Imaginary Numbers Essential Question: How do we take the square root of negative numbers?
M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once...
-
Upload
camryn-dolbeare -
Category
Documents
-
view
219 -
download
0
Transcript of M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once...
![Page 1: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/1.jpg)
M.1 U.1Complex Numbers
![Page 2: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/2.jpg)
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)
![Page 3: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/3.jpg)
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
![Page 4: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/4.jpg)
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
![Page 5: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/5.jpg)
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
![Page 6: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/6.jpg)
Imaginary Numbers Simplify these radicals:Simplify these radicals:
36x2
20y3
5a2b4c7
![Page 7: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/7.jpg)
Multiples of i Consider multiplying two imaginary numbers:Consider multiplying two imaginary numbers:
i2 1 So...So...
ii 53
![Page 8: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/8.jpg)
Multiples of i Powers of Powers of ii::
i
i2 1
i3 i2 i 1i i
i4 i2 i2 1 1 1
![Page 9: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/9.jpg)
Powers of i - Practice ii2828
ii7575
ii113113
ii8686
ii10891089
![Page 10: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/10.jpg)
Solutions Involving i Solve:Solve:
Solve:Solve:
0364 2 x
0364 2 x
![Page 11: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/11.jpg)
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
![Page 12: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/12.jpg)
Add & Subtract Like TermsLike Terms Example: Example:
((33 + + 44ii) + () + (-5-5 - 2- 2ii) =) = -2-2 + + 22ii
![Page 13: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/13.jpg)
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))
![Page 14: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/14.jpg)
Multiplying FOIL and replace FOIL and replace ii22 with -1: with -1:
ii 3124
![Page 15: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/15.jpg)
PracticeMultiply:Multiply:
55ii(3 - 4(3 - 4ii))
(7 - 4(7 - 4ii)(7 + 4)(7 + 4ii))
![Page 16: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/16.jpg)
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
![Page 17: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/17.jpg)
Rationalizing MonomialMonomial: multiply the top & : multiply the top &
bottom by bottom by ii. .
i
i
3
82
![Page 18: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/18.jpg)
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
![Page 19: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/19.jpg)
![Page 20: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/20.jpg)
Practice Simplify:Simplify:
5 i4i
2 3i
2 i5
3 4i
1 5i
4
1 8i
5
3 4i
5
![Page 21: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/21.jpg)
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
![Page 22: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/22.jpg)
Graphing Complex Numbers
![Page 23: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/23.jpg)
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
![Page 24: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/24.jpg)
Check Your Answers
![Page 25: M.1 U.1 Complex Numbers. What are imaginary numbers? n n Viewed the same way negative numbers once were – –How can you have less than zero? n n Numbers.](https://reader035.fdocuments.in/reader035/viewer/2022062417/551b0cab55034607418b50aa/html5/thumbnails/25.jpg)
Homework
Complex Numbers worksheetComplex Numbers worksheet
– For #7, remember the quadratic formula!For #7, remember the quadratic formula!