Lesson 7 I May Be Irrational, But You’re Imaginary Solidify Understanding

Learning Focus

Relate irrational numbers to physical quantities such as the hypotenuse of a right triangle.

Understand expressions that contain negative numbers inside a square root, like $\sqrt{- 1}$ .

Add, subtract, and multiply complex numbers.

Can numbers like $\sqrt{- 1}$ be rewritten?

What types of numbers make up the number system?

Open Up the Math: Launch, Explore, Discuss

Part 1: Irrational numbers

1.

Verify that $4 (x - \frac{5}{2}) (x + \frac{3}{2}) = 0$ and $4 x^{2} - 4 x - 15 = 0$ are equivalent equations (show your work), and plot the solutions to the quadratic equations on the following number line:

2.

Verify that $(x - 2 + \sqrt{2}) (x - 2 - \sqrt{2}) = 0$ and $x^{2} - 4 x + 2 = 0$ are equivalent equations (show your work), and plot the solutions to the quadratic equations on the following number line:

You may have found it difficult to locate the exact points on the number line that represent the two solutions to the second pair of quadratic equations given above. The following diagrams might help.

3.

Find the perimeter of this isosceles triangle. Express your answer with any like terms combined.

We might approximate the perimeter of this triangle with a decimal number, but the exact perimeter is $2 + \sqrt{2}$ . Note that this notation represents a single number—the distance around the perimeter of the triangle—even though it is written as the sum of two terms.

4.

Explain how you could use this diagram to locate the two solutions to the quadratic equations given in problem 2: $2 + \sqrt{2}$ and $2 - \sqrt{2}$ .

5.

Are the numbers we have located on the number line in this way rational numbers or irrational numbers? Explain your answer.

Both sets of quadratic equations given in problems 1 and 2 have solutions that can be plotted on a number line. The solutions to the first set of quadratic equations are rational numbers. The solutions to the second set of quadratic equations are irrational numbers.

Big Idea #1: The set of numbers that contains all of the rational numbers and all of the irrational numbers is called the set of real numbers. The location of all points on a number line can be represented by real numbers.

Pause and Reflect:

Part 2: Imaginary and Complex Numbers

In the previous lesson, To Be Determined, you found that the quadratic formula gives the solutions to the quadratic equation $x^{2} + 4 x + 5 = 0$ as $- 2 + \sqrt{- 1}$ and $- 2 - \sqrt{- 1}$ . Because the square root of a negative number has no defined value as either a rational or an irrational number, Euler proposed that a new number $i = \sqrt{- 1}$ be included in what came to be known as the complex number system.

6.

Based on Euler’s definition of $i$ , what would the value of $i^{2}$ be?

With the introduction of the number $i$ , the square root of any negative number can be represented. For example, $\sqrt{- 2} = \sqrt{2} \cdot \sqrt{- 1} = \sqrt{2} \cdot i$ and $\sqrt{- 9} = \sqrt{9} \cdot \sqrt{- 1} = 3 i$ .

7.

Find the values of the following expressions. Show the details of your work.

a.

${(\sqrt{2} \cdot i)}^{2}$

b.

$3 i \cdot 3 i$

Big Idea #2: Numbers like $3 i$ and $\sqrt{2} \cdot i$ are called pure imaginary numbers. Numbers like $- 2 - i$ and $- 2 + i$ that include a real term and an imaginary term are called complex numbers.

Big Idea #3: Complex numbers are not real numbers—they do not lie on the real number line that includes all of the rational and irrational numbers; also note that the real numbers are a subset of the complex numbers since a real number results when the imaginary part of $a + b i$ is $0$ , that is, $a + 0 i$ .

Operations with complex numbers follow the same properties as real numbers, but there are a few things to pay attention to, like when adding or subtracting two complex numbers, the real parts of the numbers are like terms and the imaginary part of the numbers are like terms.

For example: $(3 + 4 i) + (5 - 2 i) = 8 + 2 i$ . If you want to line them up vertically like a polynomial, it will keep the like terms organized. When complex numbers are subtracted, it helps to change the signs of the number being subtracted and add just like we did with polynomials.

For example: $(7 + 2 i) - (3 - 5 i)$ can be rewritten as $(7 + 2 i) + (- 3 + 5 i) = 4 + 7 i$ .

Now try a few yourself:

8.

a.

$(13 + 2 i) + (- 12 - 5 i)$

b.

$(- 7 + 12 i) + (3 - 6 i)$

c.

$(8 - i) - (7 - 3 i)$

d.

$(- 9 + 4 i) - (9 - 4 i)$

Multiplying complex numbers is the same as multiplying irrational numbers like $(2 + \sqrt{3}) (1 - \sqrt{5})$ or binomials like $(x + 5) (x - 2)$ with one added step. Anytime an $i^{2}$ shows up, it needs to be written as $- 1$ and then the expression can be rewritten.

For instance, let’s multiply $(4 + 2 i) (3 - 5 i)$ using the open area model method.

Adding like terms gives us: $12 - 14 i - 10 i^{2}$ . Now substitute $i^{2} = - 1$ and rewrite the expression by combining like terms: $12 - 14 i - 10 (- 1) = 12 - 14 i + 10 = 22 - 14 i$

	$4$	$+ 2 i$
$3$	$12$	$6 i$
$- 5 i$	$- 20 i$	$- 10 i^{2}$

Give it a try, using either the open area model method or distributing terms horizontally.

9.

a.

$(3 + 2 i) (5 + 7 i)$

b.

$(10 - 3 i) (4 - 6 i)$

These two complex numbers are called conjugates because they are the same except the opposite sign on the second term. Check out what happens here:

10.

a.

$(4 + 2 i) (4 - 2 i)$

b.

$(3 + 5 i) (3 - 5 i)$

c.

What do you notice about the answers? Why is this result occurring?

Now let’s go back and think about complex solutions to quadratic equations.

11.

Given the function $f (x) = x^{2} + 4 x + 5$ , find $x$ such that $f (x) = 0$ .

12.

Check one of your solutions, showing the details of your work.

13.

Are the solutions to the equation $x^{2} + 4 x + 5 = 0$ the $x$ -intercepts of $f (x)$ ? Explain your answer.

14.

Find the solutions to the equation $x^{2} - 3 x + 4 = 0$

Now that we have established complex numbers as functional, useful numbers, let’s think again about the Fundamental Theorem of Algebra.

The Fundamental Theorem of Algebra, Revisited

Remember the following information given in the previous task:

A polynomial function is a function of the form: $y = a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + \dots a_{n - 3} x^{3} + a_{n - 2} x^{2} + a_{n - 1} x + a_{n}$

where all of the exponents are positive integers and all of the coefficients $a_{0} \dots a_{n}$ are constants.

As the theory of finding roots of polynomial functions evolved, a 17th century mathematician, Girard (1595-1632) made the following claim which has come to be known as the Fundamental Theorem of Algebra: An $n^{th}$ degree polynomial function has $n$ roots.

15.

Based on your work in this task, do you believe this theorem holds for quadratic functions? That is, do all functions of the form $y = a x^{2} + b x + c$ always have two roots?

Ready for More?

The solutions to the equation $x^{2} + 4 x + 5 = 0$ can be written as $- 2 + i$ and $- 2 - i$ , and the factored form of $x^{2} + 4 x + 5$ can be written as $(x + 2 - i) (x + 2 + i)$ .

Verify that $x^{2} + 4 x + 5$ and $(x + 2 - i) (x + 2 + i)$ are equivalent by expanding and combining like terms to get the standard form. Show the details of your work.

Takeaways

$x$ -intercepts:

Roots:

If the graph of a quadratic function intersects the $x$ -axis,

If the graph of a quadratic function does not intersect the $x$ -axis,

Vocabulary

complex conjugates
complex number
imaginary number
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we connected irrational numbers to the measure of geometric figures and showed where a given irrational number is located on the number line. We found irrational solutions of quadratic equations and used the solutions to write the equation in factored form. We also learned of a new set of numbers defined in terms of $\sqrt{- 1} = i$ and $i^{2} = - 1$ . We examined how these numbers fit into the number system and performed arithmetic operations on them.

real numbers

2.

Solve: $5 x^{2} + 2 x + 6 = 0$