The multiplicative inverse of a fraction is found by swapping the numerator and the denominator; in other words, it's the reciprocal of the fraction.
Understanding Multiplicative Inverses
In mathematics, the multiplicative inverse (also known as a reciprocal) of a number x is the number that, when multiplied by x, results in the multiplicative identity, which is 1. The reference material states, "In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1." For fractions, this is a straightforward process.
Finding the Multiplicative Inverse of a Fraction
The multiplicative inverse of a fraction a/b is b/a. This is also stated in the reference: "The multiplicative inverse of a fraction a/b is b/a."
Examples
Here's a table illustrating how to find the multiplicative inverse of different fractions:
| Fraction (a/b) | Multiplicative Inverse (b/a) |
|---|---|
| 2/3 | 3/2 |
| 5/7 | 7/5 |
| 1/4 | 4/1 = 4 |
| 8/3 | 3/8 |
Explanation
To find the multiplicative inverse, simply:
- Identify the numerator (a) and the denominator (b) of the fraction.
- Swap the numerator and denominator. The new fraction will be b/a.
- Verify: Multiply the original fraction by its inverse. The result should always be 1. For example, (2/3) * (3/2) = 6/6 = 1.
Practical Insights
- The multiplicative inverse is also known as the reciprocal.
- When dealing with whole numbers, consider them as fractions with a denominator of 1 (e.g., 5 = 5/1). Therefore, the multiplicative inverse of 5 is 1/5.
