Reciprocal Calculator
The reciprocal of a number is 1 divided by that number, also called the multiplicative inverse. Choose your input type, enter your value, and see the result as a decimal and a simplified fraction with every step shown.
Formula
Worked example
Reciprocal of 4: 1/4 = 0.25. Reciprocal of 2/3: flip to get 3/2 = 1.5. Reciprocal of 3 1/4: convert to 13/4, flip to get 4/13 = 0.307692... Check: 4 x 0.25 = 1; 2/3 x 3/2 = 1; 13/4 x 4/13 = 1.
What a reciprocal is
The reciprocal of a number x is 1 divided by x, written as 1/x or x to the power of negative one. It is called the multiplicative inverse because any number multiplied by its reciprocal equals one. For example, the reciprocal of 4 is 0.25, and 4 times 0.25 = 1. The only number without a reciprocal is zero, since dividing one by zero is undefined and no value times zero can ever produce one.
Reciprocals of fractions and mixed numbers
For a fraction a/b, the reciprocal is simply the fraction flipped: b/a. So the reciprocal of 2/3 is 3/2, and the reciprocal of 5 (which is 5/1) is 1/5. For a mixed number such as 3 1/4, first convert it to an improper fraction: 3 times 4 plus 1 = 13, so the number is 13/4. Then flip it to get 4/13. This two-step process works for any mixed number. The reciprocal keeps the same sign as the original, so the reciprocal of a negative value stays negative.
How decimal places and scientific notation help
For most everyday numbers the six-decimal result is clear enough, but for very small inputs (like 0.000025) or very large ones (like 40000), the reciprocal is easier to read in scientific notation: 1/0.000025 = 4e4 and 1/40000 = 2.5e-5. This calculator automatically shows the scientific form when it is more compact. You can also choose how many decimal places you want to see in the result, from two up to ten.
Where reciprocals appear in math and science
Reciprocals are used throughout mathematics and science. In algebra they let you divide by fractions: dividing by 3/4 is the same as multiplying by 4/3. In physics, combined resistance for parallel resistors uses the formula 1/R = 1/R1 + 1/R2, and the thin-lens formula 1/f = 1/do + 1/di both rely on reciprocals. Harmonic means are computed from reciprocals, and rates (such as speed = distance / time) involve reciprocal relationships. Recognising the reciprocal structure makes many formulas easier to remember and manipulate.
Common reciprocals reference
| Number | As fraction | Reciprocal (fraction) | Reciprocal (decimal) | Scientific notation |
|---|---|---|---|---|
| 1 | 1/1 | 1/1 | 1 | |
| 2 | 2/1 | 1/2 | 0.5 | |
| 4 | 4/1 | 1/4 | 0.25 | |
| 5 | 5/1 | 1/5 | 0.2 | |
| 10 | 10/1 | 1/10 | 0.1 | |
| 100 | 100/1 | 1/100 | 0.01 | 1e-2 |
| 1000 | 1000/1 | 1/1000 | 0.001 | 1e-3 |
| 1/2 | 1/2 | 2/1 | 2 | |
| 2/3 | 2/3 | 3/2 | 1.5 | |
| 3/4 | 3/4 | 4/3 | 1.333... | |
| 1/8 | 1/8 | 8/1 | 8 | |
| 3 1/4 | 13/4 | 4/13 | 0.307692... |
Each value paired with its reciprocal as a fraction, a decimal, and (where compact) scientific notation.
Frequently asked questions
What is the reciprocal of a fraction?
Swap the numerator and denominator. The reciprocal of a/b is b/a, so the reciprocal of 3/8 is 8/3. A whole number n is n/1, so its reciprocal is 1/n. Always simplify the result by dividing both parts by their greatest common divisor.
How do I find the reciprocal of a mixed number?
First convert the mixed number to an improper fraction: multiply the whole part by the denominator and add the numerator. For 3 1/4: (3 x 4 + 1) / 4 = 13/4. Then flip it: 4/13. Convert to a decimal by dividing: 4 / 13 = 0.30769...
What is the reciprocal of a negative number?
The reciprocal keeps the same sign. The reciprocal of -4 is -1/4 = -0.25, because (-4) x (-0.25) = 1. Flipping the sign would break the defining rule that a number times its reciprocal equals one.
Why does zero have no reciprocal?
A reciprocal must give 1 when multiplied by the original number. No number times zero equals one, and 1 / 0 is undefined, so zero is the single value with no multiplicative inverse.
What is the reciprocal of 1 and -1?
Both 1 and -1 are their own reciprocals: 1 x 1 = 1 and (-1) x (-1) = 1. They are the only two real numbers that are self-reciprocal.
When would I use a reciprocal in real life?
Dividing by a fraction (multiply by its reciprocal instead), solving equations of the form ax = b (divide both sides, i.e. multiply by 1/a), calculating parallel resistances or lens focal lengths in physics, and computing harmonic means in statistics all rely on reciprocals. Any time you need to undo a multiplication, the reciprocal is the tool.