Complex Numbers C.A-1.5

Imaginary numbers i represents the square root of – 1.

Complex Numbers A Complex Number is a number that can be written in the form: Where a and b are real numbers and

The a is the Real part, and b is called the imaginary part.

Equality of Complex Numbers Two complex numbers are equal if their real parts are equal and their imaginary parts are equal. If a + bi and c + di are two complex numbers, then a + bi = c + di if and only if a = c and b = d

Addition and Subtraction of Complex Numbers Two complex numbers such as a + bi and c + di are added and subtracted as if they were binomials: (a + bi) + (c + di) = (a + c) + (b + d )i (a + bi) - (c + di) = (a - c) + (b - d )i

Multiplication of Complex Numbers The numbers a + bi and c + di are multiplied as if they were binomials, with i 2 = -1: (a + bi)(c + di) = (ac – bd) + (ad + bc)i

Errors involving signs To avoid errors in determining the sign of the result, always express numbers in a + bi form before attempting any algebraic manipulations.

Complex Conjugates The complex numbers and Are complex conjugates of each other.

Powers of i

If n is a natural number that has a remainder of r when divided by 4, then When n is divisible by 4, the remainder r is 0 and i 0 = 1.

Absolute Value of a Complex Number If a + bi is a complex number, then