Why Can't We Divide By Zero?

The Logical Problem

You can’t divide by zero because there’s no number you can multiply by 0 to get a nonzero result. Suppose you write it as a0=x\frac{a}{0} = x0a=x. Then multiplying both sides by 0 gives 0⋅x=a0 \cdot x = a0⋅x=a. But 0⋅x=00 \cdot x = 00⋅x=0 for every xxx, so unless a=0a = 0a=0, there is no value of xxx that works — the operation is undefined.

⚠️ A Special Case: 00\frac{0}{0}00

Even when a=0a = 0a=0, so you get 00\frac{0}{0}00, the expression is still not defined. It’s indeterminate: it can correspond to many different outcomes depending on how it’s approached in a limit. For example, lim⁡x→0xx=1\lim_{x \to 0} \frac{x}{x} = 1limx→0xx=1, while lim⁡x→0x2x=0\lim_{x \to 0} \frac{x^2}{x} = 0limx→0xx2=0, and lim⁡x→0xx2=∞\lim_{x \to 0} \frac{x}{x^2} = \inftylimx→0x2x=∞. So without further context, 00\frac{0}{0}00 cannot be assigned a single value.

∞ Infinity Isn’t a Number

People often say “let x=∞x = \inftyx=∞”, but infinity is not a number like 5 or −2. For instance, if you divide 1 by smaller and smaller positive numbers — say 1,0.1,0.01,…1, 0.1, 0.01, \dots1,0.1,0.01,… — the values grow without bound. We write this as lim⁡t→0+1t=∞\lim_{t \to 0^+} \frac{1}{t} = \inftylimt→0+t1=∞. That doesn’t mean 10=∞\frac{1}{0} = \infty01=∞, only that the expression diverges as the denominator approaches zero from the right. Infinity is a concept, not a concrete value.

🕳 What About Black Holes?

In general relativity, black holes involve singularities — places where curvature or inferred density becomes infinite, or →∞\to \infty→∞. That doesn’t mean division by zero is allowed; it means the theory is breaking down. Singularities tell physicists that a new framework (like quantum gravity) is needed — not that mathematical rules have changed.

Dividing by zero is like trying to divide cookies among zero people: it’s meaningless. “How many cookies does each person get?” makes no sense when there are no people. Try it on a calculator and you’ll see an error — because the operation is undefined.

📚 Tiny Classroom Box: Use Limits, Not Substitution

You must use limits to explore what happens near zero — don’t just plug in zero blindly. For example, lim⁡x→0+1x=∞\lim_{x \to 0^+} \frac{1}{x} = \inftylimx→0+x1=∞, meaning the quotient grows arbitrarily large. But lim⁡x→0xx=1\lim_{x \to 0} \frac{x}{x} = 1limx→0xx=1, and lim⁡x→0x2x=0\lim_{x \to 0} \frac{x^2}{x} = 0limx→0xx2=0, while lim⁡x→0xx2=∞\lim_{x \to 0} \frac{x}{x^2} = \inftylimx→0x2x=∞. So again, 00\frac{0}{0}00 is indeterminate and requires context.

Always ask: which limit? which path? There’s no shortcut around the logic — and no license to divide by zero.