Question

Why do we think 0! = 1?

Answer 1

$4!=4\times3\times2\times1=24\\3!=3\times2\times1=6\\2!=2\times1\\1!=1\\0!=?$

Notice that when we go from 4! to 3! , we divide by 4.

(Because $4!\div 4 = 4\times3\times2\times1\div4=3\times2\times1=3!$  the 4's cancel)

And when we go from 3! to 2! , we divide by 3, and etc.

We can use this pattern to see why 0! = 1.

$4!=4\times3\times2\times1=24\\3! = 4!\div4=24\div4=6\\2!=3!\div3=6\div3=2\\1!=2!\div2=2\div2=1\\0!=1!\div1=1\div1=1$

And so by following the pattern, we determine that 0! = 1

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Note:

There are no factorials of negative numbers, and we can use the pattern to show why:

$0!=1\\-1!=0!\div0=1\div0=???!!??!?$

You can't divide a number by 0, therefore -1! doesn't exist, so -2! doesn't exist and so on. So you can't do the factorial of any negative number.

(Or at least there no real solutions to negative factorials)

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