Magnitude (mathematics)

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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:

They had proven that the first two could not be the same, or even isomorphic systems of magnitude.

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Encyclopedia

(positive) fractions
line segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
s (ordered by length
Length

Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end....
)
Plane figures (ordered by area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
)
Solids (ordered by volume
Volume

The volume of any solid, liquid, plasma, vacuum or theoretical object is how much three-dimensional space it occupies, often quantified numerically....
)
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
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
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

Number line

In mathematics, a number line is a picture of a straight line on which every point corresponds to a real number and every real number to a point....
. For example, the modulus of −5 is 5.

Complex numbers

Similarly, the magnitude of a complex number

Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
, called the modulus
Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
, 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

Real part

In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i.e. if , or equivalently, , then the real part of is ....
and imaginary part

Imaginary part

In mathematics, the imaginary part of a complex number , is the second element of the ordered pair of real numbers representing i.e. if , or equivalently, , then the imaginary part of is ....
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

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
is most often the Euclidean 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....
, derived from 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, which can be proven by repeated application of the Pythagorean theorem....
: 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, also known as the scalar product, is an operation which takes two vector over the real numbers R and returns a real-valued scalar quantity....
of the vector with itself:

where x = [x₁, x₂, ..., x_n]. For instance, the magnitude of [4, 5, 6] is v(4² + 5² + 6²) = v77 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

Absolute value

In mathematics, the absolute value of a real number is its numerical value without regard to its Negative and non-negative numbers. So, for example, 3 is the absolute value of both 3 and -3....
of scalars and the determinant

Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
s 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

Norm (mathematics)

Vector space

File:Vector addition ans scaling.pngA vector space is a mathematical structure formed by a collection of vectors: objects that may be Vector addition together and Scalar multiplication by numbers, called scalar s in this context....
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 Vector s with real number-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any Vector space Rn....
. 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

In mathematics, the logarithm of a number to a given base is the Power or exponent to which the base must be raised in order to produce the number....
ic scale. Real-world examples include the loudness

Loudness

Loudness is the quality of a sound that is the primary psychological correlate of physical strength .Loudness, a subjective measure, is often confused with objective measures of sound pressure such as decibels or sound intensity....
of a sound

Sound

Sound is vibration transmitted through a solid, liquid, or gas, composed of frequencies within the range of hearing and of a threshold of hearing to be heard, or the sensation stimulated in organs of hearing by such vibrations....
(decibel

Decibel

The decibel is a logarithmic units of measurement that expresses the magnitude of a physical quantity relative to a specified or implied reference level....
), 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 ball of Plasma that is held together by its own gravity. 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 Richter magnitude scale, or more correctly local magnitude ML scale, assigns a single number to quantify the amount of moment magnitude scale#Radiated seismic energy released by an earthquake....
of earthquake intensity.

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

Addition

Addition is the mathematics process of putting things together. The plus sign "+" means that numbers are added together. For example, in the picture on the right, there are 3 + 2 apples?meaning three apples and two other apples?which is the same as five apples, since 3 + 2 = 5....
and subtract magnitudes.