0!??????

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wizwilzo

1987 posts

wizwilzo

#1 h Jul 6, 2016, 5:35 PM • 4 Y

Y by Adventure10, Mango247, DhruvJha, Exponent11

why is 0! "1" ??!

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BobThePotato

1182 posts

BobThePotato

#2 h Jul 6, 2016, 5:36 PM • 6 Y

Y by ninjasrule34, TheMathematicsTiger7, Adventure10, Mango247, DhruvJha, Exponent11

Recall that for all positive integers $n$ , $n!/n=(n-1)!.$ Hence we have that $1!/1=1=(1-1)!=0!.$

This post has been edited 1 time. Last edited by BobThePotato, Jul 6, 2016, 5:37 PM
Reason: Rearrangement.

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pilover123

461 posts

pilover123

#3 h Jul 6, 2016, 5:38 PM • 5 Y

Y by Adventure10, Mango247, Owen314159, DhruvJha, Exponent11

Also, one common use of factorials is to determine the number of ways we can arrange n books on a shelf. For example, the way we find out how many ways we can arrange 5 books is 5! or 120 ways. If we have 0 books, then there is only 1 way to arrange them, which is do nothing

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mcmcphie

1196 posts

mcmcphie

#4 h Jul 6, 2016, 5:40 PM • 4 Y

Y by wizwilzo, Adventure10, DhruvJha, Exponent11

Note that factorials aren't strictly defined for nonpositive numbers. Thus, $0!$ is defined to make sense in counting problems as shown in the above post.

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wizwilzo

1987 posts

wizwilzo

#5 h Jul 6, 2016, 5:41 PM • 2 Y

Y by Adventure10, DhruvJha

thanks! I understand now

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zxcv1337

1631 posts

zxcv1337

#6 h Jul 6, 2016, 5:43 PM • 2 Y

Y by Adventure10, Mango247

Just a question off the top of my head, Can negative numbers have factorials? like for instance $-1!$ or $-2!$

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KenV

1198 posts

KenV

#7 h Jul 6, 2016, 5:43 PM • 2 Y

Y by Adventure10, Mango247

Another reason is because $12$ choose $12$ is obviously $1$ , and solving for $0!$ gives you $0!=1$ .

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Darkdoger

216 posts

Darkdoger

#8 h Jul 6, 2016, 5:49 PM • 3 Y

Y by ninjasrule34, Adventure10, Mango247

ramarnath2 wrote:

Just a question off the top of my head, Can negative numbers have factorials? like for instance $-1!$ or $-2!$

This might be a bit confusing taking only MSM level into the account, but the Gamma Function is related to this.

https://en.m.wikipedia.org/wiki/Gamma_function

Taking into account the Gamma Function, it is safe to say that factorials of all negative integers are undefined.

This post has been edited 1 time. Last edited by Darkdoger, Jul 6, 2016, 5:50 PM

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First

2352 posts

First

#9 h Jul 6, 2016, 5:52 PM • 1 Y

Y by Adventure10

If I have zero things, how many ways can I arrange them. The answer would be 1, though it might be confusing.

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mcmcphie

1196 posts

mcmcphie

#10 h Jul 6, 2016, 5:55 PM • 2 Y

Y by Adventure10, Exponent11

But then $\dfrac{0}{0}$ makes perfect sense.

As Siri would say, 0 people and 0 cookies, then each of the 0 people would get 0 cookies. $0 \cdot 0 = 0$ , after all.

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pilover123

461 posts

pilover123

#11 h Jul 6, 2016, 5:59 PM • 1 Y

Y by Adventure10

ramarnath2 wrote:

Just a question off the top of my head, Can negative numbers have factorials? like for instance $-1!$ or $-2!$

If use the fact that $\frac{n!}{n}={(n-1)}!$ , we get $\frac{0!}{0}={(-1)}!$ . We know that division by 0 is undefined, so negative factorials are undefined.

This post has been edited 1 time. Last edited by pilover123, Jul 6, 2016, 6:00 PM
Reason: Problem

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tijnasreksab

509 posts

tijnasreksab

#12 h Jul 6, 2016, 6:00 PM • 2 Y

Y by T314159, Adventure10

mcmcphie wrote:

But then $\dfrac{0}{0}$ makes perfect sense.

As Siri would say, 0 people and 0 cookies, then each of the 0 people would get 0 cookies. $0 \cdot 0 = 0$ , after all.

Well, each of the 0 people could also get 15 cookies. That's why 0/0 can take on any value you give it.

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mcmcphie

1196 posts

mcmcphie

#13 h Jul 6, 2016, 6:04 PM • 2 Y

Y by Adventure10, Mango247

But you have 0 cookies...

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bkn6

700 posts

bkn6

#14 h Jul 6, 2016, 6:59 PM • 2 Y

Y by Adventure10, Mango247

And there are 0 people to share it...

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yegkatie

911 posts

yegkatie

#15 h Jul 6, 2016, 7:03 PM • 4 Y

Y by T314159, tijnasreksab, Adventure10, Mango247

$\frac{0}{0}$ is basically asking, what times 0 equals 0? But anything times zero equals 0 so $\frac{0}{0}=\text{anything}$ .

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mcmcphie

1196 posts

mcmcphie

#16 h Jul 6, 2016, 7:12 PM • 2 Y

Y by Adventure10, Mango247

Yeah, I guess.

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Littlelachy

934 posts

Littlelachy

#17 h Jul 7, 2016, 11:32 AM • 2 Y

Y by rafayaashary1, Adventure10

ramarnath2 wrote:

Just a question off the top of my head, Can negative numbers have factorials? like for instance $-1!$ or $-2!$

Technically yes but they are all undefined.

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FlyingCucumber

1675 posts

FlyingCucumber

#18 h Jul 7, 2016, 12:09 PM • 3 Y

Y by Adventure10, Mango247, Exponent11

mcmcphie wrote:

But you have 0 cookies...

You could give 0 cookies to 0 people 15 times.

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Mathguy5837

778 posts

Mathguy5837

#19 h Jul 7, 2016, 1:15 PM • 2 Y

Y by Adventure10, Mango247

Sorry if this is a bit off topic for this but $-2^2=-4$ because the minus sign isn't part of the number. Shouldn't it therefore be $(-1)!$ ? Or do we not need the brackets?

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yamyamx2

253 posts

yamyamx2

#20 h Jul 7, 2016, 1:24 PM • 1 Y

Y by Adventure10

You probably do need the brackets for factorials otherwise -1! would just be -1.

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tijnasreksab

509 posts

tijnasreksab

#21 h Jul 7, 2016, 2:42 PM • 2 Y

Y by Adventure10, Mango247

Correct, Google's calculator agrees.

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wizwilzo

1987 posts

wizwilzo

#22 h Jul 7, 2016, 11:06 PM • 2 Y

Y by Adventure10, Mango247

very irrelevant but if you type in "calc", the calculator will show up

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scurry30

302 posts

scurry30

#23 h Jul 8, 2016, 4:14 PM • 3 Y

Y by Adventure10, Mango247, iwastedmyusername

0! is 0 factorial, and that would equal to 1 .

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eric201291

208 posts

eric201291

#25 h Mar 5, 2025, 2:19 PM

Y by

Because, 1=Cn,n=n!/(n!*0!), so 0!=1

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CKracingcar

1090 posts

CKracingcar

#27 h Mar 7, 2025, 9:08 AM

Y by

$3! = 4!/4, 2! = 3!/3, 1! = 2!/2, 0! = 1!/1$

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eddie.li

32 posts

eddie.li

#28 h Mar 7, 2025, 8:27 PM

Y by

0+1=0 so 0!=1

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Bummer12345

150 posts

Bummer12345

#29 h Mar 8, 2025, 7:53 PM

Y by

bumping a 9 year old post is not $\sum$

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DhruvJha

893 posts

DhruvJha

#30 h Mar 8, 2025, 7:54 PM

Y by

0!= 0+1!=1

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greenplanet2050

1324 posts

greenplanet2050

#31 h Mar 8, 2025, 8:04 PM

Y by

DhruvJha wrote:

0!= 0+1!=1

thats not an explanation

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Thayaden

1364 posts

Thayaden

#41 h Apr 18, 2025, 5:36 PM

Y by

wizwilzo wrote:

why is 0! "1" ??!

Intuitive solution,
A factorial represents the number of ways you can arrange $n$ books on a shelf. If there are no books, then all of the books are still arranged, thus there is 1 way to arrange zero books on a shelf, starting position.

Middel School Sloution,
define for $n\in\mathbb{Z}^+_0$ we know,
$\begin{align*} 1! &=1,\\ 2! &= 2,\\ 3! &= 6,\\ 4! &= 24. \end{align*}$ Although reversing,
$\begin{align*} 4! &= 24,\\ 3! &= 6,\\ 2! &= 2,\\ 1! &= 1. \end{align*}$ So to get to the last factorial, we divided by the number, ie to get $3!$ from $4!$ , we divided $3!=24/4,$ going further $0! =1!/1=1$

High school grade solution
Consider
$\Gamma(z)=\int_0^\infty x^{z-1}e^{-x}dx=(n-1)!$ for $n\in\mathbb{Z}^+$ this is a much more formal definition for a factorial.
We find $\Gamma(1)=1$ thus $0!=1$

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