Mathematics
Why Division by Zero Is Undefined
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A calculator says ERROR for 12 ÷ 0 because no number times 0 gives 12. It is not infinity either.
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Key points
- Division undoes multiplication: a ÷ b = c means c × b = a.
- a ÷ 0 (a ≠ 0): no c gives c × 0 = a → undefined.
- 0 ÷ 0: every c works → no single answer → undefined.
- Infinity is not a number; near 0, 12/x runs to +∞ on one side and −∞ on the other.
Why there is no answer
12 ÷ 3 = 4 because 4 × 3 = 12. So 12 ÷ 0 = c would need c × 0 = 12, but every number × 0 = 0. No c exists.
Not infinity either
12 ÷ 0.1 = 120, 12 ÷ 0.01 = 1200 (towards +∞); 12 ÷ (−0.01) = −1200 (towards −∞).
What is allowed
| Expression | Value |
|---|---|
| 0 ÷ 5 | 0 |
| 5 × 0 | 0 |
| 5 ÷ 0 | Undefined |
| 0 ÷ 0 | Undefined |
That is why a rational number p/q needs q ≠ 0.
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