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.
History Ancient 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 measure) 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.
Numbers
The magnitude of any number x {\displaystyle x} is usually called its absolute value or modulus, denoted by | x | {\displaystyle |x|} .
Real numbers The absolute value of a real number r is defined by:
| r | = r , if r ≥ 0 {\displaystyle \left|r\right|=r,{\text{ if }}r{\text{ ≥ }}0}
| r | = − r , if r < 0. {\displaystyle \left|r\right|=-r,{\text{ if }}r<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.
Complex numbers 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 z = a + bi is similar to that for the Euclidean norm of a vector in a 2-dimensional Euclidean space:
| z | = a 2 + b 2 {\displaystyle \left|z\right|={\sqrt {a^{2}+b^{2}}}}
where the real numbers a and b are the real part and the imaginary part of z, respectively. For instance, the modulus of −3 + 4i is ( − 3 ) 2 + 4 2 = 5 {\displaystyle {\sqrt {(-3)^{2}+4^{2}}}=5} . Alternatively, the magnitude of a complex number z may be defined as the square root of the product of itself and its complex conjugate, z ¯ {\displaystyle {\bar {z}}} , where for any complex number z = a + b i {\displaystyle z=a+bi} , its complex conjugate is z ¯ = a − b i {\displaystyle {\bar {z}}=a-bi} .
| z | = z z ¯ = ( a + b i ) ( a − b i ) = a 2 − a b i + a b i − b 2 i 2 = a 2 + b 2 {\displaystyle \left|z\right|={\sqrt {z{\bar {z}}}}={\sqrt {(a+bi)(a-bi)}}={\sqrt {a^{2}-abi+abi-b^{2}i^{2}}}={\sqrt {a^{2}+b^{2}}}}
(where i 2 = − 1 {\displaystyle i^{2}=-1} ).
Vector spaces
Euclidean vector space
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 = [x1, x2, ..., xn]. Its magnitude or length, denoted by ‖ x ‖ {\displaystyle \|x\|} , is most commonly defined as its Euclidean norm (or Euclidean length):
… excerpt ends here. Continue reading the full article.
