Why is the Factorial of Zero Defined as One?

  • Context: High School 
  • Thread starter Thread starter ananthu017
  • Start date Start date
  • Tags Tags
    Factorial Zero
Click For Summary

Discussion Overview

The discussion revolves around the definition of the factorial of zero, specifically why it is defined as one. Participants explore various methods and definitions related to factorials, including mathematical reasoning and alternative interpretations.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • Some participants provide the definition of factorial and mention that 0! is defined as 1, referencing the Gamma function.
  • One participant presents a mathematical derivation using the Gamma function to show that 0! equals 1.
  • Another participant expresses skepticism about the clarity of the mathematical explanation provided, suggesting it may be too complex for the original poster.
  • There is a discussion about the recursive definition of factorial, where n! = n(n-1)!, leading to the conclusion that if 1! = 1, then 0! must also equal 1.
  • One participant questions the implications of defining factorial for negative integers, specifically asking about (-1)!.

Areas of Agreement / Disagreement

Participants generally agree that 0! is defined as 1, but there are differing views on the clarity and implications of the definitions and methods presented. The discussion includes some skepticism and questions regarding the understanding of these concepts.

Contextual Notes

Some participants note that the recursive definition of factorial and the relationship to the Gamma function may involve assumptions that are not fully explored, particularly regarding the factorial of negative integers.

Who May Find This Useful

This discussion may be useful for individuals interested in mathematical definitions, particularly in combinatorics or calculus, as well as those exploring the foundations of factorials and their applications.

ananthu017
Messages
4
Reaction score
0
can u give me the method to find the factorial of zero ?
 
Mathematics news on Phys.org
What is the definition of the factorial?
 
0!=\Gamma(1)=\intop_{0}^{\infty} t^{1-1}\exp(-t)\mathrm{d}t=-[\exp(-t)]_{0}^{\infty}=-(0-1)=1.

Alternatively, there's exactly one way to arrange an empty set.
 
DeIdeal said:
0!=\Gamma(1)=\intop_{0}^{\infty} t^{1-1}\exp(-t)\mathrm{d}t=-[\exp(-t)]_{0}^{\infty}=-(0-1)=1.

I highly doubt the OP would be able to make heads or tails of this.

n! = n(n-1)!

Work with this definition, it's all you need.
 
So your answer to the original question is that 0!= 0(-1)!? But what is (-1)!?

(Yes, you can write that 1!= 1(0!) and, if you know that 1!= 1, then it follows that 0!= 1. But better is just to state the basic definition of n! that asserts 0!= 1 for as part of the definition.)
 

Similar threads

Undergrad Questions regarding Kurepa's Conjecture
  • · Replies 1 ·
Replies
1
Views
3K
High School Cool fact about number of digits in n!
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Why is There No Inverse Factorial Function?
  • · Replies 13 ·
Replies
13
Views
3K
Very cool factory production line photos
  • · Replies 6 ·
Replies
6
Views
2K
Insights Why Division by Zero is a Bad Idea
  • · Replies 47 ·
2
Replies
47
Views
7K
Undergrad Define the double factorial as being a continous, non-hybrid function
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Not seeing things in physics and mathematics.
  • · Replies 5 ·
Replies
5
Views
2K
MHB Is there any sense with these squares?
  • · Replies 2 ·
Replies
2
Views
2K
Python [SOLVED] Python - Issues Taking an Input as an Integer (or Converting It)
  • · Replies 1 ·
Replies
1
Views
2K
High School Division by Zero: Is it Defined?
  • · Replies 7 ·
Replies
7
Views
2K