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Aim: How do we multiply complex numbers?

Aim: How do we multiply complex numbers?. Write an equivalent expression for. Do Now:. 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. Find the product:. 3(-2 + 3 i ).

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Aim: How do we multiply complex numbers?

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  1. Aim: How do we multiply complex numbers? Write an equivalent expression for Do Now:

  2. i0 = 1 i1 = i i2 = –1 i3 = –i i4 = 1 i5 = i i6 = –1 i7 = –i i8 = 1 i9 = i i10 = –1 i11 = –i i12 = 1 Find the product: 3(-2 + 3i) Find the product: i4(-2 + 3i) distributive property The Powers of i i2 = i2 = –1 –1 (3)(-2) + (3)(3i) distributive property -6 + 9i (i4)(-2) + (i4)(3i) -2i4 + 3i5 -2 + 3i simplify

  3. (3 + 2i)(2 + i) 6 F - (3 + 2i)(2 + i) +3i O - (3 + 2i)(2 + i) +4i + 4i I - (3 + 2i)(2 + i) 2i2 L - FOILing Complex Numbers Multiply the binomials (3 + 2i)(2 + i) = -2 6 + 7i – 2 4 + 7i

  4. distribute: Distributive Property Multiply the binomials (3 + 2i)(2 + i) 3(2 + i) + 2i(2 + i) 6 + 3i + 4i + 2i2 i2 = -1 6 + 7i + 2i2 6 + 7i + 2(-1) 4 + 7i

  5. The conjugate of a complex number a + bi is Conjugates 2x2 - 50 = 2(x – 5)(x + 5) conjugates of each other a2 – b2 = (a – b)(a + b) General Terms When conjugates are multiplied, the result is the difference between perfect squares. i2 = -1 a – bi (a + bi)(a – bi) = = a2 + b2 a2 – (bi)2 = a2 – b2i2 (5 + 2i)(5 – 2i) = = 25 + 4 52 – (2i)2 = 25 – b2i2 = 29 The product of two complex numbers that are conjugates is a real number.

  6. Express the product of and its conjugate in simplest form a = 2 b = (a + bi)(a – bi) = a2 + b2 Model Problems Express the number (4 – i)2 – 8i3 in simplest form. (4 – i)2 – 8i3 = (4 – i)(4 – i) – 8i3 = 16 – 8i + i2 – 8i3 i3 = -i = 16 – 8i – 1– 8(-i) = 15

  7. Model Problems

  8. Model Problems

  9. yi i(2 + i) = 2i + i2 5i 4i 3i 2i (-1 + 2i) i x (2 + i) 1 -5 -4 -3 -2 -1 0 2 3 4 5 6 -i -2i -3i -4i -5i -6i Graph Representation Multiply i(2 + i) = -1 + 2i Draw & compare vectors 2 + i & -1 + 2i Multiplication by i is equivalent to a counterclockwise rotation of 900 about the origin. rotational transformation Rotation of 900 about the origin R90º(x,y) = (y,-x) i(2 + i) = -1 + 2i

  10. yi 5i 4i 3i (6 + 3i) 2i i x (2 + i) 1 -5 -4 -3 -2 -1 0 2 3 4 5 6 -i -2i -3i -4i -5i -6i Graph Representation Multiply by distributing (3 + 2i)(2 + i) 3(2 + i) + 2i(2 + i) = 4 + 7i distributed: Multiplication by 3 is equivalent to a dilation of 3.

  11. yi 5i recall: i(2 + i) = -1 + 2i 4i (-2 + 4i) 3i 2i (-1 + 2i) i x (2 + i) 1 -5 -4 -3 -2 -1 0 2 3 4 5 6 -i -2i -3i -4i -5i -6i Graph Representation (con’t) Multiply by distributing (3 + 2i)(2 + i) 3(2 + i) + 2i(2 + i) = 4 + 7i distributed: (6 + 3i) Rotation of 900 about the origin R90º(x,y) = (y,-x) Multiplication by i is equivalent to a counterclockwise rotation of 900 about the origin. 2•i(2 + i) = 2(-1 + 2i) Multiplication by 2 is equivalent to a dilation of 2.

  12. 7i (-2 + 4i) 6i (4 + 7i) (6 + 3i) 5i 4i 3i 2i i 0 -i -2i -3i -4i Graph Representation (con’t) Multiply the binomials (3 + 2i)(2 + i) 3(2 + i) + 2i(2 + i) = 4 + 7i + (6 + 3i) (-2 + 4i) yi x -1 -5 -4 -3 -2 2 3 4 5 6 1

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