If we say:

12 ÷ 4 = 3

we mean precisely the same as if we had said:

12 = 3 × 4

In other words, when we divide a number (called a dividend) a by some number b, we get a quotient q. And this quotient q when multiplied by b gives us the number  a.

a ÷ b = q

or

a = q × b

Suppose b is zero, then we have:

a ÷ 0 = q

or

a = q × 0

This raises two problems:

(1) Any number multiplied by zero equals zero. Therefore,

a = q × 0

can only be true if a is equal to zero. [If a is not equal to zero, let’s say a = 3. Then what number q times zero will give us a?]

(2) Even if a = 0, then any value of q would satisfy the equation. Consequently the value of q is indeterminate.

You can read a lot more about zero, its history, and common fallacies over here (this is an external link. While it was deemed safe at the time of writing, I accept no responsibility for external content).

This article proposes a system in which division by zero is permitted. You need to pay to gain access to the article. (This is an external link. While it was deemed safe at the time of writing, I accept no responsibility for external content).

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