Additive Inverse

The additive inverse of a number is a number that when added to the original number, gives the value zero. This means the sum of a number and its additive inverse gives a value zero. So, to get the additive inverse of a number, we should think of a number that needs to be added to the given number to make the sum zero. To get the additive inverse, we take the given number and just change its sign. The number thus obtained is the additive inverse of the given number.

For example, the additive inverse of 5 is -5 because when 5 and -5 are added the sum becomes zero. Again the additive inverse of -24 is 24 following the same rule.

The given number can be a whole number, a natural number, an integer, a decimal, a fraction, or any real number for which the additive inverse can be determined. The additive inverse of a real number is the same number with the opposite sign. If the given number is positive, its additive inverse is the same number with a negative sign and if the number is negative, its additive inverse is the same number without the negative sign. Any number when added to its negative gives a result equal to zero. So we need to multiply the given number with -1 to get its additive inverse. This means, for a number M, the additive inverse will be (-1) x M.

When the sum of two numbers is zero, then each number is the additive inverse of the other. For example, 2.75 + (-2.75) = 0. Here 2.75 is the additive inverse of -2.75 and vice versa.

The concept of the additive inverse also applies to algebraic expressions. Any number can be expressed by an algebraic expression. As per the definition mentioned above, the additive inverse of an algebraic expression is another algebraic expression such that the addition of both the expressions gives the result as zero. Learn this concept in an easy way from Cuemath.

Multiplicative Inverse

The multiplicative inverse of a number is another number such that the product of these two numbers is 1. In other words, when the product of two numbers is 1, the numbers are the multiplicative inverse of each other. The multiplicative inverse of any natural number is obtained by the division of 1 by that number. It is also called the reciprocal of the other number.

For example, for any given number p, its multiplicative inverse will be 1/p because p x 1/p = 1.

The multiplicative inverse of a negative number is it’s reciprocal with the negative sign. It is to be noted that for the multiplicative inverse of a negative number, the negative sign is attached with the numerator, not with the denominator.

The multiplicative inverse of a fraction is it is reciprocally provided the numerator and denominator both are not equal to zero.

For example, for any fraction m/n, the multiplicative inverse is n/m. When we multiply m/n by n/m the result will be 1.

A unit fraction has a numerator 1 so the multiplicative inverse of a unit fraction is equal to the denominator of the given unit fraction with the same sign.

Properties of Multiplicative Inverse

For simplification of mathematical expressions, it is necessary to consider certain properties or characteristics related to multiplicative inverse which are mentioned as follows:

  • The multiplicative inverse of a fraction is a fraction with the numerator and denominator interchanged.
  • The multiplicative inverse of 1 is 1 because the reciprocal of 1 is 1.
  • The multiplicative inverse of 0 is not possible as the inverse of 0 is undefined.
  • The multiplicative inverse of a mixed fraction is obtained by converting the mixed fraction into an improper fraction and finding its reciprocal.
  • The multiplicative inverse of a negative number is always negative.

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