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Inverse Operations

In math, an "operation" refers to calculating a value using values and a math operator. The numbers used in the operation are called operands. Depending on the type of operation, different terms are assigned to the operands. The operators are the symbols indicating which math operation you are to use. The four most common operations are:

  • + for addition
  • - for subtraction
  • * for multiplication
  • ÷ for division

So what is an inverse operation?

Inverse means "opposite," so an inverse operation is an opposite operation. Specifically, it is the operation that can undo what was done by the previous operation. A set of two opposite operations is called inverse operations.

For example, if you had 3 apples and bought 5 more apples, you would have 8 apples, because 3 + 5 = 8 . Then if you ate or gave away the 5 apples, you would be back to 3, because 8 - 5 = 3 . This example shows how addition and subtraction are inverse operations.

Addition and subtraction

As we've already seen, subtraction is the inverse operation for addition. Similarly, addition is in the inverse operation for subtraction. This means that addition undoes subtraction and subtraction undoes addition.

By rearranging the numbers in the addition equation, we can get two different subtraction equations, as in the following equations:

18 + 7 = 25

From this addition equation, you can get the following two subtraction equations:

25 + 18 = 7

25 - 7 = 18

It's also true that if you take a number, say 21, and add and subtract the same number to and from it, you will get the number, 21 in this case.

21 + 9 - 9 = 21

We can also rearrange the numbers given in a subtraction equation to make two addition equations as in the example below:

28 - 19 = 9

We can create the following two addition equations:

19 + 9 = 28

9 + 19 = 28

Again, if we first subtract a number and then add the same number to a specific number, we end up with the original number.

13 - 7 + 7 = 13

Multiplication and division

Like addition and subtraction, multiplication and division are inverse operations of each other. This is because multiplication undoes division and division undoes multiplication.

We can also rearrange the numbers in a given multiplication equation to create two division equations, as follows:

3 × 4 = 12

We can rearrange these numbers to create the two following division equations:

12 ÷ 3 = 4

12 ÷ 4 = 3

When we multiply a number and later divide the same number, we get the reverse effect, essentially reaching the original number.

6 × 4 = 24

24 ÷ 4 = 6

As with the multiplication equation, we can rearrange the numbers in a division equation to make two multiplication equations, as follows:

63 ÷ 9 = 7

We can use this equation to make the following two multiplication equations:

9 × 7 = 63

7 × 9 = 63

And, as with multiplication, if we divide a number and later multiply the same number, the effect is reversed and we end up with the original number.

45 ÷; 9 = 5

9 × 5 = 45

Exponents and Logarithms

Since exponents and logarithms are not commutative, they are slightly more complicated. After raising a number to a particular power, you can use the logarithm to get back to the exponent, but not the base.

10 3 = 1000

log 3 1000 10


log 10 1000 3

Topics related to the Inverse Operations

Inverse Functions

Multiplicative Inverse

Inverse Trigonometric Functions

Flashcards covering the Inverse Operations

Precalculus Flashcards

CLEP Precalculus Flashcards

Practice tests covering the Inverse Operations

Precalculus Diagnostic Tests

Get help learning about inverse operations

Inverse operations can be confusing specifically because they are the opposite of each other. Many students find the concepts especially challenging. If your student needs help with inverse operations, having them meet with a professional math tutor is a great idea. An expert tutor can supplement your student's classroom learning by working at your student's pace, taking extra time to go over the parts your student is having a hard time with. At the same time, they can speed through the parts that your student picks up easily. This makes tutoring an efficient learning method. Tutors are there to answer your student's questions as soon as they arise, making sure they do problems right the first time. To learn more about how tutoring can help your student pick up inverse operations, contact Varsity Tutors today to speak to one of our helpful Educational Directors.

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