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. Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. ... In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ...

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. In common usage a fraction is any part of a unit. ... The geometric definition of a 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. ... Length is the long dimension of any object. ... Area is a physical quantity expressing the size of a part of a surface. ... Volume is a quantification of how much space a certain region occupies. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

Real numbers

The magnitude of a real number is usually called the absolute value or modulus. It is written | x |, and is defined by: In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...

| 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. A number line is a one-dimensional graph in which the integers are shown as specially-marked points evenly spaced on a line. ...

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. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... Mathematical meanings Especially in British/European usage, the modulus of a number is its absolute value. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... There are thousands of proofs of the Pythagorean theorem. ...

where and are the real part and imaginary part of z. For instance, the modulus of −3 + 4i is 5. In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , i. ... In mathematics, the imaginary part of a complex number z is the second element of the ordered pair of real numbers representing z, i. ...

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: In physics and in vector calculus, a spatial vector is a concept characterized by a magnitude, which is a scalar, and a direction (which can be defined in a 3-dimensional space by the Euler angles). ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, the Euclidean distance or Euclidean metric is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ... In mathematics, a square root of a number x is a number whose square (the result of multiplying the number by itself) is x. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...

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

General vector spaces

A concept of magnitude can be applied to a vector space in general. This is then called a normed vector space. The function that maps objects to their magnitudes is called a norm. Link titleIn mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

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. Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... Loudness is the quality of a sound which is high in volume (amplitude, or sound pressure). ... Sound is a disturbance of mechanical energy that propagates through matter as a wave. ... The decibel (dB) is a measure of the ratio between two quantities, and is used in a wide variety of measurements in acoustics, physics and electronics. ... Brightness is an attribute of visual perception in which a source appears to emit a given amount of light. ... The Pleiades, an open cluster of stars in the constellation of Taurus. ...

To put it another way, often it is not meaningful to simply add and subtract magnitudes. 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... In mathematics, subtraction is one of the four basic arithmetic operations. ...