Multiplicative Inverse - Math Steps, Examples & Questions

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Multiplicative inverse

Here you will learn about the multiplicative inverse, including the definition and how to find it.

Students will first learn about the multiplicative inverse in $katex is not defined$ th grade math as part of their work in number and operations with fractions when they learn how to divide fractions and extend their knowledge as they work through number systems in middle school.

What is the multiplicative inverse?

The multiplicative inverse of a number is its reciprocal. The multiplicative inverse property states that a number that is multiplied to the original number has a product of $katex is not defined$ In other words, a number and its reciprocal will always have a product of $katex is not defined$

Let’s take the number $katex is not defined$

$katex is not defined$ has a reciprocal of $katex is not defined$

Multiply $katex is not defined$ and $katex is not defined$

$katex is not defined$ is the multiplicative inverse of $katex is not defined$

To help make sense of a multiplicative inverse, let’s look at visual models.

US Web Page_ Multiplicative Inverse 1 US

Do you see a pattern?

$katex is not defined$

Not yet? Do you see a pattern now?

$katex is not defined$

Can negative numbers have multiplicative inverses?

Let’s look at $katex is not defined$

If you follow the pattern, the reciprocal of $katex is not defined$ is $katex is not defined$ (the placement of the negative sign can be in the denominator, numerator, or out in front).

$katex is not defined$

The multiplicative inverse of $katex is not defined$ is $katex is not defined$

What is the multiplicative inverse?

Common Core State Standards

How does this apply to $katex is not defined$ th grade and $katex is not defined$ th grade math?

Grade 6: Number System (6.NS.A.1)
Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, example, by using visual fraction models and equations to represent the problem.

Grade 7: Number System (7.NS.A.2a)
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as $katex is not defined$ and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

Grade 7: Number System (7.NS.A.2b)
Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If $katex is not defined$ and $katex is not defined$ are integers, then $katex is not defined$ Interpret quotients of rational numbers by describing real-world contexts.

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How to find the multiplicative inverse

In order to write the multiplicative inverse of a number:

If the number is a whole number, mixed number, decimal, or integer, write it as an improper fraction.
“Flip” the fraction by switching the numerator and denominator.
Check to make sure the product is $katex is not defined$ .

Multiplicative inverse examples

Example 1: multiplicative inverse of a whole number

Find the multiplicative inverse of $katex is not defined$ .

If the number is a whole number, mixed number, decimal, or integer, write it as an improper fraction.

$katex is not defined$

2“Flip” the fraction by switching the numerator and denominator.

$katex is not defined$

$katex is not defined$ becomes the denominator, and $katex is not defined$ becomes the numerator.

3Check to make sure the product is $katex is not defined$ .

$katex is not defined$

The multiplicative inverse of $katex is not defined$ is $katex is not defined$

Example 2: multiplicative of a fraction number

Find the multiplicative inverse of $katex is not defined$ .

If the number is a whole number, mixed number, decimal, or integer, write it as an improper fraction.

$katex is not defined$ is a proper fraction.

“Flip” the fraction by switching the numerator and denominator.

$katex is not defined$

$katex is not defined$ becomes the denominator, and $katex is not defined$ becomes the numerator.

Check to make sure the product is $katex is not defined$ .

$katex is not defined$

$katex is not defined$ is the multiplicative inverse of $katex is not defined$

Example 3: multiplicative inverse of a mixed fraction

Find the multiplicative inverse of a $katex is not defined$ .

If the number is a whole number, mixed number, decimal, or integer, write it as an improper fraction.

$katex is not defined$

“Flip” the fraction by switching the numerator and denominator.

$katex is not defined$

$katex is not defined$ becomes the denominator, and $katex is not defined$ becomes the numerator.

Check to make sure the product is $katex is not defined$ .

$katex is not defined$

$katex is not defined$ is the multiplicative inverse of $katex is not defined$ or $katex is not defined$ .

Example 4: multiplicative inverse of a decimal number

Find the multiplicative inverse of $katex is not defined$ .

If the number is a whole number, mixed number, decimal, or integer, write it as an improper fraction.

$katex is not defined$ is thirteen hundredths, which is $katex is not defined$ .

“Flip” the fraction by switching the numerator and denominator.

$katex is not defined$

$katex is not defined$ becomes the denominator, and $katex is not defined$ becomes the numerator.

Check to make sure the product is $katex is not defined$ .

$katex is not defined$

$katex is not defined$ is the multiplicative inverse of $katex is not defined$ .

Example 5: multiplicative inverse of a negative number

Find the multiplicative inverse of $katex is not defined$ .

If the number is a whole number, mixed number, decimal, or integer, write it as an improper fraction.

$katex is not defined$

“Flip” the fraction by switching the numerator and denominator.

$katex is not defined$

Check to make sure the product is $katex is not defined$ .

$katex is not defined$

The multiplicative inverse of $katex is not defined$ is $katex is not defined$ .

Example 6: multiplicative of a negative fraction

Find the multiplicative inverse of $katex is not defined$ .

If the number is a whole number, mixed number, decimal, or integer, write it as an improper fraction.

$katex is not defined$

“Flip” the fraction by switching the numerator and denominator.

$katex is not defined$

$katex is not defined$ becomes the denominator, and $katex is not defined$ becomes the numerator.

The placement of the negative number can be out in front of the fraction, $katex is not defined$

Check to make sure the product is $katex is not defined$ .

$katex is not defined$

$katex is not defined$ is the multiplicative inverse of $katex is not defined$

Teaching tips for multiplicative inverse

Reinforce that the multiplicative inverse is also the reciprocal.

Have students discover the multiplicative inverse of numbers on their own by strategically asking questions such as, “what number multiplied to $katex is not defined$ will give a product of $katex is not defined$

When working with multiplicative inverse, have students see that the reciprocal of a unit fraction is a whole number.

Although worksheets serve a purpose in the classroom and can help with skill practice and test prep practice, having students discover the mathematical concepts is more meaningful for building long lasting understanding.

Provide students with calculators or multiplication tables so that they do not get frustrated making calculations, but rather develop an understanding of the topic.

Easy mistakes to make

Trying to find the reciprocal of zero
Zero does not have a reciprocal. This is because we can think about $katex is not defined$ as being $katex is not defined$ When the numerator and the denominator are flipped it becomes $katex is not defined$ or $katex is not defined$ which is undefined (it does not exist).

Forgetting that a whole number can be written as an improper fraction
Whole numbers can be written as improper fractions. For example, $katex is not defined$ can be written as $katex is not defined$

Practice multiplicative inverse questions

katex is not defined

katex is not defined

katex is not defined

katex is not defined

Write $katex is not defined$ as an improper fraction, $katex is not defined$

Flip the numerator and the denominator. $katex is not defined$ becomes the denominator and $katex is not defined$ becomes the numerator. Check that the product of the number and its multiplicative inverse is $katex is not defined$

$katex is not defined$

katex is not defined

katex is not defined

katex is not defined

katex is not defined

$katex is not defined$ is already a proper fraction.

$katex is not defined$

The multiplicative inverse of $katex is not defined$ is $katex is not defined$ .

katex is not defined

katex is not defined

katex is not defined

katex is not defined

The multiplicative inverse of $katex is not defined$ when multiplied to $katex is not defined$ will make the product equal to $katex is not defined$

Make $katex is not defined$ into a fraction, $katex is not defined$

Flip the numerator and the denominator. $katex is not defined$ becomes the denominator and $katex is not defined$ becomes the numerator.

The multiplicative of $katex is not defined$ is $katex is not defined$ or $katex is not defined$

$katex is not defined$

katex is not defined

katex is not defined

katex is not defined

katex is not defined

$katex is not defined$ change to an improper fraction, $katex is not defined$

Flip the numerator and the denominator. $katex is not defined$ becomes the denominator and $katex is not defined$ becomes the numerator.

The multiplicative inverse is $katex is not defined$ or $katex is not defined$ .

$katex is not defined$

katex is not defined

katex is not defined

katex is not defined

katex is not defined

Change $katex is not defined$ to a fraction, $katex is not defined$

Flip the numerator and the denominator.

$katex is not defined$ becomes the denominator and $katex is not defined$ becomes the numerator, $katex is not defined$

$katex is not defined$

The multiplicative inverse of $katex is not defined$ is $katex is not defined$

katex is not defined

katex is not defined

katex is not defined

katex is not defined

Change $katex is not defined$ to an improper fraction.

$katex is not defined$

Flip the numerator and the denominator.

$katex is not defined$ becomes the denominator and $katex is not defined$ becomes the numerator.

The multiplicative inverse of $katex is not defined$ is $katex is not defined$

$katex is not defined$

Multiplicative inverse FAQs

Is there a multiplicative inverse of any real number or complex number?

Rational numbers (natural numbers, whole numbers, integers, fractions/decimals) and irrational numbers have multiplicative inverses. The only number that does not have a multiplicative inverse is $katex is not defined$
The multiplicative inverse of any number a is $katex is not defined$
For example, the multiplicative inverse of $katex is not defined$ can be written as $katex is not defined$ or $katex is not defined$
$katex is not defined$ $katex is not defined$ is an irrational number.
The multiplicative inverse of $katex is not defined$ is $katex is not defined$ or $katex is not defined$
To find the multiplicative inverse of $katex is not defined$ is $katex is not defined$ or $katex is not defined$ which does not exist because you cannot divide by $katex is not defined$

Complex numbers, which you will study in great depth in an Algebra $katex is not defined$ class, also have multiplicative inverses.
The complex number $katex is not defined$ has $katex is not defined$ as its multiplicative inverse. All types of numbers have multiplicative inverses.

Can you find the multiplicative inverse of a multiplicative inverse?

Yes, if you find the multiplicative inverse of a multiplicative inverse, you will wind up back at the original given number.
For example, the multiplicative inverse of $katex is not defined$ (written as $katex is not defined$ ) is $katex is not defined$
If you then take the multiplicative inverse of $katex is not defined$ it will be $katex is not defined$ or $katex is not defined$ is the original given number.

Is the multiplicative inverse of a number just turning the fraction upside down?

The multiplicative inverse of a number is always a number that when multiplied by the original number gives a product of $katex is not defined$ So, it is often just flipping the fraction around.

What is the reciprocal of a number?

The reciprocal of a number is the multiplicative inverse.

What is the multiplicative identity?

The multiplicative identity is $katex is not defined$ because a number times $katex is not defined$ has a product of that number.

What is the additive inverse?

The additive inverse is a number when added to the original number has a sum of $katex is not defined$ For example, the additive inverse of $katex is not defined$ is $katex is not defined$ because $katex is not defined$

What is an extended euclidean algorithm?

The euclidean algorithm or extended euclidean algorithm is used for finding the greatest common divisor (gcd) between two positive numbers and writing the gcd as an integer linear combination. This is something explored in an advanced high school or college level.

What is modular arithmetic?

Modular arithmetic is studied in advanced high school and college courses and is a branch of arithmetic where the only concern is with the remainder after dividing two numbers.

There is an operator that is used to determine the remainder of a division operation called the modulus operator or the modulo operator.

The next lessons are

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