How to Divide by Zero: a/0 = b ⟺ a = 0

TLDR: Division by zero is not as scary as it’s made out to be:

a/0 = b ⟺ a = 0


Division is multiplication, backwards. These two equations are exactly equivalent, by definition:

a/c = b

a = b×c

It’s easy to understand division by zero if we look at the equivalent multiplication.

a/0 = b

a = b×0

For any real number b:

a = b×0 = 0

a = 0

There are two cases with division by zero:

If a = 0, then a/0 = b is unconstrained, any real number b satisfies the equation. You can discard such an equation which does not constraint the result.

If a ≠ 0 then a/0 = b is contradictory. There is no real number b which satisfies that equation. This is still useful to know; “there is no answer” can be a sort of meta-answer. For example if trying to solve a system of equations of static forces, “there is no answer” might mean you need to consider a different design for your bridge!

There is no need to consider advanced concepts such as limits in order to fully understand division.

In short, a/0 = b is true if and only if a = 0.

If you see such an equation a/0 = b, you may simplify it to a = 0.

a/0 = b ⟺ a = b×0 ⟺ a = 0

a/0 = b ⟺ a = 0

I posted this here about a year ago: http://matheducators.stackexchange.com/a/5667/3287

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