Complex numbers

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 the square root of -1.

The imaginary unit 'i' is defined as i = √-1, and it is used to represent the square root of negative numbers.

i² = -1.

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

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

To multiply complex numbers, use the distributive property (FOIL). For example, (a + bi)(c + di) = ac + adi + bci + bdi², and then replace i² with -1.

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, (a + bi) / (c + di) = [(a + bi)(c - di)] / [(c + di)(c - di)].

The modulus of a complex number a + bi is given by |a + bi| = √(a² + b²). It represents the distance from the origin in the complex plane.

The argument of a complex number a + bi is the angle θ formed with the positive real axis, calculated as θ = arctan(b/a).