College Algebra 1.3 Complex Numbers; Quadratic Equations in the Complex Number System

Solve the following: x 2 = –1 (2x + 5) 2 + 7 = 0 In each problem, we encounter the problem of taking . But there is no real number equal to . Why?

x 2 = –1

(2x + 5) 2 + 7 = 0

In each problem, we encounter the problem of taking .

But there is no real number equal to . Why?

Imaginary Numbers Definition: The number i , called the imaginary unit , is the number such that i = ______ and i 2 = ________

Complex Numbers If a and b are real numbers and i is the imaginary unit, then a + bi is called a complex number . ▪ a is the real part ▪ bi is the imaginary part .

Adding or Subtracting Complex Numbers Let a + bi and c + di be complex numbers. 1. Add/Subtract the Real parts. 2. Add/Subtract the Imaginary parts. (3 – 2 i ) + (5 – 4 i ) (3 – 2 i ) - (5 – 4 i )

Multiplying Complex Numbers Let a + bi and c + di be complex numbers. 1. Multiply the binomials (FOIL). 2. Convert i 2 to -1 and add the like terms. (3 – 2 i ) (5 – 4 i )

Division Of Complex Numbers Let a + bi and c + di be complex numbers. multiply the numerator and denominator of the fraction by the Complex Conjugate of the Denominator . Then to perform the operation

Definition: For any positive real number a ,

b 2 – 4ac is called the Discriminant .