Complex Numbers Introduction
Complex Numbers
Introduction
What is a complex number?
A complex number is made up of a real number and an imaginary number.
The standard form is a+bi where a is the real term and bi is the imaginary term.
Powers of i
i =
i2 =
i3 =
i4 =
√-1
i i = √-1 √-1 = -1
i i2 = i -1 = -i
i2 i2 = -1 -1 = 1
Practice
Simplify
i5 =
i6 =
i7 =
i8 =
Any power of i that is a multiple of 4 is also one. We can find any power of i by writing it as the product of the highest power that is a multiple of 4 and i, i2, and i3.
More Practice
Simplify
i25 =
i36 =
i11 =
i19 =
Solve quadratic equations
with complex roots Ex.1
Solve quadratic equations
with complex roots Ex.2
Solve quadratic equations
with complex roots Ex.3
Practice
Page 369
# 17-24, 25-35 o, 37-43 o, 55-61 o, 63-75 o, 81-92
Operations with complex numbers
Ex. 4 Multiply or divide. Simplify.
a.) b.)
c.) d.)
Operations with complex numbers
Ex. 5 Write in standard form a+bi.
Operations with complex numbers
Addition & Subtraction
(a+bi) + (c+di)
= (a+c) + (b+d)i
(a+bi) – (c+di)
= (a+c) – (b+d)i
Operations with complex numbers
Ex. 6 Find each sum or difference.
a.) (4 – 5i) + (-5 + 8i) =
b.) (-6 + 3i) + (12 – 9i) =
c.) (-10 + 7i) – (5 – 3i) =
d.) (15 – 8i) – (-10 + 4i) + (-25 + 12i) =
Operations with complex numbers
Multiplication
(a+bi)(c+di)
= (ac – bd) + (ad + bc)i
FOI
L
Operations with complex numbers
Ex. 7 Find each product.
a.) (5 + 3i)(2 – 7i) b.) (4 – 5i)2
c.) (3 – i)(-3 + i) d.) (9 – 8i)(9 + 8i)
Operations with complex numbers
Ex. 8
Operations with complex numbers
Ex. 9
Practice
Page 369
# 95-106, 109, 110