Division can be understood as the inverse of multiplication. We say 6÷2=3 because 2×3=6. Writing 6÷0=x would therefore require 0×x=6. Yet 0×x is 0 for every real value of x, so the equation has no solution.
What about 0÷0? That asks for 0×x=0, which every real number satisfies. The problem is no longer that there is no answer, but that there are too many answers for the original expression to determine a unique one. Ordinary real-number arithmetic treats both cases as undefined so that division remains a well-defined operation, although the reasons differ.
Calling the answer “infinity” does not repair the rule. If 1 is divided by a positive number approaching zero, the values grow without bound; approach from the negative side and they decrease without bound instead. These are statements about limits from different directions, not about setting the denominator equal to zero. If 1÷0 were assigned any number, multiplying it back by zero would still produce 0 rather than 1, breaking the relationship between multiplication and division.
Division by zero is therefore not forbidden because the result is merely very large. Multiplication by zero erases distinctions: every number is sent to the same result, so the original number cannot be recovered.
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