Understanding the Controversy: Simplifying 0^0 in Calculus

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Homework Help Overview

The discussion revolves around the simplification of the expression 0^0, which is presented as a word problem. Participants explore its implications within the context of calculus, questioning whether it should be defined as 0, 1, or left undefined.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants express differing opinions on the value of 0^0, with some suggesting it is 0 or undefined, while others reference definitions that support it being 1 in certain contexts. The original poster seeks clarity on the reasoning behind these definitions.

Discussion Status

The discussion is active, with various perspectives being shared. Some participants provide references to definitions and examples that illustrate the complexity of the topic, indicating a productive exploration of the concept without reaching a consensus.

Contextual Notes

There is mention of the binomial expansion and its implications for defining 0^0, as well as references to mathematical definitions that may not universally apply. The original poster is studying independently, which may influence the depth of the discussion.

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Homework Statement



It's a word problem, but I just need to simplify 0^0.

Homework Equations



n/a

The Attempt at a Solution



0^n = 0.
n^0 = 1.




I'm just going through a calculus book on my own this summer. It's not even a calculus question, but I've never come across this before. Which is it? 0 or 1? Why?
 
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I believe it's 0 (or not even defined, such as in poles of a polynom), because from what I can remember n^0 = 1 is by definition only true for those n that are different from 0.
 
Last edited:


0^0 is undefined.
 


From Wolfram Mathworld http://mathworld.wolfram.com/ExponentLaws.html

The definition 0^0=1 is sometimes used to simplify formulas, but it should be kept in mind that this equality is a definition and not a fundamental mathematical truth (Knuth 1992; Knuth 1997, p. 56).
 


Okay, thanks guys. Wolfram is awesome!
 


Consider the binomial expansion:

[tex]1 = 1^2=(x+(1-x))^2 =\binom 2 0 x^0(1-x)^2+\binom 2 1 x^1(1-x) + \binom 2 2 x^2(1-x)^0[/tex]

This doesn't work for x = 0 or 1 unless you define 00 = 1. That is an example of why it is usually defined as 1. Also why 0! = 1 in the same example.
 

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