Inverse Property: Definition, Uses & Examples

Inverse Property: In this article, you will understand the definition of inverse property as it applies employs to different types of numbers. You will also watch examples that show you how to calculate the additive inverse and multiplicative inverse of a number.

Inverse Property of Addition responds that any number added to its opposite will equal zero. What is the opposite you might ask? All you have to do is change the sign from positive to negative or negative to positive. Inverse Property of Multiplication responds that any number multiplied by its reciprocal is equal to 1.

Inverse Property

Inverse Property

Properties of Real Numbers

Real numbers have unique properties, which makes them particularly useful in everyday life.

  1. Real numbers are an ordered set of numbers. This means real numbers are sequential. The numerical value of every real number fits between the numerical values of two other real numbers. Everyone is comfortable with this concept since all measurements (weight, the shopping power of money, the speed of a car, etc.) depend upon the fact that some numbers have a greater value than other numbers. Ten is greater than five, and five is greater than four, and so on.
  2. We never run out of real numbers. The variety of real numbers available is not fixed. There is an infinite number of values available. The availability of numbers expands without end. Real numbers are not simply a finite “row of separate points” on a number line. There is always another real number whose value lies between any two real numbers.
  3. When real numbers are added or multiplied, the outcome is always another real number. (This is not the case with all arithmetic operations. For example, the square root of a -1 yields an imaginary number.)

With these three marks in mind, the question is:

  1. How can we practice real numbers in practical calculations?
  2. What rules employ?
  3. How should numbers be added, multiplied, subtracted,  and divided?
  4. What scope do we have?
  5. Does it matter what we do first? second? third? fourth?
  6. Can we add a streak of numbers together in any order?
  7. Will the outcome be the same nevertheless of the order we choose?
  8. Can we multiply a streak of numbers together in any order?
  9. Will the concluding answer be the same nevertheless of the order we choose?

The following properties of real numbers answer these types of questions. The property characteristics which follow show how much latitude you have to change the mechanics of calculations which use real numbers without changing the results.

  • Associative Property
  • Commutative Property
  • Distributive Property
  • Identity Property
  • Inverse Property

Inverse Property

Definition

Inverse properties undo each other. The goal of the inverse property of addition is to get an outcome of zero. The goal of the inverse property of multiplication is to get an outcome of 1. We use inverse properties to solve equations.

In other words, we can say that

An inverse property is two properties that undo each other e.g. addition and subtraction or multiplication and division. You can perform the same inverse property on each side of an equivalent equation without changing the equality.

For every non-zero real number, the Inverse Property is another number (called an Inverse Number). An inverse property is not a procedure. (The Inverse Operation [explained in the below section] is a procedure.)

The entire set of non-zero real numbers has the inverse property under addition and multiplication because every element in the set has an inverse.

Inverse Property for Addition

The additive inverse of any number is the same number with the opposite sign. When a number and its additive inverse are added to one another, the result is always 0 (zero) – the identity element for addition.

The additive inverse of +16 is -16.

When these two additive inverses are added together:

16 + (-16) = 0

The additive inverse of -56 is +56.

When these two additive inverses are added together:

-56 + 56 = 0

Note: The additive inverse of +A is -A.

When these two additive inverses are added together:

+A + (-A) = 0

Inverse Property for Multiplication

The multiplicative inverse of any number is the reciprocal of that number. When a number and its multiplicative inverse are multiplied by one another, the result is always 1 (one) – the identity element for multiplication.

The multiplicative inverse of 16 is (1/16).

When these two multiplicative inverses are multiplied with each other:

16 • (1/16) = (16/1)/(1/16) = 1

The multiplicative inverse of -56 is 1/(-56).

When these two multiplicative inverses are multiplied with each other:

-56 • (1/-56) = (-56/1)/(1/-56) = 1

Note: The multiplicative inverse of A is (1/A)

When these two multiplicative inverses are multiplied with each other:

A • (1/A) = (A/1)/(1/A) = 1

Inverse Properties are important keys that can be used to simplify equations.

Examples

Inverse Property Examples

Example 1: Using the Additive Inverse Property

Solve for X

X + 3 = 5

To remove the +3, the additive inverse property of -3 can be applied because +3 – 3 = 0 (the additive identity number).

Add the additive inverse of -3 to each side of the equation.

Additive Inverse Property

Additive Inverse Property

Example 2: Using the Multiplicative Inverse Property

Solve for X

2X = 6

To eliminate the +2 (of the 2X), the multiplicative inverse property (of the number 2) of ½ can be used because (2) * ( ½ ) = 1 (the multiplicative identity element).

Multiply each side of the equation by the multiplicative inverse of ½.

Multiplicative Inverse Property

Multiplicative Inverse Property