Sec. 1.5 Complex Numbers

Imaginary Unit i = √-1

Complex numbers are made by adding real numbers to real multiples of the imaginary unit Standard form a + bi Every real number can be written as a complex number by using b = 0

Complex numbers are equal a + bi = c + di If and only if a = c and b = d

Addition and Subtraction of complex numbers (a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) – (c + di) = (a – c) + (b – d)i Combine like terms

Additive Identity Is 0 (same as real numbers)

Additive Inverse of a + bi So (a + bi) + (-a – bi) 0 + 0i Or 0

Ex. 1 (3 – i) + (2 + 3i) 2i + (-4 – 2i) 3 – (-2 + 3i) + (-5 + i)

P. 125 properties that still apply To multiply use FOIL Ex. 2 i(-3i) (2 – i)(4 + 3i) (3 + 2i)(3 – 2i) (3 + 2i)2

Complex Conjugates and Division a + bi Conjugate is a – bi Notice what happens when you multiply a complex by its conjugate

Dividing You don’t want to leave a complex number in the denominator To get rid of it in the denominator multiply both numerator and denominator by the conjugate

Ex. 3

Ex. 4

Complex solution of Quadratic Eq. When you have a negative under the radical in the quadratic formula, factor out i Ex. √-3

Principal square root √-a = i√a Ex. 5 √-3 √-12 √-48 - √-27 (-1 + √-3)2

Ex. 6 Solve 3x2 – 2x + 5 = 0 p. 128 5-47odd, 51, 53, 63, 65, 91-100