Division by Zero

Division by zero is impossible because multiplying any number by zero always yields zero.

By definition, a number “n” (the dividend) is divisible by another number “d” (the divisor) if there exists a third number “q” (the quotient) such that the product of the divisor and the quotient equals the dividend: $$ \frac{n}{d} = q \Longleftrightarrow d \cdot q = n $$

Now consider any nonzero number “n”:

$$ n \ne 0 $$

The expression \( n : 0 \) has no valid meaning.

$$ q = \frac{n}{0} $$

There is no number “q” (quotient) that, when multiplied by zero (the divisor), produces “n” (the dividend):

$$ q \cdot 0 = n $$

$$ 0 = n $$

Therefore, dividing a nonzero number by zero yields no possible result.

In other words, it is an operation that cannot be performed.

Note. Zero is the absorbing element of multiplication: multiplying any value “q” by zero always gives zero. Hence, the product can never equal “n,” which by assumption is a nonzero number.

Division of Zero by Zero

The division of zero by zero is indeterminate:

$$ q = \frac{0}{0} $$

because every number “q” multiplied by zero yields the dividend 0.

$$ q \cdot 0 = 0 $$

In this case, dividing zero by zero (0:0) admits infinitely many possible results - there is no unique value for “q.”

Hence, it is an indeterminate operation.

And so on.