What is 0 (zero) to the power of 0 (zero)?
Posted: December 17th, 2020, 4:48 am
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AsleepInYorkshire wrote:Spoiler
The spoiler is a YouTube fourteen minute explanation of how to get to the answer.
AiY
absolutezero wrote:There is no agreed value but I go with 1.
Anything to the power of 0 is 1.
scrumpyjack wrote:What is 0 to the power 0?
I assume you are familiar with powers.
The problem is similar to that with division by zero. No value can be assigned to 0 to the power 0 without running into contradictions. Thus 0 to the power 0 is undefined!
How could we define it? 0 to any positive power is 0, so 0 to the power 0 should be 0. But any positive number to the power 0 is 1, so 0 to the power 0 should be 1. We can't have it both ways.
Underlying this argument is the same idea as was used in the attempt to define 0 divided by 0. Consider a to the power b and ask what happens as a and b both approach 0. Depending on the precise way this happens the power may assume any value in the limit.
(per the University of Utah maths department) (Excel agrees and returns an error, not 1)
scrumpyjack wrote:I think you should have a very large glass of sherry. Zero in on the wine cupboard and be noughty!
Mike4 wrote:absolutezero wrote:There is no agreed value but I go with 1.
Anything to the power of 0 is 1.
On the other hand, anything multiplied by 0 is 0, including itself I'd have thought.
Mike4 wrote:On the other hand, anything multiplied by 0 is 0, including itself I'd have thought.
Dod101 wrote:Excellent youtube video. I wish I had had a guy like him to teach me! Absolute 0 to the power absolute 0 is surely 0?
Dod
scrumpyjack wrote:What is 0 to the power 0?
I assume you are familiar with powers.
The problem is similar to that with division by zero. No value can be assigned to 0 to the power 0 without running into contradictions. Thus 0 to the power 0 is undefined!
How could we define it? 0 to any positive power is 0, so 0 to the power 0 should be 0. But any positive number to the power 0 is 1, so 0 to the power 0 should be 1. We can't have it both ways.
Underlying this argument is the same idea as was used in the attempt to define 0 divided by 0. Consider a to the power b and ask what happens as a and b both approach 0. Depending on the precise way this happens the power may assume any value in the limit.
(per the University of Utah maths department) (Excel agrees and returns an error, not 1)
Gengulphus wrote:scrumpyjack wrote:What is 0 to the power 0?
I assume you are familiar with powers.
The problem is similar to that with division by zero. No value can be assigned to 0 to the power 0 without running into contradictions. Thus 0 to the power 0 is undefined!
How could we define it? 0 to any positive power is 0, so 0 to the power 0 should be 0. But any positive number to the power 0 is 1, so 0 to the power 0 should be 1. We can't have it both ways.
Underlying this argument is the same idea as was used in the attempt to define 0 divided by 0. Consider a to the power b and ask what happens as a and b both approach 0. Depending on the precise way this happens the power may assume any value in the limit.
(per the University of Utah maths department) (Excel agrees and returns an error, not 1)
That isn't quite right, because what it's run into is a discontinuity, not a contradiction: the function x^y cannot be made continuous at (x,y) = (0,0).
E.g. if I define a function f(x) by f(x) = 0 if x <= 0, 1 if x > 0, then if I approach x = 0 from below without actually getting to it, f(x) is always 0, and if I approach x = 0 from above without actually getting to it, f(x) is always 1. If the quoted argument were entirely correct, I would have to conclude that f(0) is undefined because giving it any value runs into a contradiction. But that conclusion that f(0) is undefined itself contradicts my definition of f(x), which says that f(0) is defined to be 0. I.e. the attempt to avoid a 'contradiction' itself runs into a contradiction!
The resolution of that issue is that functions do not need to have the property that as x approaches y, f(x) approaches f(y). That property (which needs to be defined a bit more formally - "approaches" is rather vague - but I won't bore people with the formal definitions!) is known as continuity, and the resolution of what is said in the last paragraph is that f(0) is defined and equal to 0, but the function f(x) is discontinuous at x = 0. Furthermore, it cannot be made continuous at x = 0 by defining f(0) differently, because f(x) approaches two different values as x approaches 0, depending on how it approaches 0. (As a contrast, consider the function g(x) defined by g(x) = 0 if x = 0, 1 if x is not equal to 0. Since g(x) is always 1 as x approaches 0, and g(0) is not 1, the function g(x) is discontinuous at x = 0. But since g(x) only ever approaches 1 as x approaches 0, it can be made continuous by choosing g(0) = 1 instead.)
Anyway, the quoted argument says that the function x^y approaches different values as (x,y) approaches (0,0), depending on how the approach is done. So x^y is discontinuous at (x,y) = (0,0), and cannot be made continuous there by choosing any particular value for 0^0. That doesn't say that 0^0 cannot be defined - it just says that whether one defines it or not, one cannot use continuity arguments about it.
So how, if at all, should one define 0^0? At this point, an issue comes up that is quite common in mathematics, but not particularly well-known outside mathematics: often, the question of how one best defines a mathematical concept depends on the mathematical area in which one is operating. In particular, in the case of 0^0, it makes a big difference whether one is working with real numbers or with integers (i.e. whole numbers).
When working with real numbers, the issue of x^y being discontinuous at (x,y) = (0,0) is a fairly important one, so that there is good reason for 0^0 to be left undefined, because one then has the fairly simple theorem "x^y is continuous everywhere where it is defined" rather than the somewhat more complex "x^y is continuous everywhere where it is defined, except at 0^0". In particular, anyone trying to apply the former theorem needs to apply it for the two cases "x^y is defined" and "x^y is not defined", rather than the three cases "x^y is 0^0", "x^y is defined but not 0^0" and "x^y is not defined", so arguments using the theorem will tend to be simpler.
When working with integers, and especially for questions in the mathematical area of combinatorics where the numbers are generally non-negative integers, continuity is essentially irrelevant, and the definition used for X^Y is basically "the number of ways one can choose an ordered sequence of Y objects from X different types of object, with duplicate choices allowed". For example, 2^3 = 8 because I can choose an ordered sequence of 3 objects from objects of types A and B in eight different ways: (A,A,A), (A,A,B), (A,B,A), (A,B,B), (B,A,A), (B,A,B), (B,B,A) and (B,B,B). And 3^2 = 9 because I can choose an ordered sequence of 2 objects from objects of types X, Y and Z in nine different ways: (X,X), (X,Y), (X,Z), (Y,X), (Y,Y), (Y,Z), (Z,X), (Z,Y) and (Z,Z).
Using that definition, n^0 = 1 for all non-negative n, because there is always just one way of making no choices, and 0^n = 0 for all positive n but not for n = 0, because we are faced with the impossible task of choosing an object from no types of object as soon as we are asked to choose an object. So 0^0 is unambiguously 1 in the mathematical area of combinatorics.
I don't know of any mathematical areas that would favour any other values for 0^0, so I think the answer to the question in this thread's subject is basically "Either 1 or undefined, depending on the mathematical area one is working in".
Gengulphus
johnhemming wrote:I am confused myself because I thought it was y=x^x that we were considering.