0! = 1? or why factorial zero is one

Long ago, back in the 10th grade (8 years ago), I accidentally discovered a rather simple explanation of why factorial zero is equal to one.

I talked about this to many teachers, but did not push anyone. Therefore, I will simply post this knowledge here, and then suddenly someone will come in handy or guide on certain thoughts. Immediately I will say I'm not a mathematician, I stumbled upon it by chance when I was playing with numbers. I didn’t even know what factorial was :)

First, let's recall the general theory:

The factorial of the number n is the product of all natural numbers up to n inclusive:
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By definition, 0 is assumed! = 1 . Factorial is defined only for non-negative integers.

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

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

• First we 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 n and write the results below.
  
  We get:
  

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

• Now subtract the last but 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

• Repeat the previous step already on the received row until one number remains (or a row of identical numbers, if the number is greater 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)

Take four numbers in degree 3 and calculate the "pyramid difference" (he came up with)

1 ³ 2 ³ 3 ^{3 3 3}
1 8 27 64
(8 - 1) (27 - 8) (64 - 27)

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

12 18
(18 - 12)

6
It all fits together!

Well and for 1 we will try (1! = 1)

1 ¹ 2 ¹
12
(2 - 1)

one

Have you guessed it? :)

Everything is very simple and for zero:

Take n + 1 numbers to the power of 0, that is, one is enough

1 ^o
one

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

But nevertheless, I think it's great :)

Thanks for attention!

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