Magnitude Map Worksheet Magnitude Map
Complex Numbers: Absolute Value & Distance
Student:
Date:
Warm-Up: Hypotenuse Hunt
Find the length of the hypotenuse \(c\) for the following right triangles with legs \(a\) and \(b\). Show your work using \(a^2 + b^2 = c^2\).
1. \(a = 3\), \(b = 4\)
2. \(a = 6\), \(b = 8\)
Video Insights
Problem 1: Basic Absolute Value (0:00 - 0:34)
How is \(|6 + 8i|\) calculated? Write down the process used in the video.
Problem 8: Advanced Simplification (7:50 - 9:02)
Simplify \(5i^{12} - 3i^{19}\) into \(a + bi\) form first, then find the absolute value.
The Connection
How does the Distance Formula relate to the Absolute Value of a complex number? Why must we simplify to \(a + bi\) form before we start our calculation?
Graph & Calculate
Plot each complex number on your Complex Plane Grid . Then, calculate its absolute value (distance from the origin) in the spaces below. Show your steps.
Point A \(3 + 2i\)
Point B \(-4 + 3i\)
Point C \(5i\)
Point D \(-2 - 5i\)
Final Prediction
Which value do you think is "larger"? Explain your reasoning based on what we've learned about magnitude.
\(|3 + 4i|\)
vs
\(|5|\)
Complex Plane Grid Complex Plane Grid
Coordinate System for Graphing Complex Numbers
Legend
Real Axis
Imaginary Axis
Real (Re)
-10
10
Imaginary (Im)
10i
-10i
How to use this grid:
The horizontal axis represents the real part (a) .
The vertical axis represents the imaginary part (bi) .
Each grid square represents 1 unit.
Mark points A, B, C, and D from your worksheet on this plane.
Magnitude Presentation Slides Algebra II: Complex Numbers
Absolute Magnitude
Calculating the "Distance" of Imaginary Numbers
Warm-Up: Hypotenuse Hunt
Triangle A
Legs: 3 & 4
Hypotenuse = ?
Triangle B
Legs: 6 & 8
Hypotenuse = ?
What formula are you using to find these lengths?
The Complex "Map"
Definition
Absolute value is the distance from the origin \((0,0)\) on the complex plane.
The Formula
|a + bi| = \(\sqrt{a^2 + b^2}\)
|z|
Magnitude
Case Study: Problem 1
Watch: 0:00 - 0:34
Embedded media
Observe how the real and imaginary parts are treated like triangle legs.
Case Study: Problem 8
Watch: 7:50 - 9:02
Embedded media
Why must we simplify the powers of \(i\) first?
The Deep Dive
How does the Distance Formula relate to the Absolute Value of a complex number?
Why is it critical to reach the \(a + bi\) form before calculating magnitude?
Graph & Calculate
Plot the numbers from your worksheet on the Complex Plane Grid .
Point A: \(3 + 2i\)
Point B: \(-4 + 3i\)
Point C: \(5i\)
Point D: \(-2 - 5i\)
Task
Calculate the distance of each point from the origin. Round to 2 decimal places.
The Big Question
Which has a larger absolute value?
|3 + 4i|
VS
|5|
Explain your reasoning!
Magnitude Map Answer Key Magnitude Map
Teacher Answer Key & Guide
Official Key
Warm-Up: Hypotenuse Hunt
1. \(a = 3\), \(b = 4\)
\(3^2 + 4^2 = c^2\)
2. \(a = 6\), \(b = 8\)
\(6^2 + 8^2 = c^2\)
Video Insights
Problem 1: \(|6 + 8i|\)
\(|6 + 8i| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\)
Problem 8: \(5i^{12} - 3i^{19}\)
Step 1: Simplify \(i\) powers
Graph & Calculate Results
POINT A: \(3+2i\)
POINT B: \(-4+3i\)
POINT C: \(5i\)
POINT D: \(-2-5i\)
Closure: Prediction Key
Both are equal !
