In mathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an ordering (or ranking) of the class of objects to which it belongs. Magnitude as a concept dates to Ancient Greece and has been applied as a measure of distance from one object to another. For numbers, the absolute value of a number is commonly applied as the measure of units between a number and zero.
In vector spaces, the Euclidean norm is a measure of magnitude used to define a distance between two points in space. In physics, magnitude can be defined as quantity or distance. An order of magnitude is typically defined as a unit of distance between one number and another's numerical places on the decimal scale.
Ancient Greeks distinguished between several types of magnitude,[1] including:
They proved 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 primarily used in contexts in which zero is either the smallest size or less than all possible sizes.
See main article: Absolute value.
x
|x|
The absolute value of a real number r is defined by:[3]
\left|r\right|=r,ifr≥0
\left|r\right|=-r,ifr<0.
Absolute value may also be thought of as the number's distance from zero on the real number line. For example, the absolute value of both 70 and −70 is 70.
A complex number z may be viewed as the position of a point P in a 2-dimensional space, called the complex plane. The absolute value (or modulus) of z may be thought of as the distance of P from the origin of that space. The formula for the absolute value of is similar to that for the Euclidean norm of a vector in a 2-dimensional Euclidean space:[4]
\left|z\right|=\sqrt{a2+b2}
\sqrt{(-3)2+42}=5
\bar{z}
z=a+bi
\bar{z}=a-bi
\left|z\right|=\sqrt{z\bar{z}}=\sqrt{(a+bi)(a-bi)}=\sqrt{a2-abi+abi-b2i2}=\sqrt{a2+b2}
i2=-1
A Euclidean vector represents the position of a point P in a Euclidean space. 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 numbers (the Cartesian coordinates of P): x = [''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>]. Its magnitude or length, denoted by
\|x\|
\|x\|=
| 2 | |
| \sqrt{x | |
| 1 |
+
| 2 | |
| x | |
| 2 |
+ … +
| 2}. | |
| x | |
| n |
\sqrt{32+42+122}=\sqrt{169}=13.
\|x\|=\sqrt{x ⋅ x
The Euclidean norm of a vector is just a special case of Euclidean distance: the distance between its tail and its tip. Two similar notations are used for the Euclidean norm of a vector x:
\left\|x\right\|,
\left|x\right|.
See main article: Normed vector space.
By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstract vector space does not possess a magnitude.
A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space. The norm of a vector v in a normed vector space can be considered to be the magnitude of v.
In a pseudo-Euclidean space, the magnitude of a vector is the value of the quadratic form for that vector.
When comparing magnitudes, a logarithmic scale is often used. Examples include the loudness of a sound (measured in decibels), the brightness of a star, and the Richter scale of earthquake intensity. Logarithmic magnitudes can be negative. In the natural sciences, a logarithmic magnitude is typically referred to as a level.
See main article: Order of magnitude.
Orders of magnitude denote differences in numeric quantities, usually measurements, by a factor of 10—that is, a difference of one digit in the location of the decimal point.