Magnitude (mathematics)
Encyclopedia
The magnitude of an object
Object
Object may refer to:* Object , a thing, being or concept** Entity, something that is tangible and within the grasp of the senses* As used in object relations theories of psychoanalysis, that to which a subject relates....

in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering
Order theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...

(or ranking) of the class of objects to which it belongs.
It also can be termed as a numerical value of that unit (or class) which it belongs.

The Greeks distinguished between several types of magnitude, including:
• (positive) fractions
• line segment
Line segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...

s (ordered by length
Length
In geometric measurements, length most commonly refers to the longest dimension of an object.In certain contexts, the term "length" is reserved for a certain dimension of an object along which the length is measured. For example it is possible to cut a length of a wire which is shorter than wire...

)
• Plane figures (ordered by area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...

)
• Solids (ordered by volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

)
• 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.

## Numbers

The magnitude of any number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

x is usually called its "absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

" or "modulus", denoted by |x|.

### Real numbers

The absolute value of a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

r is defined by:
|r| = r, if r ≥ 0
|r| = -r, if r < 0.

It may be thought of as the number's distance from zero on the real number line
Number line
In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point. Often the integers are shown as specially-marked points evenly spaced on the line...

. For example, the absolute value of both 7 and −7 is 7.

### Complex numbers

A complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

z may be viewed as the position of a point P in a 2-dimensional space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, called the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

. The absolute value of z may be thought of as the distance of P from the origin of that space. The formula for the absolute value of z is similar to that for the Euclidean norm of a vector in a 2-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

:

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

A Euclidean vector represents the position of a point P in a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

. Geometrically, it can be described as an arrow from the origin of the space (vector tail) to that point (vector tip). Mathematically, a vector x in an n-dimensional Euclidean space can be defined as an ordered list of n real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s (the Cartesian coordinates of P): x = [x1, x2, ..., xn]. Its magnitude or length is most commonly defined as its Euclidean norm (or Euclidean length):
For instance, in a 3-dimensional space, the magnitude of [4, 5, 6] is √(42 + 52 + 62) = √77 or about 8.775.
This is equivalent to the square root
Square root
In mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

of the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

of the vector by itself:

The Euclidean norm of a vector is just a special case of Euclidean distance
Euclidean distance
In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. By using this formula as distance, Euclidean space becomes a metric space...

: the distance between its tail and its tip. Two similar notations are used for the Euclidean norm of a vector x:

The second notation is generally discouraged, because it is also used to denote the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

of scalars and the determinant
Determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

s of matrices.

## Normed vector spaces

By definition, all Euclidean vectors have a magnitude (see above). 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
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...

. A vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

endowed with a norm, such as the Euclidean space, is called a normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following properties of "vector length" are crucial....

. 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 logarithm
Logarithm
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

ic scale. Real-world examples include the loudness
Loudness
Loudness is the quality of a sound that is primarily a psychological correlate of physical strength . More formally, it is defined as "that attribute of auditory sensation in terms of which sounds can be ordered on a scale extending from quiet to loud."Loudness, a subjective measure, is often...

of a sound
Sound
Sound is a mechanical wave that is an oscillation of pressure transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a level sufficiently strong to be heard, or the sensation stimulated in organs of hearing by such vibrations.-Propagation of...

(decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...

), the brightness
Brightness
Brightness is an attribute of visual perception in which a source appears to be radiating or reflecting light. In other words, brightness is the perception elicited by the luminance of a visual target...

of a star
Star
A star is a massive, luminous sphere of plasma held together by gravity. At the end of its lifetime, a star can also contain a proportion of degenerate matter. The nearest star to Earth is the Sun, which is the source of most of the energy on Earth...

, or the Richter scale
Richter magnitude scale
The expression Richter magnitude scale refers to a number of ways to assign a single number to quantify the energy contained in an earthquake....

of earthquake intensity.

To put it another way, often it is not meaningful to simply add