Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Nobody's Confidant on 20/09/2007 17:55:48
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What is it? Do they have scientists who just sit around counting until they get another three zeros and make up some new "-illion?"
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It is not really a meaningful question.
I think the highest number that has a name is a Googolplex, which is 1010100, but you can create a number that is arbitrarily large by just making numbers powers of each other:
http://en.wikipedia.org/wiki/Googolplex
In pure mathematics, the magnitude of a googolplex is not as large as some of the specially defined extraordinarily large numbers, such as those written with tetration, Knuth's up-arrow notation, Steinhaus-Moser notation, or Conway chained arrow notation. Even more simply, one can name numbers larger than a googolplex with fewer symbols, for example,
1099999
is much larger. This last number can be expressed more concisely as 69 using tetration, or 9⇈6 using Knuth's up-arrow notation.
Some sequences grow very quickly; for instance, the first two Ackermann numbers are 1 and 22=4; but then the third is 333, a power tower of threes more than seven trillion high. Yet, much larger still is Graham's number, perhaps the largest natural number mathematicians actually have a use for.
As for the largest number of digits written down for a number, π (pi has been written to at least 1 billion digits (http://www.starqwest.org/page5.html) (although I don't know if it is in a single document, but there is a document for π to 4 million digits (http://www.zenwerx.com/pi.php)).
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It is interesting to note that these very large numbers are much bigger than the number represented by all the possible ways of arranging all the subatomic particles in our universe which is very large and is probably the largest number associated with a physically real situation.
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Scientists also use "infinity" quite a bit. Of course, infinity can never be stored in a computer or written down, since there aren't enough particles in the universe to do so, but it is a perfectly legitimate tool, and it's something most physicists are comfortable slinging around.
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strangly some series of numbers such a 1+1/2+1/4+1/8 can be extended to infinity and still have a finite value i.e 2
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Scientists also use "infinity" quite a bit. Of course, infinity can never be stored in a computer or written down, since there aren't enough particles in the universe to do so, but it is a perfectly legitimate tool, and it's something most physicists are comfortable slinging around.
Infinity is not a number, it is a concept - it can have no numerical value.
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I used to annoy my teachers at school by insisting with what seemed perfect logic to me that the product of zero (0/1) and infinity (1/0) was by simple cancelation of the zero's was one .........0/1 * 1/0 = 1/1 = 1
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Scientists also use "infinity" quite a bit. Of course, infinity can never be stored in a computer or written down, since there aren't enough particles in the universe to do so, but it is a perfectly legitimate tool, and it's something most physicists are comfortable slinging around.
Infinity is not a number, it is a concept - it can have no numerical value.
Yes and no. It's not a number in the traditional sense of a real or complex number. However, it is a number in the sense that it can be used to count and measure things. (I think the details go well beyond the scope of this question). Generally as used in physics, it is not a number, but a concept of making something arbitrarily large.
In the sense of this question, though, infinity (or technically limit:x->infinity) pretty much describes the whole concept of the scientist sitting in a room constantly counting higher and higher...
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Despite the vast inventory of numbers opened up by these studies, it is remarkable that the arguably 8 most important numbers in mathematics lie within a small zone near the number 2. They are:
0
1
sqrt(2) = 1.41421....
sqrt(3) = 1.732.....
2 (basis of computers)
e = 2.7182818....
pi = 3.1415926.....
i , absolute value = 1
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Yes and no. It's not a number in the traditional sense of a real or complex number. However, it is a number in the sense that it can be used to count and measure things.
If you can count it, it is not infinite. If you can measure it, it is not infinite. In certain cases, you can say you can do algebra with it, but it is neither physically countable nor physically measurable.
Generally as used in physics, it is not a number, but a concept of making something arbitrarily large.
If that is how you are using the term infinity, then it is the wrong usage. Infinity is not a number that is arbitrarily large, it is a number that is provably uncountably large (i.e. if you believe it is a countable number, merely too large to be counted for practical reasons, then it is not infinity).
In the sense of this question, though, infinity (or technically limit:x->infinity) pretty much describes the whole concept of the scientist sitting in a room constantly counting higher and higher...
You can prove that pi is an infinitely long number when expressed as a decimal value, but that is because you can show theoretically that you cannot ever possibly (not merely impractical to, but it is physically impossible to) count the total number of digits of pi in decimal notation.
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Despite the vast inventory of numbers opened up by these studies, it is remarkable that the arguably 8 most important numbers in mathematics lie within a small zone near the number 2. They are:
0
1
sqrt(2) = 1.41421....
sqrt(3) = 1.732.....
2 (basis of computers)
e = 2.7182818....
pi = 3.1415926.....
i , absolute value = 1
Only really e and pi can be regarded as independent numbers.
It is though interesting that you regard i (imaginary 1) as more important than -1 (negative 1), but in reality it is not the size of the number (that it is a unitary size) but the domain the operate in (the imaginary or negative domains) that makes them important, so noting that their size is 1 is of little importance.
That 2 is used as the basis of computing is merely because it is the first integer that is larger than 1 - so it is not surprising that 1 and 2 are in close proximity (ofcourse, it depends on what you regard as proximity - to someone whose whole world is in investigating numbers between one and two, the number two might seem very far away).
To regard sqrt(2) and sqrt(3) as important seems somewhat arbitrary to me (sqrt(2) comes up quite often, but sqrt(3) is somewhat rarer).
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I feel that 10 should rank as the most important number as all arithmetic is based on it.
Some arithmetic to the base of 2, 8 or 16 is used in the computer world but in the real word 10 is always used
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I feel that 10 should rank as the most important number as all arithmetic is based on it.
Some arithmetic to the base of 2, 8 or 16 is used in the computer world but in the real word 10 is always used
Talk to the Babylonians about that. In fact, even today, 12, and 60, are still very important numbers.
Even 2, 8, and 16, are still used in a number of imperial weights and measures; but even though imperial units are beginning to fall out of favour, there is no evidence that anybody in the foreseable future will use units of time that are not based on multiples of 12 (24 and 60 are both multiples of 12).
In many ways, 10 is a very arbitrary and not necessarily most convenient base for our numbering system, it just happens to be convenient for counting on our fingers and thumbs, but is that really the appropriate basis upon which to choose a numbering system in the 21st century?
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Yes and no. It's not a number in the traditional sense of a real or complex number. However, it is a number in the sense that it can be used to count and measure things.
If you can count it, it is not infinite. If you can measure it, it is not infinite. In certain cases, you can say you can do algebra with it, but it is neither physically countable nor physically measurable.
I agree completely that you can't "physically" count infinite things. You can, however, count infinite things in the sense of the mathematical theory of sets. The common example is the set natural numbers, which is countably infinite. (http://mathworld.wolfram.com/CountablyInfinite.html)
Generally as used in physics, it is not a number, but a concept of making something arbitrarily large.
If that is how you are using the term infinity, then it is the wrong usage. Infinity is not a number that is arbitrarily large, it is a number that is provably uncountably large (i.e. if you believe it is a countable number, merely too large to be counted for practical reasons, then it is not infinity).
I have been very careful not to call infinity a number, since it isn't one. Number implies that it's contained in the set of natural numbers, which it isn't. It's a concept or quantity that is arbitrarily large. By arbitrarily large, I mean something that is larger than every real number. You can also define two types of infinities used in counting "countably infinite" (what you get if you count the natural numbers) or "uncountably infinite" (what you get if you count the real numbers). This is way beyond the scope of the question, but it shows you can use infinity to measure things.
In the sense of this question, though, infinity (or technically limit:x->infinity) pretty much describes the whole concept of the scientist sitting in a room constantly counting higher and higher...
You can prove that pi is an infinitely long number when expressed as a decimal value, but that is because you can show theoretically that you cannot ever possibly (not merely impractical to, but it is physically impossible to) count the total number of digits of pi in decimal notation.
Again, I agree completely that one cannot physically count to infinity. I never claimed anyone could. The idea of counting forever implies that you have infinite time in which to count, in which case if you sat down and chose any particular digit of pi, the counter would pass it eventually. In fact, this is an example of a countably infinite set, since you can arrange the decimal expansion of pi to map onto the natural numbers.
In the spirit of the original question, all this theory on infinity is probably way too detailed. There are three answers that pretty well sum up the question of trying to name the largest number:
(1) The largest number isn't well defined, since someone can come along and add 1 to anything you propose. (another_someone)
(2) The largest number we can represent is limited by the number of particles or amount of energy in the universe, since we'll run out of "stuff" to represent it with. (Soul Surfer)
(3) The largest quantity (though not a number) we can come up with is infinity.
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I understand that you can have infinity for instance the number of all the integers then an even larger number by interposing all the possible fractional numbers between the integers (something to do with moving guests about in a hotel I recall).
We have Hilbert to thank for this idea
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
Not being a mathematician I cannot visualise the purpose of such concepts
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Even 2, 8, and 16, are still used in a number of imperial weights and measures; but even though imperial units are beginning to fall out of favour, there is no evidence that anybody in the foreseable future will use units of time that are not based on multiples of 12 (24 and 60 are both multiples of 12).
Indicating, I gather, that we probably will not be needing the googolplex for this purpose.
It is though interesting that you regard i (imaginary 1) as more important than -1 (negative 1),
How perceptive of you; -1 is indeed an important number and should have been listed.
To regard sqrt(2) and sqrt(3) as important seems somewhat arbitrary to me (sqrt(2) comes up quite often, but sqrt(3) is somewhat rarer).
Both of these numbers are important in some of the most common geometry.
You can also define two types of infinities used in counting "countably infinite" (what you get if you count the real numbers) or "uncountably infinite" (what you get if you count the rational numbers).
No, this is inaccurate. The type of infinity associated with integers and that which is associated with rational numbers is the same. The proof is elementary: To the field of rational numbers, line up all the numerators along the x axis and all the denominators along the y axis. Now create a path starting from (0,0) and zigzagged back and forth diagonally, taking in row after row of diagonal dots. The order in which those dots occurs on the zigzag path constitutes an assignment of each to an integer, in order. However, because each represents a rational number, you end up with a one to one correspondence between rational numbers and integers. In fact, it is even more remarkable because many of those dots actually represent the same rational number, but not reduced to lowest terms. So that according to that, one would expect the number of integers to exceed that of rational numbers, but that cannot be of course because integers themselves are rational numbers. This is bizarre and counterintuitive, but demonstrates that integers and rational numbers are of the same order of infinity. Now there is indeed a higher order of infinity, and I believe it belongs to the set of all real numbers. I further understand that it is possible to create sets having even higher distinct orders of infinity, I think they are called the Alephs, which can be arranged in order from Aleph 0 on up to Aleph as large an integer as we want, Aleph 0 corresponding to the level of enumerable sets such as integers and rational numbers.
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"aleph zero" is the name given to the number at the end of infinity.
some infinities are larger than others,well make up your own mind,the idea is that the infinity of 1,2,3,4,5,etc is smaller than that of 2,4,6,8 etc.
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You can also define two types of infinities used in counting "countably infinite" (what you get if you count the real numbers) or "uncountably infinite" (what you get if you count the rational numbers).
No, this is inaccurate. The type of infinity associated with integers and that which is associated with rational numbers is the same. The proof is elementary: To the field of rational numbers, line up all the numerators along the x axis and all the denominators along the y axis. Now create a path starting from (0,0) and zigzagged back and forth diagonally, taking in row after row of diagonal dots. The order in which those dots occurs on the zigzag path constitutes an assignment of each to an integer, in order. However, because each represents a rational number, you end up with a one to one correspondence between rational numbers and integers. In fact, it is even more remarkable because many of those dots actually represent the same rational number, but not reduced to lowest terms. So that according to that, one would expect the number of integers to exceed that of rational numbers, but that cannot be of course because integers themselves are rational numbers. This is bizarre and counterintuitive, but demonstrates that integers and rational numbers are of the same order of infinity. Now there is indeed a higher order of infinity, and I believe it belongs to the set of all real numbers. I further understand that it is possible to create sets having even higher distinct orders of infinity, I think they are called the Alephs, which can be arranged in order from Aleph 0 on up to Aleph as large an integer as we want, Aleph 0 corresponding to the level of enumerable sets such as integers and rational numbers.
D'oh! Thanks for pointing that out. I reversed the cases in my post. Countably infinite should refer to natural numbers, integers, and rational numbers, which can be demonstrated to be the same "size,} as you showed. Uncountably infinite should refer to real numbers.