Table of contents

0. Review of College Algebra4h 43m
1. Measuring Angles40m
2. Trigonometric Functions on Right Triangles2h 5m
3. Unit Circle1h 19m
4. Graphing Trigonometric Functions1h 19m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
6. Trigonometric Identities and More Equations2h 34m
7. Non-Right Triangles1h 38m
8. Vectors2h 25m
9. Polar Equations2h 5m
10. Parametric Equations1h 6m
11. Graphing Complex Numbers1h 7m

11. Graphing Complex Numbers

Polar Form of Complex Numbers

Trigonometry

11. Graphing Complex Numbers

Polar Form of Complex Numbers

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2:01 minutes

Problem 9

Textbook Question

In Exercises 1–10, plot each complex number and find its absolute value.z = −3 + 4i

Verified step by step guidance

Step 1: Identify the real and imaginary parts of the complex number. Here, the complex number is given as $ z = -3 + 4i $. The real part is $-3$ and the imaginary part is $4$.

Step 2: Plot the complex number on the complex plane. The horizontal axis (x-axis) represents the real part, and the vertical axis (y-axis) represents the imaginary part. Plot the point $(-3, 4)$ on this plane.

Step 3: To find the absolute value (or modulus) of the complex number $ z = a + bi $, use the formula $ |z| = \sqrt{a^2 + b^2} $.

Step 4: Substitute the real part $a = -3$ and the imaginary part $b = 4$ into the formula: $ |z| = \sqrt{(-3)^2 + (4)^2} $.

Step 5: Simplify the expression under the square root: $ |z| = \sqrt{9 + 16} $. This will give you the absolute value of the complex number.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form z = a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i'. In this case, z = -3 + 4i has a real part of -3 and an imaginary part of 4. Understanding complex numbers is essential for visualizing them on the complex plane.

Plotting on the Complex Plane

The complex plane is a two-dimensional plane where the x-axis represents the real part and the y-axis represents the imaginary part of complex numbers. To plot the complex number z = -3 + 4i, you would locate the point at (-3, 4) on this plane. This visualization helps in understanding the geometric interpretation of complex numbers.

Absolute Value of a Complex Number

The absolute value (or modulus) of a complex number z = a + bi is calculated using the formula |z| = √(a² + b²). This value represents the distance of the point (a, b) from the origin (0, 0) in the complex plane. For z = -3 + 4i, the absolute value is |z| = √((-3)² + (4)²) = √(9 + 16) = √25 = 5.