Precalculus by Richard Wright

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2-01 Complex Numbers

Summary: In this section, you will:

• Express square roots of negative numbers as multiples of i.
• Plot complex numbers on the complex plane.
• Add and subtract complex numbers.
• Multiply and divide complex numbers.

SDA NAD Content Standards (2018): PC.5.6

Mathematics continually builds on itself. Young child are taught counting numbers, then the knowledge expands to include negative numbers. Next fractions and rational numbers fill the spaces between the integers. Finally students learn about irrational numbers such as π and $$\sqrt{2}$$. Together all these sets of numbers make the real number set. However, some problems are not solvable on the set of real numbers. For example, the following equation has no real solution.

x2 + 4 = 0

x2 = −4

$$x = ±\sqrt{-4}$$

In grade school, students are taught that there are no such things as square roots of negative numbers because squaring a real number always produces a positive number. But what if that was not the case. There are more number sets, called imaginary numbers and complex numbers that will solve this problem.

The picture in figure 1 is part of the Mandelbrot Set which is created using complex numbers.

Complex Numbers

Finding the square root of a negative number is similar to finding the square root of a positive number. The only change is what to do with the negative sign. Imaginary numbers are square roots of negative numbers. The imaginary unit, i is defined as the square root of −1.

$$i = \sqrt{-1}$$

Squaring both sides produces,

i2 = −1

The name "imaginary" is somewhat of a bad name. They follow all the rule of "real" numbers. They are not just made up, but come from solving math problems. Don't let the name fool you.

Complex numbers are the combined real set and imaginary set of numbers. They are written with a real and an imaginary part in the form of a + bi such as the number below. The blue part is real and the red part with i is imaginary.

2 + 3i

Real   Imaginary

Imaginary and Complex Numbers:

A complex number is a number of the form a + bi where

• a is the real part of the complex number.
• bi is the imaginary part of the complex number.

If b = 0 then a + bi is a real number. If a = 0 and b ≠ 0, the complex number is called an imaginary number.

Express a Complex Number in Standard Form
1. Write $$\sqrt{-a}$$ as $$\sqrt{a}\sqrt{-1}$$.
2. Express $$\sqrt{-1}$$ as i.
3. Write $$\sqrt{a}i$$ in simplest form.

Example 1: Evaluate the Square Root of a Negative Number

Evaluate $$\sqrt{-18}$$.

Solution

Start by expressing the square as $$\sqrt{a}\sqrt{-1}$$.

$$\sqrt{-18} = \sqrt{18}\sqrt{-1}$$

Replace the $$\sqrt{-1}$$ with i.

$$= \sqrt{18}i$$

Simplify the $$\sqrt{18}$$.

$$= \sqrt{9}\sqrt{2}i$$

$$= 3\sqrt{2}i$$

Try IT 1

Evaluate $$\sqrt{-24}$$.

$$2\sqrt{6}i$$

Plot a Complex Number on the Complex Plane

Real numbers are graphed on a number line; however, complex numbers cannot be graphed on the real number line. To graph complex numbers, a second number line is drawn perpendicular to the real number line. This is the complex plane in which the horizontal axis is the real number line and the vertical axis is the imaginary number line. Complex number in the form a + bi are graphed by plotting the point (a, b) on the complex plane.

Complex Plane:

In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis as shown in figure 2.

Graph a Complex Number on the Complex Plane

For a complex number, a + bi,

1. Determine the real part, a, and the imaginary part, b, of the complex number.
2. Move along the horizontal real axis the distance a.
3. Move up or down parallel to the vertical imaginary axis the distance b.
4. Plot the point.

Example 2: Plot a Complex Number on the Complex Plane

Plot the complex numbers (a) 2 + 3i, (b) −1 + 2i, and (c) −5 − 4i on the complex plane.

Solution
1. 2 + 3i would be like plotting (2, 3).
2. −1 + 2i would be like plotting (−1, 2).
3. −5 − 4i would be like plotting (−5, −4).
Try IT 2

Plot the complex number 3 − 4i on the complex plane.

All the normal arithmetic operations work with complex number like they do with real numbers. Add or subtract complex numbers by combining the corresponding parts, similar to combining like terms.

(a + bi) + (c + di) = (a + c) + (b + d)i

Subtracting complex numbers:

(a + bi) − (c + di) = (ac) + (b + d)i

Essentially, this is combining like terms.

Simplify (2 + 3i) + (−3 + 5i).

Solution

Combine like terms. Add the real parts, then add the imaginary parts.

(2 + 3i) + (−3 + 5i)

(2 + (−3)) + (3i + 5i)

−1 + 8i

Try IT 3

Simplify (2 + 4i) − (−3 + 5i).

5 − i

Multiply Complex Numbers

Multiplying complex numbers is similar to multiplying binomials. The major difference is to replace any i2 to −1. Use the distributive property to multiply each term in the first complex number with each term in the second complex number. If i2 results, change it to −1. Then collect like terms.

(a + bi)(c + di)

ac + adi + bci + bdi2

Complex Numbers: Multiplying
1. Use the distributive property.
2. Replace any i2 with −1
3. Simplify.

Example 4: Multiply a Complex Number by a Real Number

Simplify 3(5 + 2i).

Solution

Distribute the 3.

3(5 + 2i)

= (3·5) + (3·2)i

= 15 + 6i

Try IT 4

Simplify −2(1 + 5i).

−2 − 10i

Example 5: Multiply a Complex Number by a Complex Number

Simplify (2 + 3i)(1 − 4i).

Solution

Use the distributive property.

(2 + 3i)(1 − 4i)

= (2·1) + (2·−4i) + (3i·1) + (3i·−4i)

= 2 − 8i + 3i − 12i2

Replace the i2 with −1.

= 2 − 8i + 3i − 12(−1)

= 2 − 8i + 3i + 12

Collect like terms.

= 14 − 5i

Try It 5

Simplify (3 − i)(5 + 2i).

17 + i

Divide Complex Numbers

Division of two complex numbers is more complicated than addition, subtraction, and multiplication because the imaginary unit, i, is technically a radical, and no radicals are allowed in the denominator of a number. However, division by complex numbers is the same as division by radicals. To divide complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator. The complex conjugate is the same number, but with the opposite sign on the imaginary part. The complex conjugate of a + bi is abi.

To divide a + bi by c + di where neither c or d equals zero, first write the division as a fraction. Then find the complex conjugate of the denominator, and multiply both the numerator and the denominator.

$$\frac{a+bi}{c+di}$$

Multiply the numerator and denominator by the complex conjugate of the denominator.

$$\frac{a+bi}{c+di}·\frac{c-di}{c-di}$$

Apply the distributive property.

$$\frac{ac - adi + bci - bdi^2}{c^2 - cdi + cdi - d^2i^2}$$

Simplify, remembering that i2 = −1.

$$\frac{ac - adi + bci - bd(-1)}{c^2 - cdi + cdi - d^2(-1)}$$

$$\frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$$

Complex Conjugate

The complex conjugate of a complex number a + bi is abi. It is the same number with the opposite sign on the imaginary part.

• When a complex number is multiplied by its complex conjugate, the result is a real number.
• When a complex number is added to its complex conjugate, the result is a real number.

Example 6: Find Complex Conjugates

Find the complex conjugate of each number.

1. 2 + 8i
2. −3i
Solution
1. The number is already in the form a + bi. The complex conjugate is abi, or 2 − 8i.
2. Rewrite this number in the form a + bi as 0 − 3i. The complex conjugate is abi or 0 + 3i. Usually this is written as 3i.
Complex Numbers: Division
1. Write the division problem as a fraction.
2. Determine the complex conjugate of the denominator.
3. Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.
4. Simplify.

Example 7: Divide Complex Numbers

Divide (1 − 4i) by (3 + i).

Solution

Write the problem as a fraction.

$$\frac{1 - 4i}{3 + i}$$

Multiply the numerator and denominator by the complex conjugate of the denominator.

$$\frac{1 - 4i}{3 + i}⋅\frac{3 - i}{3 - i}$$

Multiply the complex numbers using the distributive property.

$$\frac{(1 - 4i)(3 - i)}{(3 + i)(3 - i)}$$

$$\frac{1 - i - 12i + 4i^2}{9 - 3i + 3i - i^2}$$

Replace i2 with −1 and simplify.

$$\frac{1 - i - 12i + 4(-1)}{9 - 3i + 3i - (-1)}$$

$$\frac{-3 - 13i}{10}$$

Separate into real and imaginary parts.

$$-\frac{3}{10} - \frac{13}{10}i$$

Try It 6

Simplify $$\frac{2 - i}{3i}$$.

$$-\frac{1}{3} - \frac{2}{3}i$$

Example 8: Substitute a Complex Number into a Polynomial Function

Let f(x) = x2 + 3x. Evaluate f(2 − i).

Solution

Substitute x = 2 − i into the function f(x) = x2 + 3x and simplify.

f(2 − i) = (2 − i)2 + 3(2 − i)

Use the distributive property. Remember that (2 − i)2 = (2 − i)(2 − i).

= 4 − 4i + i2 + 6 − 3i

Replace i2 with −1.

= 4 − 4i + (−1) + 6 − 3i

Collect like terms.

= 9 − 7i

Try It 7

Let f(x) = 2x2x. Evaluate f(3 − 2i).

2 − 10i

Example 9: Substitute an Imaginary Number in a Rational Function

Let $$f(x) = \frac{x}{x + 1}$$. Evaluate f(2 + i).

Solution

$$f(x) = \frac{x}{x + 1}$$

Substitute x = 2 + i.

$$f(2 + i) = \frac{2 + i}{(2 + i) + 1}$$

$$= \frac{2 + i}{3 + i}$$

Because this is division, multiply the numerator and denominator by the conjugate of the denominator.

$$= \frac{(2 + i)}{(3 + i)}⋅\frac{(3 - i)}{(3 - i)}$$

Use the distributive property.

$$= \frac{6 - 2i + 3i - i^2}{9 - 3i + 3i - i^2}$$

Replace i2 with −1.

$$= \frac{6 - 2i + 3i - (-1)}{9 - 3i + 3i - (-1)}$$

Combine like terms.

$$= \frac{7 + i}{10}$$

Put the number in standard form by splitting it into real and imaginary parts.

$$= \frac{7}{10} + \frac{1}{10}i$$

Try It 8

Let $$f(x) = \frac{1 - x}{x + 2}$$. Evaluate 1 − 4i.

$$-\frac{16}{25} + \frac{12}{25}i$$

Simplify Powers of i

The powers of i follow a pattern. The list below are increasing powers of i. What is the pattern?

i1 = i

i2 = −1

i3 = i2·i = (-1)·i = i

i4 = i3·i = −i·i = −i2 = −(−1) = 1

i5 = i4·i = 1·i = i

i6 = i5·i = i·i = i2 = −1

i7 = i6·i = (−1)·i = i

i8 = i7·i = −i·i = −i2 = −(−1) = 1

i9 = i8·i = 1·i = i

The powers of i follow a four part cycle. To find the value of a power of i, divide the exponent by 4 and find the remainder. The value of the power of i is the same as the value of iremainder.

Powers of i

To evaluate a power of i,

1. Divide the exponent by 4 and find the remainder.
2. Choose the value of iremainder from the list below.
• i0 = 1
• i1 = i
• i2 = −1
• i3 = −i

Example 10 Simplify Powers of i

Evaluate i51.

Solution

Divide 51 by 4 and find the remainder.

$$\frac{51}{4} = 12 R 3$$

Since the remainder is 3, i51 = i3 = −i.

Lesson Summary

Imaginary and Complex Numbers:

A complex number is a number of the form a + bi where

• a is the real part of the complex number.
• bi is the imaginary part of the complex number.

Express a Complex Number in Standard Form
1. Write $$\sqrt{-a}$$ as $$\sqrt{a}\sqrt{-1}$$.
2. Express $$\sqrt{-1}$$ as i.
3. Write $$\sqrt{a}i$$ in simplest form.

Complex Plane:

In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis as shown in figure 2.

Graph a Complex Number on the Complex Plane

For a complex number, a + bi,

1. Determine the real part, a, and the imaginary part, b, of the complex number.
2. Move along the horizontal real axis the distance a.
3. Move up or down parallel to the vertical imaginary axis the distance b.
4. Plot the point.

(a + bi) + (c + di) = (a + c) + (b + d)i

Subtracting complex numbers:

(a + bi) − (c + di) = (ac) + (b + d)i

Essentially, this is combining like terms.

Complex Numbers: Multiplying
1. Use the distributive property.
2. Replace any i2 with −1
3. Simplify.

Complex Conjugate

The complex conjugate of a complex number a + bi is abi. It is the same number with the opposite sign on the imaginary part.

• When a complex number is multiplied by its complex conjugate, the result is a real number.
• When a complex number is added to its complex conjugate, the result is a real number.

Complex Numbers: Division
1. Write the division problem as a fraction.
2. Determine the complex conjugate of the denominator.
3. Multiply the numerator and denominator of the fraction by the complex conjugate of the denominator.
4. Simplify.

Powers of i

To evaluate a power of i,

1. Divide the exponent by 4 and find the remainder.
2. Choose the value of iremainder from the list below.
• i0 = 1
• i1 = i
• i2 = −1
• i3 = −i

Practice Exercises

1. Explain how to add complex numbers.
2. Plot the complex numbers on the complex plane.

3. (a) 4 − 3i, (b) −5i
4. (a) −2 + 5i, (b) 3
5. Simplify. Write the result in standard form.

6. $$\sqrt{-25} + 2\sqrt{-9}$$
7. $$\frac{\sqrt{-50} - \sqrt{-8}}{4}$$
8. (1 − 9i) + (−3 − 5i)
9. (4 − 6i) − (2 − 7i)
10. (2i)(3 − i)
11. (2 + 4i)(1 + i)
12. (−1 + 2i)(4 + 3i)
13. $$\frac{3 - 5i}{2i}$$
14. $$\frac{-4 + i}{2 - 9i}$$
15. Evaluate the function for the given complex number.

16. If f(x) = x2 + x, evaluate f(−2i).
17. If f(x) = x2 − 2x + 7, evaluate f(3 − 4i).
18. If $$f(x) = \frac{x + 1}{x - 2}$$, evaluate f(2 + 5i).
19. Mixed Review

20. (1-05) Find zeros of f(x) = x2 − 9.
21. (1-05) Find the (a) domain, (b) range, (c) interval where the graph is increasing and (d) decreasing.
22. (1-07) Describe how the graph of the function is a transformation of the graph of a parent function: f(x) = −(x + 2)2 − 5
23. (1-10) Draw a scatter plot for the data provided. Find the equation of the best fitting line.
 1 2 3 4 5 10 8 6 4 2
24. (1-10) y varies inversely with the square of x. When x = 2, then y = 1. Find y when x is 8.

1. Add the real parts together and the imaginary parts together; combine like terms.
2. 11i
3. $$\frac{3\sqrt{2}}{4}i$$
4. −2 − 14i
5. 2 + i
6. 2 + 6i
7. −2 + 6i
8. −10 + 5i
9. $$-\frac{5}{2} - \frac{3}{2}i$$
10. $$-\frac{1}{5} - \frac{2}{5}i$$
11. −4 − 2i
12. −6 − 16i
13. $$1 - \frac{3}{5}i$$
14. −3, 3
15. (−∞, ∞); [−1, ∞); (2, ∞); (−∞, 2)
16. Reflected over the x-axis, shifted left 2 and down 5.
17. y = −2x + 12
18. $$\frac{1}{16}$$