Complex numbers: Operations

A complex number is a number that can be expressed in the form $$a + bi$$ , where $$a$$ is the real part, $$b$$ is the imaginary part, and $$i$$ is the imaginary unit, defined as $$i = \frac{1}{\text{sqrt}(-1)}$$ .

The imaginary unit $$i$$ is defined as $$i = \text{sqrt}(-1)$$ . It is used to extend the real number system to include solutions to equations that do not have real solutions.

To add two complex numbers, combine their real parts and their imaginary parts separately. For example, if $$z_1 = a + bi$$ and $$z_2 = c + di$$ , then $$z_1 + z_2 = (a+c) + (b+d)i$$ .

To subtract two complex numbers, subtract their real parts and their imaginary parts separately. For example, if $$z_1 = a + bi$$ and $$z_2 = c + di$$ , then $$z_1 - z_2 = (a-c) + (b-d)i$$ .

To multiply two complex numbers, use the distributive property (FOIL method). For example, if $$z_1 = a + bi$$ and $$z_2 = c + di$$ , then $$z_1 z_2 = (ac - bd) + (ad + bc)i$$ .

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For example, to divide $$\frac{z_1}{z_2}$$ , where $$z_2 = c + di$$ , multiply by $$\frac{c - di}{c - di}$$ .

The conjugate of a complex number $$a + bi$$ is $$a - bi$$ . It is used in division to eliminate the imaginary part from the denominator.

$$\frac{11i}{9}$$ .

The square root of $$-36$$ is $$6i$$ .

$$12 + 3i$$ .