Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real...
Transcript of Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real...
![Page 1: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/1.jpg)
Chapter 22-9 Operations with complex
numbers
![Page 2: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/2.jpg)
Complex PlaneO Just as you can represent real
numbers graphically as points on a number line, you can represent complex numbers in a special coordinate plane.
O The complex plane is a set of coordinate axes in which the horizontal axis represents real numbers and the vertical axis represents imaginary numbers.
![Page 3: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/3.jpg)
Graphing complex numbers
The real axis corresponds to the x-axis, and the imaginary axis corresponds to the y-axis. Think of a + bi as x + yi
![Page 4: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/4.jpg)
Example#1O Graph each complex number.O A. 2 – 3iO B. –1 + 4iO C. 4 + IO D. –i
![Page 5: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/5.jpg)
Solution to Example#1
![Page 6: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/6.jpg)
Student guided practice
O Go to page 130 and do problems 2-5
![Page 7: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/7.jpg)
Absolute valueO Recall that absolute value of a real
number is its distance from 0 on the real axis, which is also a number line. Similarly, the absolute value of an imaginary number is its distance from 0 along the imaginary axis.
![Page 8: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/8.jpg)
Absolute value
![Page 9: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/9.jpg)
Example#2O Find each absolute valueO A. |3 + 5i|O B. |–13|O C. |–7i|
![Page 10: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/10.jpg)
Student guided practice
O Do problems 6-10 on your book page 130
![Page 11: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/11.jpg)
Adding and Subtracting complex numbers
O Adding and subtracting complex numbers is similar to adding and subtracting variable expressions with like terms. Simply combine the real parts, and combine the imaginary parts.
O The set of complex numbers has all the properties of the set of real numbers. So you can use the Commutative, Associative, and Distributive Properties to simplify complex number expressions.
![Page 12: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/12.jpg)
Example#3O Add or subtract. Write the result
in the form a + bi.O (4 + 2i) + (–6 – 7i)
![Page 13: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/13.jpg)
Example#4O Add or subtract. Write the result
in the form a + bi.O (5 –2i) – (–2 –3i)
![Page 14: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/14.jpg)
Example#5O Add or subtract. Write the result
in the form a + bi.O (1 – 3i) + (–1 + 3i)
![Page 15: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/15.jpg)
Student guided practice
O Do problems 12-17 on your book page 130
![Page 16: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/16.jpg)
Graph complex numbers
O You can also add complex numbers by using coordinate geometry.
![Page 17: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/17.jpg)
Example#5O Find (3 – i) + (2 + 3i) by graphingO Step 1 Graph 3 – i and
2 + 3i on the complex plane. Connect each of these numbers to the origin with a line segment.
![Page 18: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/18.jpg)
Example#5
![Page 19: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/19.jpg)
Example#5 continueO Step 2 Draw a parallelogram that
has these two line segments as sides. The vertex that is opposite the origin represents the sum of the two complex numbers, 5 + 2i. Therefore, (3 – i) + (2 + 3i) = 5 + 2i.
![Page 20: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/20.jpg)
Example#5
![Page 21: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/21.jpg)
Example#6O Find (3 + 4i) + (1 – 3i) by
graphing.
![Page 22: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/22.jpg)
Multiplying complex numbers
O You can multiply complex numbers by using the Distributive Property and treating the imaginary parts as like terms. Simplify by using the fact i2 = –1.
![Page 23: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/23.jpg)
Example#7O Multiply. Write the result in the
form a + bi.O –2i(2 – 4i)
![Page 24: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/24.jpg)
Example#8O Multiply. Write the result in the
form a + bi.O (3 + 6i)(4 – i)
![Page 25: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/25.jpg)
Example#9O Multiply. Write the result in the
form a + bi.O (2 + 9i)(2 – 9i)
![Page 26: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/26.jpg)
Student guided practice
O Do problems 21-23 from book page 130
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Imaginary numbersO The imaginary unit i can be raised to
higher powers as shown below.
![Page 28: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/28.jpg)
Example#10O Simplify –6i14.
![Page 29: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/29.jpg)
Example#11O Simplify i63
![Page 30: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/30.jpg)
Example#12O Simplify i42
![Page 31: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/31.jpg)
Dividing complex numbers
O Recall that expressions in simplest form cannot have square roots in the denominator (Lesson 1-3). Because the imaginary unit represents a square root, you must rationalize any denominator that contains an imaginary unit. To do this, multiply the numerator and denominator by the complex conjugate of the denominator.
![Page 32: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/32.jpg)
Example#13O Simplify
![Page 33: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/33.jpg)
Example#14O Simplify
![Page 34: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/34.jpg)
Student guided practice
O Do problems 30-32 from book page 130
![Page 35: Chapter 2 2-9 Operations with complex numbers. Complex Plane O Just as you can represent real numbers graphically as points on a number line, you can.](https://reader036.fdocuments.in/reader036/viewer/2022062407/56649dc85503460f94abdd81/html5/thumbnails/35.jpg)
Homework!!O Do problems
36,38,40,43,45,47,50,53,55,56,64,65O Page 130 and 131
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closureO Today we learned bout how we can
graph, add ,subtract ,multiply and divide complex numbers.
O Next class we are going to start third chapter