2. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. Aim: How do we multiply complex numbers? Do Now: Write an equivalent expression for

3. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. The Powers of i –1 i 2 = –1 i 2 = Find the product: 3(-2 + 3i) distributive property (3)(-2) + (3)(3i) -6 + 9i Find the product: i 4 (-2 + 3i) distributive property (i 4 )(-2) + (i 4 )(3i) -2i 4 + 3i 5 -2 + 3i simplify i 0 = 1 i 1 = i i 2 = –1 i 3 = –i i 4 = 1 i 5 = i i 6 = –1 i 7 = –i i 8 = 1 i 9 = i i 10 = –1 i 11 = –i i 12 = 1

4. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. (3 + 2i)(2 + i) FOILing Complex Numbers (3 + 2i)(2 + i) 2i22i2 F - O - I - L - Multiply the binomials (3 + 2i)(2 + i) 6 + 7i – 2 6 = -2 4 + 7i +3i+3i +4i+4i + 4i

5. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. Distributive Property Multiply the binomials (3 + 2i)(2 + i) distribute: 3(2 + i) 6 + 3i + 4i + 2i 2 6 + 7i + 2i 2 i 2 = -1 6 + 7i + 2(-1) 4 + 7i + 2i(2 + i)

6. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. Conjugates General Terms 2x 2 - 50 = 2(x – 5)(x + 5) conjugates of each other (a – b)(a + b) a 2 – b 2 = When conjugates are multiplied, the result is the difference between perfect squares. The conjugate of a complex number a + bi is a – bi (a + bi)(a – bi) = a 2 – (bi) 2 = a 2 – b 2 i 2 i 2 = -1 = a 2 + b 2 (5 + 2i)(5 – 2i) = 5 2 – (2i) 2 = 25 – b 2 i 2 = 25 + 4 = 29 The product of two complex numbers that are conjugates is a real number.

7. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. Model Problems Express the number (4 – i) 2 – 8i 3 in simplest form. (4 – i) 2 – 8i 3 = (4 – i)(4 – i) – 8i 3 = 16 – 8i + i 2 – 8i 3 = 16 – 8i – 1 – 8(-i) = 15 i 3 = -i Express the product of and its conjugate in simplest form a = 2 b = (a + bi)(a – bi) = a 2 + b 2

8. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. Model Problems

9. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. Model Problems

10. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. x 1 23456 -5-4-3-2 0 i 2i 3i 4i 5i -4i -3i -2i -i -5i -6i yi Graph Representation Multiply i(2 + i) (2 + i) Multiplication by i is equivalent to a counterclockwise rotation of 90 0 about the origin. i(2 + i) = 2i + i 2 = -1 + 2i (-1 + 2i) rotational transformation Draw & compare vectors 2 + i & -1 + 2i i(2 + i) = -1 + 2i Rotation of 90 0 about the origin R 90º (x,y) = (y,-x)

11. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. x 1 23456 -5-4-3-2 0 i 2i 3i 4i 5i -4i -3i -2i -i -5i -6i yi Graph Representation Multiply by distributing (3 + 2i)(2 + i) 3(2 + i) + 2i(2 + i) distributed: (2 + i) (6 + 3i) Multiplication by 3 is equivalent to a dilation of 3. = 4 + 7i

12. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. x 1 23456 -5-4-3-2 0 i 2i 3i 4i 5i -4i -3i -2i -i -5i -6i yi Graph Representation (con’t) distributed: 2i(2 + i) = 2(-1 + 2i) (-1 + 2i) (-2 + 4i) Multiplication by 2 is equivalent to a dilation of 2. Multiply by distributing (3 + 2i)(2 + i) (2 + i) i(2 + i) = -1 + 2i recall: (6 + 3i) Rotation of 90 0 about the origin R 90º (x,y) = (y,-x) Multiplication by i is equivalent to a counterclockwise rotation of 90 0 about the origin. 3(2 + i) + 2i(2 + i)= 4 + 7i

13. Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig. x 1 23456 -5-4-3-2 0 -4i -3i -2i -i i 2i 3i 4i 5i 7i 6i yi Graph Representation (con’t) Multiply the binomials (3 + 2i)(2 + i) 3(2 + i) (6 + 3i) (-2 + 4i) = 4 + 7i + 2i(2 + i) (4 + 7i) (6 + 3i)(-2 + 4i) +