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Magic of Additive Inverse and Multiplicative Inverse

What is an inverse in daily life? If you eat good food you get energy, or else you won’t. So the inverse is just the opposite for something. It might be amount, health, happiness, etc.


Suppose there are 5 toys, you have to give them to 5 kids equally. What is left with you? Nothing right. Similarly in mathematics, the additive inverse is just changing the sign of a number; so that when you add it to the original number, it gets nullified. For example, additive inverse of 1 is -1 because 1+(-1) = 0.

Multiplicative Inverse

The multiplicative inverse is the same as the additive inverse. Here, the operation opposite to multiplication is division. So the multiplicative inverse is a number which when multiplied with the given number gives the value 1.

Ex. The multiplicative inverse of 100 is 1/100, because 100 * 1/100 = 1.

Additive Inverse of Different Types of Numbers

The numbers are of different types, The given number might be a real number, natural number, whole number, fraction, etc. The additive inverse of any form of number is the number with an opposite sign.

Ex: Additive inverse of fraction 1/6 is -1/6, Additive inverse of negative number -4 is +4, Additive inverse of complex number 1+i3 is -(1+i3) because 1+i3 -(1+i3) = 0.

The Additive Inverse in Modular Arithmetic

The modular additive inverse of A is defined as, a number y such that y + A ⩭ 0(mod n).

Ex: The additive inverse of 2 modulo 7 is 5 since it satisfy the equation 2 + A ⩭ 0(mod 7)

The Multiplicative Inverse of Various Kinds of Numbers

    • The multiplicative inverse of the real number 5 is 1/5.
    • The multiplicative inverse of the faction number -2/5 is -5/2.
    • The multiplicative inverse of the unit fraction number 1/3 is 3/1.
    • Finding the multiplicative inverse of a complex number is a bit tricky. If you have a square root at the denominator, it makes it still more complicated.
    • In a complex number, you have two parts. Real part and the imaginary part which is represented as x + iy. Where x represents the real part and iy represents the imaginary part.
    • Ex: Find the multiplicative inverse of 1 + i
    • To find this we have to multiply and divide it with 1 -i.

1+i= =

So the additive inverse of 1 + i is .

Multiplicative inverse of mixed fraction 3⅔: 3⅔ = 3/11 so the inverse is 11/3.

Multiplicative Inverse in Modular Arithmetic

In modular arithmetic, the modular multiplicative inverse of an integer is an integer such that the product of a given integer and its inverse is congruent to 1 with respect to the modulus m.

Let p be the given integer and x be its inverse such that p x ⩭ 1(Mod m). Modular inverse exists if and only if p and m are relatively prime.

Ex: The multiplicative inverse for 2 if m = 5 is 3

Because 2x ⩭ 1(Mod 5) = [(2 3) – 1]/5 = 5/5 = 1

Can You Differentiate Additive Inverse and Multiplicative Inverse Numbers?

In additive inverse, the sum of the number and its inverse is equal to zero. Additive inverse of zero is zero only. Sign of the number is changed. Whereas in the multiplicative inverse, the given number is multiplied with its inverse to get 1. Reciprocal of the original number is used here. Still if you want to learn more about these topics log on to cuemath website for more curious information.