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!
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