Properties of Complex Numbers – Part 2

In this section, we explore the identity and inverse properties of complex numbers. These are core ideas that reinforce how operations behave — ensuring that there's always a way to "do nothing" (identity) or "undo" an operation (inverse).

Additive Identity

The additive identity is the number that, when added to any other number, leaves it unchanged. For complex numbers, this is $0 + 0i$. A complex number with both its real and imaginary part equal to 0.

Multiplicative Identity

The multiplicative identity is the number that, when multiplied with any other number, leaves it unchanged. For complex numbers, this is simply $1 + 0i$. A complex number with 1 as its real part and 0 as its imaginary part. Be vigilant that the multiplicative identity for complex numbers is not $1 + 1i$.

Additive Inverse

The additive inverse of a number is what you add to it to get zero. For a complex number $z = a + bi$, the inverse is $-z = -(a - bi) = -a - bi$.

Multiplicative Inverse

The multiplicative inverse of a number what you multiply to it get the multiplictive identity, which is 1. For real numbers, the multiplicative inverse is simply the reciprocal of the number. The multiplicative inverse of $2$ is $\dfrac{1}{2}$. In general, the multiplicative inverse of $a$ is $\dfrac{1}{a}$ with the exception that zero does not have a multiplicative inverse.

In the same vein, the multiplicative inverse of a complex number $z$ is $\dfrac{1}{z}$.

If $z = a + bi$, then its multiplicative inverse is given by $\dfrac{1}{a+bi}$. However this fraction is not a good representation of a complex number. Square roots and iota in the denominators are ugly and have to be resolved through division which will be covered in the future lessons.