notice the question at the bottom -

[hamdani yusuf (OP)]

And here's Wikipedia's answer.

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

[hamdani yusuf (OP)]

Here's an unconventional approach.

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.

[chiralSPO]

(edited to correct inequality direction)

y = x^x 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 = 0^x 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 = x^a as y = 1 × x^a, 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.

[chiralSPO]

Here's an unconventional approach.

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.

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?)

[Petrochemicals]

Garbage.

[hamdani yusuf (OP)]

Garbage.

Someone's trash is someone else's treasure.

[hamdani yusuf (OP)]

y = x^x is well-defined for all x < 0.

What does it mean? Doesn't it mean all negative x?

[evan_au]

Here is a video about it.... This exact question starts around 3 minutes...

[hamdani yusuf (OP)]

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.

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.

(-0.0001)^(-0.0001) = 1.00092141 - 0.000314448745 i

[Petrochemicals]

Garbage.

Someone's trash is someone else's treasure.

4^0.5=2
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?

[chiralSPO]

y = x^x is well-defined for all x < 0.

What does it mean? Doesn't it mean all negative x?

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! :-)

[chiralSPO]

Garbage.

Someone's trash is someone else's treasure.

4^0.5=2
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.

...ok.... so?

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
3^2.7 = 1 x 3 x 3 x 3^0.7 = 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).

[Zer0]

0 + 0 = 0.

0 * 0 = 0.

0 / 0 = 1.

0 ^ 0 = Undefined.

0 - 0 = Nothing.

Ps - Dyscalculia!
👻

[chiralSPO]

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

[hamdani yusuf (OP)]

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 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