Zero to the zeroth power

  • Thread starter PFuser1232
  • Start date
  • Tags
    Power Zero
In summary, when representing the function f(x) as a power series, we can interpret x^0 as 1 in order to make the function differentiable. This convention is supported by the binomial theorem and is a common practice, although some mathematicians choose to leave 0^0 undefined. This topic has been widely discussed and debated.
  • #1
PFuser1232
479
20
Suppose we wish to represent ##f(x)## as a power series:

$$f(x) = \sum_{k=0}^{∞} a_k x^k = a_0 x^0 + a_1 x + a_2 x^2 + ...$$

How is it that ##f(0) = a_0## if ##x^0 = 1## only for nonzero ##x##?
 
Mathematics news on Phys.org
  • #2
You just have to be mindful of your definitions. In this case ##x^0## should really be interpreted as 1. It is simply the only way of doing it here, you could also just pull the term out of the sum. If it was not interpreted as one your function would not even be differentiable.
 
  • #3
Writing f(x) as a power series implies that f(x) is analytic (and therefore continuous) within its circle of convergence, so f(0)=a0.
 
Last edited:
  • #4
Many authors use the convention that ##0^0 = 1##, and there are many good reasons. For example, Knuth (who invented LaTeX) wrote in his (very beautiful) book "concrete mathematics:

Some textbooks leave the quantity ##0^0## undefined, because the functions ##0^x## and ##x^0## have different limiting values when ##x## decreases to ##0##. But this is a mistake. We must define ##x^0=1## for all ##x## , if the binomial theorem is to be valid when ##x=0## , ##y=0## , and/or ##x=-y## . The theorem is too important to be arbitrarily restricted! By contrast, the function ##0^x## is quite unimportant.

On the other hand, there are also many mathematicians leaving it undefined. The resolution I take is that ##0^0## is ##1## if the exponent is only allowed to be integers. So ##a^n## where ##n\in \mathbb{Z}## only. If the exponent is allowed to be more general real numbers, then it's best to leave it undefined.

That said, if you choose to follow the convention of ##0^0 = 1##, then there is nothing wrong with that as long as you're consistent.
 
  • Like
Likes Mark44

What is "Zero to the zeroth power"?

In mathematics, "zero to the zeroth power" is an indeterminate form, meaning that it does not have a definite value. It is often represented as 0^0 and can be interpreted in different ways depending on the context.

Why is "Zero to the zeroth power" an indeterminate form?

This is because any number raised to the power of 0 is equal to 1. However, when both the base and the exponent are 0, there is no clear definition for the result. It can be thought of as an expression that is undefined or has infinite solutions.

Can "Zero to the zeroth power" ever have a defined value?

In certain cases, it is possible to assign a value to 0^0, but it depends on the context and the rules being used. In some fields of mathematics, it is defined as 1, while in others it is left undefined. It is important to note that this value is not universally accepted.

What are some real-world applications of "Zero to the zeroth power"?

In some fields of mathematics, 0^0 can be used in probability and combinatorics to represent the number of ways to choose 0 objects from 0 objects. It can also be seen in calculus when evaluating limits and in physics when dealing with zero-mass particles.

How do mathematicians handle "Zero to the zeroth power" in equations and formulas?

There are different conventions and rules for handling 0^0 in equations and formulas. Some mathematicians choose to leave it undefined, while others may use the value of 1. It is important to understand the context and rules being used in order to properly interpret the result.

Similar threads

Replies
3
Views
530
Replies
3
Views
2K
  • General Math
Replies
4
Views
673
  • General Math
Replies
2
Views
1K
Replies
4
Views
253
  • Calculus and Beyond Homework Help
Replies
5
Views
472
  • Advanced Physics Homework Help
Replies
1
Views
710
Replies
4
Views
676
  • General Math
Replies
4
Views
2K
Replies
1
Views
779
Back
Top