Magnitude Formula

The magnitude formula depends on what you're measuring: for a vector v = a, b, magnitude is |v| = sqrt(a^2 + b^2).

The Formula

∣x∣={xif x≥0−xif x<0|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 −7-7 and +7+7 have magnitude 7. Temperature of −10°-10° is 'bigger' in magnitude than +5°+5°.

Notation

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

What This Formula Means

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.

Formal View

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

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?

Common Mistakes

  • Reporting a magnitude as negative - size and distance are never negative; drop the sign.
  • Forgetting magnitude discards direction - 5 east and 5 west have the same magnitude.
  • Adding vector components instead of using a2+b2\sqrt{a^2+b^2} - magnitude of (a,b)(a,b) is the Pythagorean length, not a+ba+b.

Why This Formula Matters

Magnitude separates 'how much' from 'which way', a split that runs through physics (speed vs velocity), distance, and error. It is also always non-negative, which is the key fact that distinguishes it from a signed coordinate. Recognizing it by "Am I asking how big or how far, with the sign or direction thrown away (so the answer can't be negative)?" — rather than by familiar numbers — is what lets a student tell it apart from absolute value and signed integer / coordinate and vector itself in a mixed problem set.

Frequently Asked Questions

What is the Magnitude formula?

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

Why is the Magnitude formula important in Math?

Magnitude separates 'how much' from 'which way', a split that runs through physics (speed vs velocity), distance, and error. It is also always non-negative, which is the key fact that distinguishes it from a signed coordinate. Recognizing it by "Am I asking how big or how far, with the sign or direction thrown away (so the answer can't be negative)?" — rather than by familiar numbers — is what lets a student tell it apart from absolute value and signed integer / coordinate and vector itself in a mixed problem set.

What do students get wrong about Magnitude?

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.

What should I learn before the Magnitude formula?

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

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