Why can't you divide by zero?

Division is defined through multiplication, and zero breaks that definition. Dividing 6 by 3 means finding the number that, multiplied by 3, gives 6. For 6 divided by 0, you would need a number that multiplied by 0 gives 6, but any number times zero is zero. No such number exists, so the quotient is left undefined. [1] [2]

Division undoes multiplication. Multiplying and then dividing by the same nonzero quantity returns the original number. A USC lecture note makes the formal version: in the axioms of arithmetic, the multiplicative inverse exists only for nonzero numbers, and a divided by b is defined as a times the inverse of b. Since zero has no inverse, 1/0 is simply not defined. [1] [3]

The same note gives an intuitive picture. Multiplying by 5 loses no information, because you can recover the original number by dividing by 5. Multiplying by zero always gives zero, so the original number cannot be recovered. The operation cannot be reversed. [3]

Suppose some real number z were chosen as the inverse of zero. By definition, 0 times z would equal 1. But a theorem provable from the basic axioms says that zero times any real number is zero. That would make 1 equal to 0, a contradiction. So no real number can serve as 1/0. [3]

The case of 0 divided by 0 fails differently. The equivalent statement is that some unknown times 0 equals 0, which is true for every number. No single answer is picked out, so it is also left undefined. [1]

Allowing division by zero while using ordinary algebra rules can also produce fallacies, which are subtle mistakes that lead to absurd results. Leaving it undefined prevents these. [1]

Some systems do assign a result. The quotient can be defined as a single point at infinity, or as a positive or negative infinity depending on the sign of the dividend. These systems are not fields like the real numbers. Division by zero is no longer a special exception in them, but the infinite points bring their own exceptional behavior. [1]

The USC note makes a related point: one could introduce a new non-real symbol for the inverse of zero, but it would not obey ordinary arithmetic rules, and its usefulness is unclear, so the simpler choice is to leave it out. [3]

In computing, division by zero can trigger a fatal condition known as a division by zero error. [4]