Multiplicative inverses

Key Notes:

Definition:

What are Multiplicative Inverses?

In mathematics, the multiplicative inverse of a number is the value that, when multiplied by the original number, equals 1. Another name for the multiplicative inverse is the reciprocal. Thinking of it another way: To find the multiplicative inverse of a number, simply flip it! This works for fractions and whole numbers (because any whole number can be written as a fraction over 1).

For any number x, its multiplicative inverse is 1/x, as long as x is not zero. Zero does not have a multiplicative inverse because dividing by zero is undefined.

Examples of Multiplicative Inverses

Example 1: The multiplicative inverse of 5 is 1/5 because 5 * (1/5) = 1. Example 2: The multiplicative inverse of 1/2 is 2 because (1/2) * 2 = 1. Example 3: The multiplicative inverse of 3/4 is 4/3 because (3/4) * (4/3) = 1.

Example 4: The multiplicative inverse of -2 is -1/2 because -2 * (-1/2) = 1. Example 5: The multiplicative inverse of -5/3 is -3/5 because (-5/3) * (-3/5) = 1. Key Point: The multiplicative inverse has the same sign as the original number!

Symbol:

Introduce the notation a⁻¹ for the multiplicative inverse of a.

Non-zero Condition:

Emphasize that the multiplicative inverse exists only if a≠0, because division by zero is undefined.

Finding Inverses:

Teach methods to find multiplicative inverses, such as:

• For fractions: Flip the fraction (reciprocal).
• For whole numbers: Identify the reciprocal that, when multiplied, gives 1.

Applications:

Discuss practical applications, like solving equations involving fractions or ratios.

Practice Exercises

Provide plenty of practice problems to reinforce understanding, ensuring students can confidently find multiplicative inverses of various numbers.

Answers to Practice Exercises

let’s practice!