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:
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)
Let's try to calculate factorial 3 in this way (3! = 1 * 2 * 3 = 6)
Well, for 1 try (1! = 1)
Have you already guessed? :)
Everything is very simple and for scratch:
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.
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.