0! = 1? or why the factorial of zero is equal to one

A long time ago, back in the 10th class (about 8 years ago), I accidentally discovered a rather simple explanation of why the factorial of zero is equal to one.

I talked about this to many teachers, but no one stuck. Therefore, I will simply post this knowledge here, otherwise suddenly someone will come in handy or will lead to certain thoughts. I must say right away that I am not a mathematician, I came across this by accident when I was playing with numbers. I didn’t even know what factorial is then :)

First, let's recall the general theory:

The factorial of n is the product of all natural numbers up to n inclusive:



By definition, they put 0! = 1 . Factorial is defined only for non-negative integers.

In fact, the factorial of zero is completely computable!
To do this, we need to do a simple sequence of ordinary mathematical operations.

Let's try in action using the example of the factorial n = 4 (4! = 1 * 2 * 3 * 4 = 24)

• First, take a sequence of n + (1 or more) numbers, where each subsequent number is greater than the previous one by 1.
  
  For example:
  

  1 2 3 4 5
  
  

• Then we raise each number to the power of n and write the results below.
  
  We get:
  

  1 ⁴ 2 ⁴ 3 ⁴ 4 ⁴ 5 ⁴
  
  
  1 16 81 256 625

• Now we subtract the penultimate one from the last number, and so on.
  
  At the output, we get a series of numbers whose number is less by 1:
  

  (16 - 1) (81 - 16) (256 - 81) (625 - 256)
  
  15 65 175 369

• We repeat the previous step already on the resulting series until one number remains (or a series of identical numbers if the number is more than n + 1)
  

  (65 - 15) (175 - 65) (369 - 175)
  
  50 110 194
  (110 - 50) (194 - 110)
  
  60 84
  (84 - 60)
  
  24

  
  As a result, we get the factorial of the number four.
  

Let's try to calculate factorial 3 in this way (3! = 1 * 2 * 3 = 6)

We take four numbers to the power of 3 and calculate the “pyramidal difference” (he invented it)

1 ³ 2 ³ 3 ³ 4 ³
1 8 27 64
(8 - 1) (27 - 8) (64 - 27)

7 19 37
(19 - 7 ) (37 - 19)

12 18
(18 - 12)

6
Everything fits together!

Well, for 1 try (1! = 1)

1 ¹ 2 ¹
1 2
(2 - 1)

1

Have you already guessed? :)

Everything is very simple and for scratch:

We take n + 1 numbers to the power of 0, i.e. one

1 ^o 1 is enough

Voila! Any number in degree 0 is equal to 1. In this, by the way, the weakness of my method, it uses the definition.

Nevertheless, I think this is great :)

Thank you for your attention!

PS:
As many have noticed this is not proof, but just a fun pattern.