Two numbers are multiplicative inverses (also called reciprocals) when their product results in .
For example:
and are multiplicative inverses because
Multiplicative Inverse Practice Problems & Exercises
Master multiplicative inverses with step-by-step practice problems. Learn reciprocals, fraction division, and special cases involving 0 and 1.
📚What You'll Master in This Practice Session
- Find multiplicative inverses of whole numbers, fractions, and decimals
- Apply the rule a × (1/a) = 1 to solve complex problems
- Convert division by fractions into multiplication by reciprocals
- Solve multi-step problems involving multiplicative inverses and mixed operations
- Understand why zero has no multiplicative inverse and special properties of 1
- Work with fraction line notation and complex fraction expressions
Understanding Multiplicative Inverse
Complete explanation with examples
Formulation of the Rule for Multiplicative Inverse Numbers:
Whenever is different from , it follows that
Note: Zero does not have a multiplicative inverse because division by zero is undefined.
Multiplication and Division of Multiplicative Inverses
Division is equivalent to multiplication by its multiplicative inverse,
That is:
Because is the multiplicative inverse of
Generally:
Practice Multiplicative Inverse
Test your knowledge with 19 quizzes
\( \frac{6}{3}\times1=\text{ ?} \)
Examples with solutions for Multiplicative Inverse
Step-by-step solutions included
Exercise #1
Solve the following exercise:
Step-by-Step Solution
According to the order of operations rules, since the exercise only involves addition and subtraction operations, we will solve the problem from left to right:
Answer:
Video Solution
Exercise #2
Solve the following exercise:
Step-by-Step Solution
According to the order of operations rules, since the exercise only involves addition and subtraction, we will solve the problem from left to right:
Answer:
9.5
Video Solution
Exercise #3
Step-by-Step Solution
According to the order of operations rules, we first divide and then add:
Answer:
Video Solution
Exercise #4
Step-by-Step Solution
According to the order of operations rules, we first divide and then add:
Answer:
Video Solution
Exercise #5
Solve the following exercise:
Step-by-Step Solution
According to the order of operations rules, we first divide and then add:
Answer:
Frequently Asked Questions
What is a multiplicative inverse and how do I find it?
+A multiplicative inverse (or reciprocal) is a number that when multiplied by the original number equals 1. To find it: for whole numbers like 5, the inverse is 1/5; for fractions like 3/7, flip it to get 7/3; for decimals like 0.25, convert to fraction (1/4) then flip to get 4.
Why doesn't zero have a multiplicative inverse?
+Zero has no multiplicative inverse because there's no number that when multiplied by 0 gives 1. Since 0 × any number = 0 (never 1), division by zero is undefined in mathematics.
How do I divide by a fraction using multiplicative inverses?
+To divide by a fraction, multiply by its multiplicative inverse instead. For example: 2 ÷ (1/3) = 2 × 3 = 6, because 3 is the multiplicative inverse of 1/3.
What are common mistakes when working with multiplicative inverses?
+Common errors include: 1) Confusing additive inverse (-a) with multiplicative inverse (1/a), 2) Forgetting that 1 is its own multiplicative inverse, 3) Trying to find the inverse of zero, 4) Not simplifying fractions before finding inverses.
How do multiplicative inverses help with complex fractions?
+Complex fractions like (a)/(1/b) become simple multiplication: a × b. This works because dividing by 1/b is the same as multiplying by b (the multiplicative inverse of 1/b).
What's the difference between reciprocal and multiplicative inverse?
+Reciprocal and multiplicative inverse mean exactly the same thing - they're interchangeable terms. Both refer to the number that when multiplied by the original gives a product of 1.
How do I check if two numbers are multiplicative inverses?
+Simply multiply them together. If the product equals 1, they are multiplicative inverses. For example: 2/5 × 5/2 = 10/10 = 1, so 2/5 and 5/2 are multiplicative inverses.
Can negative numbers have multiplicative inverses?
+Yes, negative numbers have multiplicative inverses. For -3, the multiplicative inverse is -1/3 because (-3) × (-1/3) = 1. The inverse of a negative number is also negative.