Lesson Objectives

- Demonstrate an understanding of the Pythagorean Formula
- Learn how to find the absolute value of a complex number

## How to Find the Absolute Value of a Complex Number

In this lesson, we will learn how to find the absolute value of a complex number. At this point, we should fully understand the concept of absolute value with real numbers. We know the absolute value of a number such as 5 is 5 because the number 5 is 5 units away from zero on the number line. Starting at 5, we can count 5 units to get to zero. When we talk about the absolute value of a complex number, we use the same concept. The absolute value of a complex number is also a measure of its distance from zero. The only difference is we are measuring the distance on the complex plane.

Example #1: Find the absolute value of the given complex number. $$|5 + 7i|$$ Let’s begin by plotting this complex number 5 + 7i on the complex plane. We know we would move five units right on the real axis and 7 units up on the imaginary axis. When we ask for the absolute value of a complex number, also known as the modulus, we are asking for the distance from the origin to the complex number on the complex plane. We are really familiar with our Pythagorean formula at this point and we know we can use it to find the distance here. Let’s create a right triangle. To do this, we will have a point at the origin, a point at 5 + 7i, and a point at 5 + 0i. What’s the measure of the vertical leg here? It's the vertical distance from the point 5 + 7i down to the real axis or the point 5 + 0i. We can simply subtract the imaginary parts: 7 - 0 = 7 and get a distance of 7. What's the measure of the horizontal leg here? It’s the horizontal distance from the point 5 + 0i over to the imaginary axis or the point 0 + 0i. We can simply subtract the real parts: 5 - 0 = 5 and get a distance of 5. This tells us the vertical leg of the right triangle has length 7 and the horizontal leg of the right triangle has length 5. Let's plug into the Pythagorean Formula: $$a^2 + b^2=c^2$$ $$a=5, b=7$$ $$5^2 + 7^2=c^2$$ $$25 + 49=c^2$$ $$c^2=74$$ $$c=\sqrt{74}$$ This tells us the absolute value of 5 + 7i is the square root of 74. $$|5 + 7i|=\sqrt{74}$$ Let's now consider a shortcut to this process. Since one point on our right triangle will always be the origin: 0 + 0i, our vertical leg will be |b - 0| or just |b| and our horizontal leg will be |a - 0| or just |a|.

Note: a and b here refer to the real part (a) and the imaginary part (b) of the complex number. $$|a|^2 + |b|^2=c^2$$ We can drop the absolute value bars since squaring makes our answer non-negative. We end up with: $$a^2 + b^2=c^2$$ $$c=\sqrt{a^2 + b^2}$$ Since c is the absolute value of our complex number, we can replace it: $$|a + bi|=\sqrt{a^2 + b^2}$$ If we repeat our problem with this simpler approach, we get: $$|5 + 7i|$$ $$a=5$$ $$b=7$$ $$|5 + 7i|=\sqrt{5^2 + 7^2}$$ $$|5 + 7i|=\sqrt{25 + 49}$$ $$|5 + 7i|=\sqrt{74}$$ Example #2: Find the absolute value of the given complex number. $$|5 - 12i|$$ Let's use our formula: $$|a + bi|=\sqrt{a^2 + b^2}$$ $$a=5$$ $$b=-12$$ Plug in 5 for a and -12 for b: $$|5 - 12i|=\sqrt{5^2 + (-12)^2}$$ $$|5 - 12i|=\sqrt{25 + 144}$$ $$|5 - 12i|=\sqrt{169}$$ $$|5 - 12i|=13$$

Example #1: Find the absolute value of the given complex number. $$|5 + 7i|$$ Let’s begin by plotting this complex number 5 + 7i on the complex plane. We know we would move five units right on the real axis and 7 units up on the imaginary axis. When we ask for the absolute value of a complex number, also known as the modulus, we are asking for the distance from the origin to the complex number on the complex plane. We are really familiar with our Pythagorean formula at this point and we know we can use it to find the distance here. Let’s create a right triangle. To do this, we will have a point at the origin, a point at 5 + 7i, and a point at 5 + 0i. What’s the measure of the vertical leg here? It's the vertical distance from the point 5 + 7i down to the real axis or the point 5 + 0i. We can simply subtract the imaginary parts: 7 - 0 = 7 and get a distance of 7. What's the measure of the horizontal leg here? It’s the horizontal distance from the point 5 + 0i over to the imaginary axis or the point 0 + 0i. We can simply subtract the real parts: 5 - 0 = 5 and get a distance of 5. This tells us the vertical leg of the right triangle has length 7 and the horizontal leg of the right triangle has length 5. Let's plug into the Pythagorean Formula: $$a^2 + b^2=c^2$$ $$a=5, b=7$$ $$5^2 + 7^2=c^2$$ $$25 + 49=c^2$$ $$c^2=74$$ $$c=\sqrt{74}$$ This tells us the absolute value of 5 + 7i is the square root of 74. $$|5 + 7i|=\sqrt{74}$$ Let's now consider a shortcut to this process. Since one point on our right triangle will always be the origin: 0 + 0i, our vertical leg will be |b - 0| or just |b| and our horizontal leg will be |a - 0| or just |a|.

Note: a and b here refer to the real part (a) and the imaginary part (b) of the complex number. $$|a|^2 + |b|^2=c^2$$ We can drop the absolute value bars since squaring makes our answer non-negative. We end up with: $$a^2 + b^2=c^2$$ $$c=\sqrt{a^2 + b^2}$$ Since c is the absolute value of our complex number, we can replace it: $$|a + bi|=\sqrt{a^2 + b^2}$$ If we repeat our problem with this simpler approach, we get: $$|5 + 7i|$$ $$a=5$$ $$b=7$$ $$|5 + 7i|=\sqrt{5^2 + 7^2}$$ $$|5 + 7i|=\sqrt{25 + 49}$$ $$|5 + 7i|=\sqrt{74}$$ Example #2: Find the absolute value of the given complex number. $$|5 - 12i|$$ Let's use our formula: $$|a + bi|=\sqrt{a^2 + b^2}$$ $$a=5$$ $$b=-12$$ Plug in 5 for a and -12 for b: $$|5 - 12i|=\sqrt{5^2 + (-12)^2}$$ $$|5 - 12i|=\sqrt{25 + 144}$$ $$|5 - 12i|=\sqrt{169}$$ $$|5 - 12i|=13$$

#### Skills Check:

Example #1

Find the absolute value. $$|4 - 2i|$$

Please choose the best answer.

A

$$2\sqrt{5}$$

B

$$2\sqrt{3}$$

C

$$5$$

D

$$\sqrt{17}$$

E

$$2\sqrt{13}$$

Example #2

Find the absolute value. $$|3 - 2i|$$

Please choose the best answer.

A

$$\sqrt{26}$$

B

$$5$$

C

$$\sqrt{5}$$

D

$$\sqrt{13}$$

E

$$2\sqrt{13}$$

Example #3

Find the absolute value. $$\left|-3\sqrt{2}+ 3i\sqrt{2}\right|$$

Please choose the best answer.

A

$$\sqrt{5}$$

B

$$2$$

C

$$\sqrt{10}$$

D

$$6$$

E

$$3$$

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