What is the Reciprocal of a Number?
The reciprocal of a number is a number that, when multiplied by it, equals the number one. This is also called the multiplicative inverse.
For example, 2 * 0.5 = 1[How to Read Math in Code in This Website], so 2 is the multiplicative inverse of 0.5, and 0.5 is the multiplicative inverse of 2. Similarly:
- 4 * 0.25 = 1.
- 5 * 0.2 = 1.
- 10 * 0.1 = 1.
How to Find the Reciprocal of a Number?
The reciprocal of a number is always the number one divided by it.
For example, 1 / 2 = 0.5. So 0.5 is the reciprocal of 2.
The opposite is also true: 1 / 0.5 = 2. So 2 is the reciprocal of 0.5.
Fractions as Reciprocals
The reciprocal can also be just a fraction (a division).
The reciprocal of 2 is 1 / 2.
That means 2 * 1 / 2 is "2" being multiplied by its reciprocal, "1 / 2."
Which means 2 * 1 / 2 = 1.
If we rewrite this as 2 / 2 * 1 = 1, we can see that we have 2 being divided by itself.
Any number divided by itself is 1. So 2 / 2 = 1.
Which means 2 / 2 * 1 = 1 * 1. And 1 * 1 is 1.
Cancelling Reciprocals
Whenever you have a multiplication where two factors are reciprocal factors, we can replace those factors by the factor one.
For example, in 2 * 7 * 0.5, we have three factors (2, 7, and 0.5), and two of them are reciprocal: 2 and 0.5.
We can replace them by the factor one: 1 * 7.
Any number multiplied by 1 is itself, so 1 * 7 = 7.
Numerators and Denominators of Reciprocals
In the expression 4 * 1 / 4, we have reciprocal factors (4 and 1 /4). One of them is a whole number (4) while the other is a fraction (1 / 4).
We can rewrite the whole number as a fraction by pretending we are dividing it by 1. Because:
4 = 4 / 1.
Like this:
4 / 1 * 1 / 4.
Observe that the numerator (4) of the first division is the denominator (/ 4) of the second division, while the denominator (/ 1) of the first division is the numerator (4) of the second division.
If we change the denominator on one side and the numerator on the other side, the factors remain reciprocal.
4 / 1 * 1 / 4 = 1.
4 / 2 * 2 / 4 = 1.
4 / 3 * 3 / 4 = 1.
4 / 4 * 4 / 4 = 1.
4 / 5 * 5 / 4 = 1.
Reciprocals of Exponentiations
The reciprocal of an exponentiation is always the same exponentiation with the exponent inverted from positive to negative.
2^8 = 256.
2^(-8) = 1 / 256.
This is easier to understand by thinking of positive exponents as multiplications and negative exponents as divisions.
2^2 = 1 * 2 * 2 = 1 * 4 = 4.
2^(-2) = 1 / 2 / 2 = 1 / 4 = 0.25
Reciprocal of Zero
The number zero doesn't have a reciprocal.
Any number multiplied by zero is zero, so it's not possible to multiply a number by zero and get the number one.
Similarly, we can't do 1 / 0 to find its reciprocal, since we can't divide by zero.
References
- https://mathworld.wolfram.com/Reciprocal.html, accessed 2026-03-09.