Complex Numbers

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 $$i = \sqrt{-1}$$ .

The imaginary unit $$i$$ is defined as $$i = \sqrt{-1}$$ . It is used to represent the square root of negative numbers.

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

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

The product of two imaginary numbers is given by $$(ai)(bi) = ab(i^2) = -ab$$ , since $$i^2 = -1$$ .

The square of a complex number is given by $$(a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 - b^2 + 2abi$$ .

To multiply complex numbers, use the distributive property (FOIL). For example, $$(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$$ .

The conjugate of a complex number $$a + bi$$ is $$a - bi$$ . It is used to simplify division of complex numbers.

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

The modulus of a complex number is given by $$|a + bi| = \sqrt{a^2 + b^2}$$ . It represents the distance from the origin in the complex plane.