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Teachers, Textbooks, Articles, the Internet always say that any real number is forbidden to be divided by 0. Some of us cannot understand why, some of us just accept it as a fact because we want to get good grades. In this article, we are going to discuss why it is forbidden.
One critical reason that it is a mathematical error is the definition of "Division"
Definition 1 :
To find the Quantity of How many times do we need to minus a certain number from the product until it reaches 0.
Example : 20 / 4 = ?
20 - 4 = 16
16 - 4 = 12
12 - 4 = 8
8 - 4 = 4
4 - 4 = 0
it takes 5 times
to minus 4 from the product 20,
so 20/4 = 5
What if : 20 / 0 = ?
20 - 0 = 20
20 - 0 = 20
20 - 0 = 20
... and so on
as you can see, if we keep going on doing this infinitely, we are not even closer to 19 or something smaller than 20
Definition 2 :
To find the Quantity of How many elements inside a group where all elements in all groups add up to the product
For example: if we need to put 20 apples into 4 bags and all bags must have the same amount of apples, how many apples does each bag has?
20 / 4 = 5
Classically, 5 apples each
What if: we need to put 20 apples into 0 bags and all bags must have the same amount of apples.
At first, we don't have any bags since there are 0 bags, then how come we can put any apples inside them? Therefore this is a logical error.
Definition 3 :
To find the other factor of the product when one factor is given.
The problem is, there are no numbers that can multiply with 0 and end up with a number that is not 0 since any number that multiplies with 0 ends up being 0
By using different definitions, you will end up with different answers
But other than using definitions to explain, are there any other ways to prove that dividing a number by 0 is not a valid move? And the answers are obviously yes.
Proof 1 : Prove by Contradiction
Actually a lot of math pranks uses this technique to false proof something, but here is the general idea.
Usually in algebra we "move" terms by using Addition, Subtraction, Multiplycation or Division or any other legitimate ways to arrange what is on both sides of an equation.
for example : ? - 8 = 2, we can add 8 on both sides, then we will have ? - 8 + 8 = 2 + 8, tidy it up then we can have ? = 10.
Let's just assume that
1) we are allowed to divide by 0,
2) and let A and B be any real numbers which are not equal,
since A x 0 = 0 and B x 0 = 0,
then we can conclude A x 0 = B x 0,
but if we divide both sides by 0, we will have A = B, which contradicts with what we have assume earlier. So dividing a number by 0 is not a legitimate move.
Proof 2 : Prove by Contradiction again
First we all know that N x 0 = 0 no matter what N is.
But let's just assume that a number, N, when divided by 0, is K, and we will have
N/0 = K
This is where we can end up with a lot of weird things...
If we were to multiply 0 on both sides, we will have
N/0 x 0 = K x 0
N = K x 0
This means "there is a number , multiply by 0, is not 0, is N instead", which does make any sense.
If we continue, we can divide K on both sides, and we will have
N / K = K x 0 / K
N / K = 0
seriously... what the heck is that ? O_O
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