M.1 U.1Complex Numbers

What are imaginary numbers?

Viewed the same way negative numbers once were– How can you have less than zero? 

Numbers which square to give negative real numbers.– “I dislike the term “imaginary number” — it was considered an insult, a slur,

designed to hurt i‘s feelings. The number i is just as normal as other numbers, but the name “imaginary” stuck so we’ll use it.”

Imaginary numbers deal with rotations, complex numbers deal with scaling and rotations simultaneously

(we’ll discuss this further later in the week)

Imaginary Numbers

What is the square root of 9?What is the square root of 9?

9 ?

What is the square root of -9?What is the square root of -9?

9 3 because 33 9

Imaginary Numbers The constant, The constant, ii, is defined as the , is defined as the

square root of negative 1: square root of negative 1:

i 1

Imaginary Numbers The square root of -9 is an imaginary The square root of -9 is an imaginary

number...number...

9 9 1 3 i 3i

Imaginary Numbers Simplify these radicals:Simplify these radicals:

36x2

20y3

5a2b4c7

Multiples of i Consider multiplying two imaginary numbers:Consider multiplying two imaginary numbers:

i2 1 So...So...

ii 53

Multiples of i Powers of Powers of ii::

i

i2 1

i3 i2 i 1i i

i4 i2 i2 1 1 1

Powers of i - Practice ii2828

ii7575

ii113113

ii8686

ii10891089

Solutions Involving i Solve:Solve:

Solve:Solve:

0364 2 x

0364 2 x

Complex Numbers Have a real and imaginary part .Have a real and imaginary part . Write complex numbers as Write complex numbers as a + bia + bi Examples: 3 - 7Examples: 3 - 7ii, -2 + 8, -2 + 8ii, -4, -4ii, 5 + 2, 5 + 2ii

Real = aImaginary = bi

Add & Subtract Like TermsLike Terms Example: Example:

((33 + + 44ii) + () + (-5-5 - 2- 2ii) =) = -2-2 + + 22ii

PracticeAdd these Complex Numbers:Add these Complex Numbers:

(4 + 7(4 + 7ii) - (2 - 3) - (2 - 3ii)) (3 - (3 - ii) + (7) + (7ii)) (-3 + 2(-3 + 2ii) - (-3 + ) - (-3 + ii))

Multiplying FOIL and replace FOIL and replace ii22 with -1: with -1:

ii 3124

PracticeMultiply:Multiply:

55ii(3 - 4(3 - 4ii))

(7 - 4(7 - 4ii)(7 + 4)(7 + 4ii))

A complex number is in A complex number is in standard form standard form when when there is no there is no i i in the denominator.in the denominator.

RationalizeRationalize any fraction with any fraction with ii in the in the

denominator.denominator.

8i

1 3i

Binomial Binomial Denominator:Denominator:

Monomial Monomial Denominator:Denominator:

2 8i

3i

Division/Standard Form

Rationalizing MonomialMonomial: multiply the top & : multiply the top &

bottom by bottom by ii. .

i

i

3

82

Complex #: Rationalize BinomialBinomial: multiply the numerator and : multiply the numerator and

denominator by the denominator by the conjugateconjugate of the of the denominator ...denominator ...

8i

1 3i1 3i

1 3i

conjugate is formed by negating the imaginary term of a binomial

Practice Simplify:Simplify:

5 i4i

2 3i

2 i5

3 4i

1 5i

4

1 8i

5

3 4i

5

Absolute Value of Complex Numbers

Absolute Value is a numbers distance Absolute Value is a numbers distance from zero on the coordinate plane. from zero on the coordinate plane.

– a = x-axisa = x-axis– b = y axisb = y axis

– Distance from the origin (0,0) = Distance from the origin (0,0) = » |z| = √x|z| = √x22+y+y22

Graphing Complex Numbers

Exit Ticket SimplifySimplify

(-2+4(-2+4ii) –(3+9) –(3+9i)i)

Write the following in standard formWrite the following in standard form8+78+7ii

3+43+4ii

Find the absolute valueFind the absolute value4-54-5ii

Check Your Answers

Homework

Complex Numbers worksheetComplex Numbers worksheet

– For #7, remember the quadratic formula!For #7, remember the quadratic formula!