Magnitude — Definition, Formula & Examples

Magnitude

The amount of a quantity. Magnitude is never negative.

See also

Key Formula

|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \qquad \text{and} \qquad \|\vec{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}

Where:

$|x|$ = Absolute value (magnitude) of a real number x
$\|\vec{v}\|$ = Magnitude (length) of a vector v
$v_1, v_2, \ldots, v_n$ = Components of the vector

Worked Example

Problem: Find the magnitude of the number −7 and the magnitude of the vector ⟨3, −4⟩.

Step 1: For the real number −7, magnitude means absolute value. Since −7 is negative, negate it to get the magnitude.

|-7| = 7

Step 2: For the vector ⟨3, −4⟩, magnitude means the length of the vector. Apply the formula using its components.

\|\langle 3, -4 \rangle\| = \sqrt{3^2 + (-4)^2}

Step 3: Compute the squares and add them.

= \sqrt{9 + 16} = \sqrt{25}

Step 4: Take the square root.

= 5

Answer: The magnitude of −7 is 7, and the magnitude of the vector ⟨3, −4⟩ is 5. Both results are non-negative.

Frequently Asked Questions

Is magnitude the same as absolute value?

For real numbers, yes — magnitude and absolute value mean the same thing. For vectors and complex numbers, magnitude refers to the length or modulus, which generalizes the idea of absolute value to multiple dimensions. In every case, the result is non-negative.

Can magnitude ever be zero?

Yes. Magnitude is zero when the quantity itself is zero. For example, |0| = 0, and the zero vector ⟨0, 0⟩ has magnitude 0. Magnitude cannot be negative, but zero is perfectly valid.

Magnitude vs. Direction

Magnitude tells you how large a quantity is (the 'how much'), while direction tells you where it points (the 'which way'). A vector like ⟨3, −4⟩ has both: a magnitude of 5 and a specific direction. A scalar, by contrast, has magnitude but no direction. Together, magnitude and direction fully describe a vector.

Why It Matters

Magnitude appears throughout mathematics and science whenever you need to describe the size of something without worrying about sign or direction. In physics, speed is the magnitude of velocity, and distance is the magnitude of displacement. In everyday math, comparing magnitudes lets you determine which of two quantities is larger in an absolute sense, which is essential in error analysis, optimization, and measurement.

Common Mistakes

Mistake: Stating that the magnitude of a negative number is negative (e.g., saying the magnitude of −5 is −5).

Correction: Magnitude is always non-negative. The magnitude of −5 is 5. Applying the absolute value removes the sign.

Mistake: Adding vector components directly instead of using the square root of the sum of squares (e.g., saying the magnitude of ⟨3, 4⟩ is 3 + 4 = 7).

Correction: Use the Pythagorean-based formula: √(3² + 4²) = √25 = 5. Simply adding components does not give the correct length.

Related Terms

Magnitude of a Vector — Specific formula for vector magnitude
Vector — Quantity described by magnitude and direction
Scalar — Quantity with magnitude but no direction
Absolute Value — Magnitude of a real number
Distance Formula — Uses magnitude to find distance between points
Norm — Generalized notion of magnitude in linear algebra