aim: multiply complex numbers course: adv. alg. & trig. aim: how do we multiply complex numbers?...
TRANSCRIPT
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
Aim: How do we multiply complex numbers?
Do Now:
Write an equivalent expression for
7 4
6 2
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
The Powers of i
1 2
–1i2 =
1 2
–1i2 =
Find the product: 3(-2 + 3i)
distributive property (3)(-2) + (3)(3i)
-6 + 9i
Find the product: i4(-2 + 3i)
distributive property (i4)(-2) + (i4)(3i)
-2i4 + 3i5
-2 + 3isimplify
i0 = 1i1 = ii2 = –1i3 = –ii4 = 1i5 = ii6 = –1i7 = –ii8 = 1i9 = ii10 = –1i11 = –ii12 = 1
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
(3 + 2i)(2 + i)
(3 + 2i)(2 + i)
(3 + 2i)(2 + i)
FOILing Complex Numbers
(3 + 2i)(2 + i)
2i2
F -
O -
I -
L -
Multiply the binomials(3 + 2i)(2 + i)
6 + 7i – 2
6
= -2
4 + 7i
+3i
+4i+ 4i
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
Distributive Property
Multiply the binomials(3 + 2i)(2 + i)
distribute: 3(2 + i)
6 + 3i + 4i + 2i2
6 + 7i + 2i2 i2 = -1
6 + 7i + 2(-1)
4 + 7i
+ 2i(2 + i)
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
Conjugates
General Terms
2x2 - 50 = 2(x – 5)(x + 5)
conjugates of each other
(a – b)(a + b)a2 – b2 =When conjugates are multiplied, the result is the difference
between perfect squares.
The conjugate of a complex number a + bi isa – bi
(a + bi)(a – bi) = a2 – (bi)2 = a2 – b2i2
i2 = -1
= a2 + b2
(5 + 2i)(5 – 2i) = 52 – (2i)2 = 25 – b2i2 = 25 + 4
= 29The product of two complex numbersthat are conjugates is a real number.
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
Model Problems
Express the number (4 – i)2 – 8i3 in simplestform.
(4 – i)2 – 8i3 = (4 – i)(4 – i) – 8i3
= 16 – 8i + i2 – 8i3
= 16 – 8i – 1 – 8(-i)= 15
i3 = -i
Express the product of and its conjugate in simplest form
2 i 5
2 i 5 2 i 5
4 5 9
a = 2b =
5(a + bi)(a – bi) = a2 + b2
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
Model Problems
23 4i
6 2 8 3 5 7i i
6 4 6 4i i
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
Model Problems
8 11 8 11
2
5 2i
4 3 2 5 4 3i i i
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
x1 2 3 4 5 6-5 -4 -3 -2 -1 0
i
2i
3i
4i
5i
-4i
-3i
-2i
-i
-5i
-6i
yi
Graph Representation
Multiply i(2 + i)
(2 + i)
Multiplication by iis equivalent to a counterclockwiserotation of 900 aboutthe origin.
i(2 + i) = 2i + i2 = -1 + 2i
(-1 + 2i)
rotational transformation
Draw & compare vectors
2 + i & -1 + 2i
i(2 + i) = -1 + 2i
Rotation of 900 about the origin R90º(x,y) = (y,-x)
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
x1 2 3 4 5 6-5 -4 -3 -2 -1 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 3is equivalent to a dilation of 3.
= 4 + 7i
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
x1 2 3 4 5 6-5 -4 -3 -2 -1 0
i
2i
3i
4i
5i
-4i
-3i
-2i
-i
-5i
-6i
yi
Graph Representation (con’t)
distributed:
2•i(2 + i) = 2(-1 + 2i)
(-1 + 2i)
(-2 + 4i)
Multiplication by 2is 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 900 about the origin
R90º(x,y) = (y,-x) Multiplication by iis equivalent to a counterclockwise
rotation of 900 aboutthe origin.
3(2 + i) + 2i(2 + i) = 4 + 7i
Aim: Multiply Complex Numbers Course: Adv. Alg. & Trig.
x1 2 3 4 5 6-5 -4 -3 -2 -1 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)+