Objectives

1
Complex Numbers

• Objectives
• Students will learn

Basic Concepts of Complex Numbers Operations on
Complex Numbers
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Basic Concepts of Complex Numbers

• There are no real numbers for the solution of the
  equation

To extend the real number system to include such
numbers as,
the number i is defined to have the following
property
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Basic Concepts of Complex Numbers

• So

The number i is called the imaginary
unit. Numbers of the form a bi, where a and b
are real numbers are called complex numbers. In
this complex number, a is the real part and b is
the imaginary part.
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Nonreal complex numbers a bi, b ? 0
Complex numbers a bi, a and b real
Irrational numbers
Real numbers a bi, b 0
Integers
Rational numbers
Non-integers
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Basic Concepts of Complex Numbers

• Two complex numbers are equal provided that their
  real parts are equal and their imaginary parts
  are equal

if and only if
and
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Basic Concepts of Complex Numbers

• For complex number a bi, if b 0, then
• a bi a
• So, the set of real numbers is a subset of
  complex numbers.

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Basic Concepts of Complex Numbers

• If a 0 and b ? 0, the complex number is pure
  imaginary.
• A pure imaginary number or a number, like 7 2i
  with a ? 0 and b ? 0, is a nonreal complex
  number.
• The form a bi (or a ib) is called standard
  form.

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Example 1

• WRITING AS

Write as the product of a real number and i,
using the definition of
a.
Solution
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Example 1

• WRITING AS

Write as the product of a real number and i,
using the definition of
b.
Solution
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Example 1

• WRITING AS

Write as the product of a real number and i,
using the definition of
c.
Solution
Product rule for radicals
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Operations on Complex Numbers

• Products or quotients with negative radicands are
  simplified by first rewriting
• for a positive number.

Then the properties of real numbers are applied,
together with the fact that
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Operations on Complex Numbers
Caution When working with negative
radicands, use the definition before using
any of the other rules for radicands.
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Operations on Complex Numbers
Caution In particular, the rule is
valid only when c and d are not both negative.
while
so
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Example 2

• FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE
  RADICALS

Multiply or divide, as indicated. Simplify each
answer.
a.
Solution
First write all square roots in terms of i.
i 2 -1
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Example 2

• FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE
  RADICALS

Multiply or divide, as indicated. Simplify each
answer.
b.
Solution
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Example 2

• FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE
  RADICALS

Multiply or divide, as indicated. Simplify each
answer.
c.
Solution
Quotient rule for radicals
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Example 3

• SIMPLIFYING A QUOTIENT INVOLVING A NEGATIVE
  RADICAND

Write
in standard form a bi.
Solution
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Example 3

• SIMPLIFYING A QUOTIENT INVOLVING A NEGATIVE
  RADICAND

Write
in standard form a bi.
Solution
Be sure to factor before simplifying
Factor.
Lowest terms
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For complex numbers a bi and c di,
and
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Example 4

• ADDING AND SUBTRACTING COMPLEX NUMBERS

Find each sum or difference.
a.
Add imaginary parts.
Add real parts.
Solution
Commutative, associative, distributive properties
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Example 4

• ADDING AND SUBTRACTING COMPLEX NUMBERS

Find each sum or difference.
b.
Solution
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Example 4

• ADDING AND SUBTRACTING COMPLEX NUMBERS

Find each sum or difference.
c.
Solution
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Example 4

• ADDING AND SUBTRACTING COMPLEX NUMBERS

Find each sum or difference.
d.
Solution
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Multiplication of Complex Numbers

• The product of two complex numbers is found by
  multiplying as if the numbers were binomials and
  using the fact that i2 1, as follows.

FOIL
Distributive property i 2 1
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For complex numbers a bi and c di,
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Example 5

• MULTIPLYING COMPLEX NUMBERS

Find each product.
a.
Solution
FOIL
i2 -1
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Example 5

• MULTIPLYING COMPLEX NUMBERS

Find each product.
b.
Solution
Square of a binomial
Remember to add twice the product of the two
terms.
i 2 -1
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Example 5

• MULTIPLYING COMPLEX NUMBERS

Find each product.
c.
Solution
Product of the sum and difference of two terms
i 2 -1
Standard form
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Simplifying Powers of i

• Powers of i can be simplified using the facts

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Example 6

• SIMPLIFYING POWERS OF i

Simplify each power of i.
a.
Solution
Since i 2 1 and i 4 1, write the given
power as a product involving i 2 or i 4. For
example,
Alternatively, using i4 and i3 to rewrite i15
gives
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Example 6

• SIMPLIFYING POWERS OF i

Simplify each power of i.
b.
Solution
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and so on.
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• Ex 5c. showed that

The numbers differ only in the sign of their
imaginary parts and are called complex
conjugates. The product of a complex number and
its conjugate is always a real number. This
product is the sum of squares of real and
imaginary parts.
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For real numbers a and b,
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Example 7

• DIVIDING COMPLEX NUMBERS

Write each quotient in standard form a bi.
a.
Solution
Multiply by the complex conjugate of the
denominator in both the numerator and the
denominator.
Multiply.
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Example 7

• DIVIDING COMPLEX NUMBERS

Write each quotient in standard form a bi.
a.
Solution
Multiply.
i 2 -1
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Example 7

• DIVIDING COMPLEX NUMBERS

Write each quotient in standard form a bi.
a.
Solution
i 2 -1
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Example 7

• DIVIDING COMPLEX NUMBERS

Write each quotient in standard form a bi.
a.
Solution
Lowest terms standard form
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Example 7

• DIVIDING COMPLEX NUMBERS

Write each quotient in standard form a bi.
b.
Solution
i is the conjugate of i.
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Example 7

• DIVIDING COMPLEX NUMBERS

Write each quotient in standard form a bi.
b.
Solution
Standard form
i 2 -1(-1) 1