Magnitude Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Magnitude.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Magnitude measures the size or length of a quantity — for a vector (a, b), it is sqrt(a^2 + b^2). For a single number, magnitude is its absolute value: how far it is from zero, ignoring sign or direction.

How big something is, regardless of which way it points—5 miles east and 5 miles west are the same distance.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Magnitude is the size of a quantity stripped of sign or direction — distance from zero, never negative.

Common stuck point: The procedure for magnitude is the easy part; the trap is reporting a magnitude as negative. Asking "Am I asking how big or how far, with the sign or direction thrown away (so the answer can't be negative)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I asking how big or how far, with the sign or direction thrown away (so the answer can't be negative)?

Worked Examples

Example 1

easy
Find the magnitude (absolute value) of each: ∣−9∣|{-9}|, ∣7∣|7|, ∣0∣|0|.

Answer

∣−9∣=9,∣7∣=7,∣0∣=0|-9|=9, \quad |7|=7, \quad |0|=0

First step

1
∣−9∣=9|-9| = 9 because −9-9 is 9 units from zero on the number line.

Full solution

  1. 2
    ∣7∣=7|7| = 7 because 77 is already 7 units from zero (positive, so unchanged).
  2. 3
    ∣0∣=0|0| = 0 because 0 is 0 units from zero.
The magnitude (absolute value) measures distance from zero, ignoring direction. It is always non-negative. Negative inputs have their sign removed; positive and zero inputs are unchanged.

Example 2

medium
Which has greater magnitude: −15-15 or 1212? Then determine which is greater as a signed number.

Example 3

medium
Two displacements: (3,4)(3, 4) then (4,−3)(4, -3). What is the magnitude of the total displacement?

Example 4

hard
A hiker walks 33 km left then 55 km right. What is the magnitude of total displacement?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Evaluate ∣3−8∣|3 - 8|.

Example 2

medium
Solve ∣x∣=6|x| = 6. How many solutions are there?

Example 3

easy
What is ∣−7∣|-7|?

Example 4

easy
What is ∣5∣|5|?

Example 5

easy
Which has greater magnitude, −3-3 or 22?

Example 6

easy
What is the magnitude of 00?

Example 7

easy
Find the magnitude of the vector (3,4)(3, 4).

Example 8

easy
What is ∣−100∣|-100|?

Example 9

easy
Is magnitude ever negative?

Example 10

easy
How far apart are 00 and −8-8 on the number line?

Example 11

medium
Compute the distance between −2-2 and 55 on the number line.

Example 12

medium
Find the magnitude of the vector (−6,8)(-6, 8).

Example 13

medium
Solve ∣x∣=5|x| = 5 for all real xx.

Example 14

medium
Order by magnitude, smallest first: −9,4,−1,6-9, 4, -1, 6.

Example 15

medium
The magnitude of a complex number a+bia+bi is ∣a+bi∣=a2+b2|a+bi|=\sqrt{a^2+b^2}. Find ∣8+6i∣|8 + 6i|.

Example 16

medium
If a temperature change is −12-12 degrees, how large is the change in magnitude?

Example 17

medium
Two displacements are 5 m east and 5 m west. Compare their magnitudes.

Example 18

medium
Solve ∣x−3∣=2|x - 3| = 2 for all real xx.

Example 19

challenge
A vector goes (3,4)(3, 4) then (0,−1)(0, -1). What is the magnitude of the total displacement?

Example 20

challenge
For which real numbers xx is ∣x∣<3|x| < 3?

Example 21

challenge
A complex number has magnitude 13 and real part 5 with a negative imaginary part. Find the number.

Example 22

medium
Compute ∣3−8∣|3 - 8|.

Example 23

easy
Compute ∣−15∣|{-15}| and ∣20∣|20|, then state which is larger as a magnitude.

Example 24

easy
Find the magnitude of the vector (5,12)(5, 12).

Example 25

easy
Find the magnitude of the vector (−3,−4)(-3, -4).

Example 26

easy
What is ∣3−(−5)∣|3-(-5)|?

Example 27

easy
Solve ∣x∣=10|x| = 10.

Example 28

easy
Find ∣9+12i∣|9+12i|.

Example 29

medium
Find the magnitude of the vector (−7,24)(-7, 24).

Example 30

medium
Solve ∣2x∣=8|2x| = 8.

Example 31

medium
Solve ∣x+1∣=4|x+1| = 4.

Example 32

medium
Find the magnitude of the vector (1,1)(1, 1).

Example 33

medium
For which real xx is ∣x∣>3|x| > 3?

Example 34

medium
Order by magnitude, smallest first: −12, 5, −8, 3-12,\ 5,\ -8,\ 3.

Example 35

medium
A particle goes 66 km north then 88 km east. Find the magnitude of total displacement.

Example 36

hard
Find ∣7−24i∣|7-24i|.

Example 37

hard
Solve ∣2x−6∣=4|2x-6|=4.

Example 38

hard
Find the magnitude of (8,15)(8, 15).

Example 39

hard
How many integers satisfy ∣x∣≤4|x| \leq 4?

Example 40

hard
Find the distance between −3-3 and 99 on the number line.

Example 41

hard
Solve ∣x+2∣=6|x+2| = 6.

Example 42

challenge
A complex number has magnitude 1313 and imaginary part 55 with a positive real part. Find the number.

Example 43

challenge
For real xx, find all xx with ∣x−2∣+∣x+2∣=10|x-2| + |x+2| = 10.
← Back to Magnitude Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

more lessintegers