Magnitude (mathematics)

The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs.

The Greeks distinguished between several types of magnitude, including:

• (positive) fractions
• line segments (ordered by length)
• Plane figures (ordered by area)
• Solids (ordered by volume)
• Angles (ordered by angular magnitude)

They had proven that the first two could not be the same, or even isomorphic systems of magnitude. They did not consider negative magnitudes to be meaningful, and magnitude is still chiefly used in contexts in which zero is either the lowest size or less than all possible sizes.

Real numbers

The magnitude of a real number is usually called the absolute value or modulus. It is written | x |, and is defined by:

| x | = x, if x ≥ 0

| x | = −x, if x < 0.

This gives the number's distance from zero on the real number line. For example, the modulus of −5 is 5.

Complex numbers

Similarly, the magnitude of a complex number, called the modulus, gives the distance from zero in the Argand diagram. The formula for the modulus is the same as that for Pythagoras' theorem.

where ℜ(z) and ℑ(z) are the respectively real part and imaginary part of z. For instance, the modulus of −3 + 4i is 5.

Euclidean vectors

The magnitude of a vector x of real numbers in a Euclidean n-space is most often the Euclidean norm, derived from Euclidean distance: the square root of the dot product of the vector with itself:

where x = [x₁, x₂, ..., x_n]. For instance, the magnitude of [4, 5, 6] is √(4² + 5² + 6²) = √77 or about 8.775.

Two similar notations are used for the magnitude or Euclidean norm of a vector x:

However, the second notation is generally discouraged, because it is also used to denote the absolute value of scalars and the determinants of matrices.

General vector spaces

By definition, all Euclidean vectors have a magnitude (see above). More generally, however, the notion of magnitude cannot be applied to all kinds of vectors.

A function that maps objects to their magnitudes is called a norm. A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space. In high mathematics, not all vector spaces are normed.

Practical math

A magnitude is never negative. When comparing magnitudes, it is often helpful to use a logarithmic scale. Real-world examples include the loudness of a sound (decibel), the brightness of a star, or the Richter scale of earthquake intensity.

To put it another way, often it is not meaningful to simply add and subtract magnitudes.