2.5 Complex Numbers and Roots

Objectives: N.CN.1: Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real. N.CN.2: Use the relation i2 = -1, and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. N.CN.7: Solve quadratic equations with real coefficients that have complex solutions.

For the Board: You will be able to define and use imaginary and complex numbers. You will be able to solve quadratic equations with complex roots.

Bell Work 2.5: Simplify each expression. 42 1. 108 2. 6  24 3.  3 Find the zeros of each function. 4. f(x) = x2 – 18x + 16 5. f(x) = x2 + 6x – 24

Anticipatory Set: f(x) = x2 + 1 has no real zeros. Solve for the zeros using square rooting. x2 + 1 = 0 x2 = -1 x =  1

We can find solutions if we define the square root of a negative number. This is why imaginary numbers were invented. The imaginary unit i is defined as 1 .

An imaginary number is the square root of a negative number. 1 = I  2 = 1 ∙ 2 = i 2  4 = 1 ∙ 4 = 2i

Imaginary numbers can be written in the form bi, where b is a real number and i is the imaginary unit. If b is a positive real number, then  b = i b .

The square of an imaginary number is the original negative number. ( )2 = i2 = -1  b 2 = 1  b2 = bi ( )2 = (i )2 = -b.

Open the book to page 95 and read example 1. Example: Express each number in terms of i. a. 5 121 b. -  96 5 1 ∙ 121 - 1 ∙ 96 5i ∙ 11 -i 16 ∙ 6 55i -4i 6 White Board Activity: Practice: Express each number in terms of i. 1 a. 12 b. 2  36 c.   63 3 1 1 ∙ 12 2 1 36  1 63 3 1 i 4 ∙ 3 2i ∙ 6  i  9  7 3 1 2i 12i  i 3 7 3  i 7

Open the book to page 95 and read example 2. Example: Solve each equation. Use square rooting. a. x2 = -144 b. 5x2 + 90 = 0 x =  144 5x2 = -90 x2 = -18 x =  1 144 x =  18 x =  1  18 x = ±12i x = ±i 9  2 x = ±3i 2

White Board Activity: Practice: Solve each equation. Use square rooting. a. x2 = -36 b. x2 + 48 = 0 c. 9x2 + 25 = 0 x = ±  36 x2 = -48 9x2 = -25 25 x = ± 1 36 x = ±  48 x2 =  9 25 x = ±6i x = ± 1 48 x = ±  9 25 x = ±i 16  3 x = ± 1  9 5 x = ±4i 3 x =  i 3

A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i = 1 . Examples: 3 - 5i -5 + i 2 ± 4i

The set of real numbers is a subset of the set of complex numbers C. Open the book to page 95 and study the Venn Diagram.

Every complex number has a real part “a” and an imaginary part “b”. Real numbers are complex numbers where b = 0.

Imaginary numbers are complex numbers where a = 0 and b ≠ 0. These are sometimes called pure imaginary numbers. Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal .

Open the book to page 95 and read example 3. Example: Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i true. 4x + 10i = 2 + (-4y)i 4x = 2 10 = -4y X = ½ y = 10/-4 y = -2.5

White Board Activity: Practice: Find the values of x and y that make each equation true. a. 2x – 6i = -8 + (20y)i b. -8 + (6y)i = 5x - i 6 2x + (-6)I = -8 + (20y)I -8 + (6y)I = 5x + (- )i 2x = -8 -6 = 20y -8 = 5x 6y = - 6 x = -4 y = -6/20 x = -8/5 y =  6 = -3/10

Open the book to page 96 and read example 4. Example: Find the zeros of each function using completing the square. a. f(x) = x2 + 10x + 26 b. g(x) = x2 + 4x + 12 x2 + 10x + 26 = 0 x2 + 4x + 12 = 0 (x2 + 10x + _____) = -26 + _____ (x2 + 4x + _____) = -12 + _____ ½ (10) = 5; 52 = 25 ½ (4) = 2; 22 = 4 (x2 + 10x + 25) = -26 + 25 = -1 (x2 + 4x + 4) = -12 + 4 = -8 (x + 5)2 = -1 (x + 2)2 = -8 x + 5 =  1 x + 2 =   8   1 8  i 4  2 x = -5 ± i x = -2 ± 2i 2

White Board Activity: Practice: Find the zeros of each function using completing the square. a. f(x) = x2 + 4x + 13 b. g(x) = x2 – 8x + 18 x2 + 4x + 13 = 0 x2 - 8x + 18 = 0 (x2 + 4x + _____) = -13 + _____ (x2 - 8x + _____) = -18 + _____ ½ (4) = 2; 22 = 4 ½ (8) = 4; 42 = 16 (x2 + 4x + 4) = -13 + 4 = -9 (x2 - 8x + 16) = -18 + 16 = -2 (x + 2)2 = -9 (x - 4)2 = -2 x + 2 =  1  9 x - 4 =   2   1 2 x = -2 ± 3i x = 4 ± i

Solutions like all of the above are called complex conjugate pairs. -5 + I and -5 – I -2 + 2i and -2 - 2i -2 + 3i and -2 - 3i 4 + i and 4 - i Their real parts are equal and their imaginary parts are opposites. The complex conjugate of a + bi is a – bi.

Note: If a quadratic equation has non-real roots, then these roots are complex conjugates.

Open the book to page 96 and read example 5. Example: Find each complex conjugate. a. 8 + 5i b. 6i 8 – 5i -6i

White Board Activity: Practice: Find each complex conjugate. a. 9 – i b. i + 2 c. -8i 9 + i -I + 8i

Assessment: Question student pairs.

Independent Practice: Text: pg. 97 – 98 prob. 2 – 35, 43 - 57.

For a Grade: Text: pg. 97 – 98 prob. 4, 8, 16, 24, 34.