Complex Numbers and Their Operations

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

The imaginary unit is denoted as $$i$$ and is defined as $$i = \sqrt{-1}$$ .

To add two complex numbers, add 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). For example, if $$z_1 = a + bi$$ and $$z_2 = c + di$$ , then $$z_1 z_2 = (ac - bd) + (ad + bc)i$$ .

The conjugate of a complex number $$a + bi$$ is $$a - bi$$ .

To divide two complex numbers, multiply the numerator and denominator by the conjugate of the denominator. For example, to divide $$z_1 = a + bi$$ by $$z_2 = c + di$$ , compute $$\frac{z_1}{z_2} = \frac{(a + bi)(c - di)}{c^2 + d^2}$$ .

The modulus of a complex number $$a + bi$$ is given by $$|z| = \sqrt{a^2 + b^2}$$ .

$$x = \pm 2\sqrt{5}$$ .

$$i\sqrt{23}$$ .