lesson 1: a surprising boost from geometry – introduction...

Hart Interactive – Honors Algebra 1 M4+ Lesson 1 HONORS ALGEBRA 1

Lesson 1: A Surprising Boost from Geometry – Introduction to Complex Numbers Module 4+: Complex Numbers

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Lesson 1: A Surprising Boost from Geometry – Introduction to Complex Numbers

Classwork

Opening Exercise

1. Solve each equation for 𝑥𝑥.

A. x – 1 = 0 B. x + 1 = 0 C. x2 – 1 = 0 D. x2 + 1 = 0

Although the first three problems are easily solved, the fourth one probably gave you some trouble. From the time of the ancient Greeks through the Renaissance period, mathematicians were also perplexed by such problems. Cardano, also known as Cordan, considered to be the greatest mathematician of the Renaissance, ran into this problem: Find two numbers, a and b, whose sum is 10 and whose product is 40. 2. Write two equations to represent Cardano’s problem.

3. Solve your system of equations from Exercise 2.

Cardano was the most outstanding mathematician of his time. Italian physician, mathematician, and astrologer who gave the first clinical description of typhus fever and whose book Ars magna (The Great Art; or, The Rules of Algebra) is one of the cornerstones in the history of algebra.

[source: http://www.britannica.com/biography/Girolamo-Cardano]



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The answers to Cardano’s problem use a new number system – complex numbers. The complex numbers share many properties with the real numbers with which you are familiar. We will take a geometric approach to introducing complex numbers.

Discussion

We can use geometry to help us think about multiplying by −1. If you take any positive number and multiply it by -1, you are essentially rotating the number line in the plane by 180° about the point 0.

The same is true for any negative number as shown in the picture above.

Is there a number we can multiply by that corresponds to a 90° rotation? Such a number does not map the number line to itself, so we have to imagine another number line that is a 90° rotation of the original.

This is similar to the coordinate plane, but how should we label the points on the vertical axis?



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Well, since we imagined such a number existed, let’s call it the imaginary axis and subdivide it into units of something called 𝑖𝑖. Then, the point 1 on the number line rotates to 1 ∙ 𝑖𝑖 on the rotated number line and so on, as follows:

4. What happens if we multiply a point on the vertical number line by i ?

5. When we perform two 90° rotations, it is the same as performing a 180° rotation, so multiplying by 𝑖𝑖 twice results in the same rotation as multiplying by −1.

𝑖𝑖2 ∙ 𝑥𝑥 = −1 ∙ 𝑥𝑥

for any real number 𝑥𝑥; thus, 𝑖𝑖2 = _________

and 𝑖𝑖 = _________.



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6. Now we have the following:

________i = 2 ________i = 3 ________i = 4 ________i =

7. Determine the value of each expression below.

A. 5 ________i = B. 6 ________i = C. 7 ________i = D. 8 ________i = E. 9 ________i =

8. What rule could we use to find 123i or 57i ?

9. Recall from the Opening Exercise that there are no real solutions to the equation

𝑥𝑥2 + 1 = 0. However, this new number i is a solution. What is the other solution to this equation? How can you check

your answer?



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Complex Numbers

In problem 3, you used the Quadratic Formula to get the solutions 5 15a = ± − . This can be rewritten as 5 15a i= + and 5 15a i= − . Notice that there are two parts to these answers.

5 15a i= +

5 15a i= −

These two parts together make up a complex number. We use the notation a + bi to show the generic version of a complex number where a and b are real numbers. 10. Fill in the table below to identify the real and imaginary part of each complex number or to write the

complex number.

Complex Number Real Part Imaginary Part

A. -2 + 7i

B. 3 – 4i

C. -2 5i

D. -7 0

E. 0 -i

Real Number

Imaginary Number



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Graphing Complex Numbers

Just as we saw in the Discussion on pages 2 and 3, the horizontal axis we’ll use to represent the real numbers and the vertical axis will represent the imaginary numbers. Below is a graph of the solutions for the equation 𝑥𝑥2 + 2𝑥𝑥 + 5 = 0.

11. What are the solutions to the equation 𝑥𝑥2 + 2𝑥𝑥 + 5 = 0 based on the graph?

Lesson Summary Every complex number is in the form a + bi, where a is the real part and bi is the imaginary part of the number. Real numbers are also complex numbers. The real number a can be written as the complex number a + 0i. Numbers of the form bi, for real numbers b, are called imaginary numbers. You can graph complex numbers by using the a and b parts as shown on the right.



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Homework Problem Set 1. Locate the point on the complex plane corresponding to the complex number given in parts (A)–(H). On

one set of axes, label each point by its identifying letter. For example, the point corresponding to 5 + 2𝑖𝑖 should be labeled 𝑎𝑎. A. 5 + 2𝑖𝑖 B. 3 − 2𝑖𝑖 C. −2 − 4𝑖𝑖 D. −𝑖𝑖 E. 12 + 𝑖𝑖

F. √2 − 3𝑖𝑖 G. 0

H. −32 + √3

2 𝑖𝑖 2. Find the real values of 𝑥𝑥 and 𝑦𝑦 in each of the following equations using the fact that if 𝑎𝑎 + 𝑏𝑏𝑖𝑖 = 𝑐𝑐 + 𝑑𝑑𝑖𝑖,

then 𝑎𝑎 = 𝑐𝑐 and 𝑏𝑏 = 𝑑𝑑. A. 5𝑥𝑥 + 3𝑦𝑦𝑖𝑖 = 20 + 9𝑖𝑖

B. 3(7 − 2𝑥𝑥)− 5(4𝑦𝑦 − 3)𝑖𝑖 = 𝑥𝑥 − 2(1 + 𝑦𝑦)𝑖𝑖

C. 2(5 9) (10 3 )x y i+ = −



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3. Since 𝑖𝑖2 = −1, we see that

𝑖𝑖3 = 𝑖𝑖2 ⋅ 𝑖𝑖 = −1 ⋅ 𝑖𝑖 = −𝑖𝑖 𝑖𝑖4 = 𝑖𝑖2 ⋅ 𝑖𝑖2 = −1 ⋅ −1 = 1.

Plot 𝑖𝑖, 𝑖𝑖2, 𝑖𝑖3, and 𝑖𝑖4 on the complex plane, and describe how multiplication by each rotates points in the complex plane.

4. Express each of the following in 𝑎𝑎 + 𝑏𝑏𝑖𝑖 form.

A. 𝑖𝑖5 B. 𝑖𝑖6 C. 𝑖𝑖7 D. 𝑖𝑖8 E. 𝑖𝑖102

5. Evaluate the four products below.

A. √9 ⋅ √4 B. √9 ⋅ √−4

C. √−9 ⋅ √4 D. √−9 ⋅ √−4

6. Suppose 𝑎𝑎 and 𝑏𝑏 are positive real numbers. Determine whether the following quantities are equal or not equal.

A. √𝑎𝑎 ∙ √𝑏𝑏 and √−𝑎𝑎 ∙ √−𝑏𝑏 B. √−𝑎𝑎 ∙ √𝑏𝑏 and √𝑎𝑎 ∙ √−𝑏𝑏

lesson 1: a surprising boost from geometry – introduction...

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