Apr 30, 2026
Why is dividing by zero undefined and not infinity simple explanation
In school they told us "you cant divide by zero" but they never explained WHY.
Like if you divide a number by a really small number you get a really big number right?
So as the denominator gets closer to 0, the result goes to infinity. So why isnt ?
I heard its because would imply which is nonsense. But also from the negative side, , so it approaches negative infinity.
Is there a way to make division by zero work with the "Riemann sphere" or "projectively extended real line"? Or is it fundamentally impossible?
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1 Answer
The fundamental reason: Division is defined as the inverse of multiplication.
$a \div b = c$ means $b \times c = a$.
So $1 \div 0 = c$ would mean $0 \times c = 1$. But $0 \times c = 0$ for ANY $c$. Theres no number $c$ satisfying $0 \times c = 1$.
**What about the limit argument?**
$$\lim_{x \to 0^+} \frac{1}{x} = +\infty, \quad \lim_{x \to 0^-} \frac{1}{x} = -\infty$$
The left and right limits dont agree, so even in the extended real line, the limit doesnt exist as a single value.
**The Riemann sphere** does define $1/0 = \infty$ (a single point at infinity), but this requires adding a new element $\infty$ to the complex numbers and changing the definition of division. Its useful in complex analysis but not in standard arithmetic.
**In short:** Division by zero is undefined because it would contradict the definition of division itself.
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