A fraction is a division, and dividing by zero has no result. That is why 3/0 has no value: no number times 0 gives 3. With the zero on top everything changes, because 0/5 equals 0, and 0/0 can’t be settled. If you need to divide fractions, the divide fractions calculator stops at a divisor equal to zero, and for expressions with x the guide on algebraic fractions and domain shows how to find the excluded values.
What happens in each case with a zero
Dividing a by b means finding the number q that satisfies b × q = a. That check decides all three cases with zero. If you don’t remember which is which, review the numerator and denominator first.
| Expression | Equation for q | Conclusion |
|---|---|---|
| 8/0 | 0 × q = 8 | No q satisfies it: no value |
| 0/0 | 0 × q = 0 | Every q satisfies it: indeterminate |
| 0/5 | 5 × q = 0 | Only q = 0: equals 0 |
For 8/0 you would need a q with 0 × q = 8, but zero times any number is 0. So the result can’t be 8, and no other number works either. With zero on top and bottom the opposite happens: q = −2, q = 0 and q = 5 all work equally, and without a single value there is no quotient to pick. With 0/5 the equation holds only for q = 0. That is why “zero divided by something is zero” is true only if that something isn’t zero.
Worked example: 4/5 divided by zero and zero divided by 4/5
The two calculations use the same numbers, but only one of them has a result.
- Write the division as a multiplication. Saying 0 ÷ 4/5 equals q means q times that fraction gives 0.
- Solve it. A product is zero only if one factor is, and the fraction isn’t, so q = 0.
- Repeat with zero as the divisor. Saying the fraction divided by 0 equals q means 0 × q is that fraction, and no number satisfies that.
- Look at it with the reciprocal. Dividing by 4/5 is multiplying by 5/4, and 0 × 5/4 = 0. Dividing by 0 would require multiplying by 1/0, the forbidden fraction.
Result. 0 ÷ 4/5 = 0, and 4/5 ÷ 0 has no value.
Check. Multiplying the fraction by 0 gives 0, which confirms the first. For the second there is no check, and that is the sign that there is no result.
A divisor that looks like an ordinary fraction can also be zero. In 3/4 ÷ 0/5 the divisor 0/5 equals 0, so the calculation fails just the same. To see why zero has no reciprocal, look at the reciprocal of a fraction.
Simplifying without losing the excluded value
In (x² − 1)/(x − 1), factor the numerator as (x − 1)(x + 1) and cancel x − 1. What remains is x + 1, but only for x ≠ 1, because at x = 1 the original fraction becomes zero over zero. At x = 3 the two forms agree, since 8/2 = 4 and 3 + 1 = 4. At x = 1 the original doesn’t exist, while x + 1 would give 2. That is why the graph is the line y = x + 1 with a hole at the point (1, 2). Write down the restriction before canceling, by setting the original denominator equal to zero.
There are specific tools for working with these restrictions. The one for simplifying algebraic fractions keeps the exclusions next to the result, and the one for equations with fractions discards solutions that make a denominator zero.
A small divisor is not a zero divisor
When you divide by smaller and smaller numbers, the results grow: 1 ÷ 1/10 = 10 and 1 ÷ 1/100 = 100. That is where the idea that 1/0 is infinite comes from. But with negative divisors the opposite happens, because 1 ÷ (−1/100) = −100 and the quotients drop without limit. Since the behavior depends on which side you approach zero from, no single value works. Infinity describes where an expression heads in a limit. Division itself doesn’t give it as a result, and 1/0 stays undefined.
Practice
- What is 6/0?
- What is 0/(−6)?
- What is 2/3 ÷ 0?
- Simplify (x² − 9)/(x − 3) and write the restriction.
See the answers
- It has no value, because no q satisfies 0 × q = 6.
- It equals 0, because (−6) × 0 = 0.
- It has no value, because the divisor is zero.
- x + 3, with x ≠ 3.