Magnitude Formula

The Formula

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}

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

Quick Example

Both -7 and +7 have magnitude 7. Temperature of -10° is 'bigger' in magnitude than +5°.

Notation

|x| denotes the magnitude (absolute value) of x

What This Formula Means

The size or absolute value of a quantity, considering only how large it is and ignoring direction or sign.

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

Formal View

|x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases} satisfying |x| \geq 0, |xy| = |x||y|, and the triangle inequality |x + y| \leq |x| + |y|

Worked Examples

Example 1

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

Solution

  1. 1
    |-9| = 9 because -9 is 9 units from zero on the number line.
  2. 2
    |7| = 7 because 7 is already 7 units from zero (positive, so unchanged).
  3. 3
    |0| = 0 because 0 is 0 units from zero.

Answer

|-9|=9, \quad |7|=7, \quad |0|=0
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 or 12? Then determine which is greater as a signed number.

Common Mistakes

  • Saying -3 has a smaller magnitude than 2 because -3 < 2 — magnitude ignores sign, so |-3| = 3 > 2
  • Confusing magnitude with the number itself — the magnitude of -8 is 8, not -8
  • Thinking magnitude can be negative — magnitude (absolute value) is always non-negative

Why This Formula Matters

Magnitude is essential for comparing sizes, understanding absolute value, and measuring distances. It underpins physics (force magnitude), engineering (signal strength), and finance (size of gains or losses regardless of direction).

Frequently Asked Questions

What is the Magnitude formula?

The size or absolute value of a quantity, considering only how large it is and ignoring direction or sign.

How do you use the Magnitude formula?

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

What do the symbols mean in the Magnitude formula?

|x| denotes the magnitude (absolute value) of x

Why is the Magnitude formula important in Math?

Magnitude is essential for comparing sizes, understanding absolute value, and measuring distances. It underpins physics (force magnitude), engineering (signal strength), and finance (size of gains or losses regardless of direction).

What do students get wrong about Magnitude?

Thinking -10 is 'less' than 5 in every sense (position yes, magnitude no).

What should I learn before the Magnitude formula?

Before studying the Magnitude formula, you should understand: more less, integers.

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