Warm up Simplify the following without a calculator

Warm up Simplify the following without a calculator: 5. Define real numbers ( in your own words). Give 2 examples.

Vocabulary • Square root: A number r is a square root of a number s if r² = s. For instance 4 is a square root of 16 because 4²=16 • Radical: The expression is called a radical. The symbol is a radical sign. • Radicand: The number s beneath the radical sign. • Imaginary unit: The imaginary unit i is defined as where i²=-1

• Complex number: A complex number written in standard form is a number a + bi , where a and b are real numbers; a is the real part and bi is the imaginary part • Imaginary number : The number a + bi if b≠ 0 Pure imaginary number : If a = 0 in the number a + bi, then bi is a pure imaginary number.

Equality of complex numbers • Two complex numbers are equal if their real parts are equal and their imaginary parts are equal. For instance, if , then x=8 and y=-1 Example: Find real numbers x and y to make the equation true a) 4 x-4 yi=8 -12 i b)5 x + 3 yi = 10 + 18 i

Add and Subtract complex numbers • Sum of Complex numbers: To add two complex numbers, add their real parts and their imaginary parts separately. • (2 + 3 i) + (5 + 4 i) = (2 + 5) + (3 i + 4 i) =7+7 i Difference of complex numbers: To subtract two complex numbers, subtract their real parts and their imaginary parts separately. (3 + 4 i) – (5 + 6 i) = (3 – 5) + (4 i – 6 i) = -2 -2 i

Examples 1 a. (6 -2 i) + (4 + i) 1 b. (5 + 5 i) +( -2 - 3 i) 2 a. (8 -3 i) – (2 + 4 i) 2 b. (9 + 5 i) – (7 – 6 i)

Additional practice • Write the following as complex numbers in standard form 1. (-1 + 19 i) – (4 – i) 2. 6 – (-11 +9 i) 3. (8 + 2 i) + ( 2 – 3 i) 4. 18 + (-2 + 18 i) 5. -8 i + (9 + 16 i) – 16 Homework: page 9 #s 1 -5, 9 -17 (odd only).

Multiply and Divide Complex numbers Vocabulary Complex conjugates: two complex numbers of the form. The product of complex conjugates is always a real number. To Multiply two complex numbers, use the distributive property or the FOIL method. To divide two complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator and simplify.

Examples 1. Multiply ( 4 + 3 i) (4 – 3 i) Do Checkpoint exercise. 2 a. 4 i(6 + 2 i) b. (3 – 2 i)(-1 + 4 i) c. ( 5+7 i)(2 +i) Do checkpoint exercise.

Examples contd. • Divide a. b. c. • Practice: even numbers worksheet. • Homework : odd numbers on workseet. • Additional problems on 14 of your textbook.