Reciprocal Calculator
Calculate the reciprocal of a number or fraction. Enter a whole number or numerator/denominator — get the reciprocal as a fraction and decimal.
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How to use this calculator
The reciprocal of a number x is 1/x. For a fraction a/b, the reciprocal is b/a (simply flip numerator and denominator). Any number times its reciprocal equals 1.
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Enter the numerator (or just the whole number if you have a simple number).
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Enter the denominator — use 1 if your number is a whole number.
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The reciprocal is shown as a simplified fraction and a decimal.
- 4
The verification row confirms that original × reciprocal = 1.
Frequently asked questions
What is the reciprocal of a whole number?
The reciprocal of a whole number n is 1/n. For example, the reciprocal of 5 is 1/5 = 0.2, and the reciprocal of 8 is 1/8 = 0.125. Enter the number in the "Numerator" field and leave "Denominator" as 1.
What is the reciprocal of a fraction?
Flip the fraction: the reciprocal of 3/4 is 4/3 (= 1.333...), and the reciprocal of 7/2 is 2/7 (≈ 0.2857). The original fraction times its reciprocal always equals 1: (3/4) × (4/3) = 12/12 = 1.
Why is the reciprocal of zero undefined?
1/0 is undefined in standard arithmetic because no finite number multiplied by 0 gives 1. In calculus, as x approaches 0 from the positive side, 1/x approaches +∞; from the negative side, −∞. The reciprocal function has a vertical asymptote at x = 0.
Where are reciprocals used in real life?
Reciprocals appear in resistance (total resistance of parallel resistors: 1/R = 1/R₁ + 1/R₂), optics (lens formula: 1/f = 1/v + 1/u), speed (time = distance / speed, where 1/speed is a reciprocal factor), and dividing fractions (dividing by a/b is the same as multiplying by b/a).
Reciprocal calculator — 1/x for numbers and fractions
Reciprocals and division
Dividing by a number is identical to multiplying by its reciprocal. This is why "keep, change, flip" works for fraction division: to compute (2/3) ÷ (5/7), keep 2/3, change ÷ to ×, and flip 5/7 to 7/5, giving (2/3) × (7/5) = 14/15. The underlying mechanism is that dividing by x equals multiplying by 1/x — the multiplicative inverse.
Multiplicative inverses in algebra
A number and its reciprocal are called multiplicative inverses of each other. Their defining property is that their product is the multiplicative identity: x × (1/x) = 1. In abstract algebra, a field is a number system where every nonzero element has a multiplicative inverse — which is why 0 has no reciprocal (it would violate the field axioms). Matrices have analogous inverses: A × A⁻¹ = I.
Physics applications: parallel resistors and optics
In electrical circuits, resistors in parallel combine as 1/R_total = 1/R₁ + 1/R₂ + ... The total resistance is the reciprocal of the sum of reciprocals. Similarly, the thin lens equation 1/f = 1/d_o + 1/d_i relates focal length f and object/image distances using reciprocals. These formulas arise because the relevant physical quantities combine reciprocally rather than directly.
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