Inverses: Additive & Multiplicative Inverse

Every number has an opposite number. Similarly, there are various types of properties that apply to numbers. Such properties include associative property, distributive property, and much more. One such property of numbers is the inverse property. Generally, every number has two inverses: an additive inverse and a multiplicative inverse. But don’t be scared with these two terms. Finding the opposites of a number is actually quite simple and easy to understand.

This article will teach us about how to find the multiplicative and additive inverse of numbers.

The Additive Inverse

The very basic type of signs in the number system you might know is positive and negative. For example, the opposite of a positive sign in numbers +2 is a negative sign in numbers i.e. -2. This type of property of numbers is also called the additive inverse.

One thing to keep in mind is that when you add the additive inverse of the number with the number itself, it should always be equal to 0. Likewise, -2 + 2 = 0. 

Similarly, the additive inverse of 8 is a negative 8 (-8). This can be justified because 8 + (-8) = 0. The same can be applied to variables as well. For example, x + (-x) = 0. 

Surprisingly, every number has an additive inverse. No matter what the number or the variable is, adding the same to its additive inverse will always yield the answer as 0.

To find the Additive Inverse:

1. For negative numbers and negative variables ( -6 or -x): Simply remove the negative sign: – 6 → 6; -x → x.
2. For positive numbers and positive variables ( 6 or x): Simply add a negative sign (-) before the number: 6 → -6, x → -x.

The Multiplicative Inverse

This type of inverse is related to basic calculations such as division and multiplication. Such a property is called the reciprocal, which is commonly known as the multiplicative inverse. You can learn more about this topic in an interesting way from Cuemath.

To understand the concept of this type of inverse, you should know the very basic rule: every whole number can be expressed as an improper fraction with the help of the number 1 i.e. the numerator is greater than the denominator. For example, the whole number 4 can be written as 4/1. Similarly, variables like x can be written as x/1.

Considering the example of the number 4 (fraction 4/1). Here, 4 is termed as the numerator, and 1 is termed as the denominator. When we interchange the positions of the numerator and denominator, this property is called the reciprocal. 

In this example of the number 4 (fraction 4/1), the denominator 1 will be placed on the top and the numerator 4 will be placed on the bottom, the new fraction will be 1/4. So, the reciprocal of 4 is 1/4 because 4 = 4/1, and 1/4 is the reciprocal or multiplicative inverse. 

Below is a list of properties of the multiplicative inverse:

• There are multiplicative inverses for any non-zero real or complex integer.
  
• You get a product of 1 when you multiply a number by its multiplicative inverse.
  
• If you find the multiplicative inverse of a multiplicative inverse, you’ll get the number itself. For example, the multiplicative inverse of 6 is 1/6. Furthermore, the multiplicative inverse of 1/6 is 6/1 i.e. the number itself.
  
• Positive numbers have a positive multiplicative inverse, whereas negative numbers have a negative multiplicative inverse.
  
• As numbers decrease in size, their multiplicative inverses become larger.
  
• The whole number 0 does not have any multiplicative inverse.
  
• Dividing two numbers is equivalent to multiplying the first by the multiplicative inverse of the second. For example, let’s consider two numbers 2 & 4. The multiplicative inverse of 4 is 1/4. Further ahead, when we multiply 2 X 1/4, we get 1/2. On the other hand, when we divide 2 by 4 (2/4), we get 1/2.
  
• In the number system, only two numbers 1 and -1 have a multiplicative inverse that equals the number itself.

To learn about how to find the multiplicative and additive inverses of numbers in a fun way, visit Cuemath.