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Information about Magnitude (mathematics)

For other senses of this word, see magnitude.


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. 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 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. For example, the modulus of −5 is 5.

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.



where ℜ(z) and ℑ(z) are the real part and imaginary part 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 is most often the Euclidean norm, derived from Euclidean distance: the square root of the dot product of the vector with itself:

where x = [x1, x2, ..., xn]. 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.

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.

To put it another way, often it is not meaningful to simply add and subtract magnitudes.
Magnitude may refer to:
  • Magnitude (mathematics), a measure of the size of a mathematical object:
  • A vector object has both magnitude and direction as its defining characteristics.

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Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering. This article gives a detailed introduction to the field and includes some of the most basic definitions.
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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.
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fraction (from the Latin fractus, broken) is a concept of a proportional relation between an object part and the object whole. Each fraction consists of a denominator (bottom) and a numerator (top), representing (respectively) the number of equal parts that an object is
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line segment is a part of a line that is bounded by two end points, which have a finite length, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square.
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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. This may be distinguished from height, which is vertical extent, and width or breadth
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Area is a physical quantity expressing the size of a part of a surface. The term Surface area is the summation of the areas of the exposed sides of an object.

Units

Units for measuring surface area include:
square metre = SI derived unit

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The volume of a solid object is the three-dimensional concept of how much space it occupies, often quantified numerically. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.
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In mathematics, an isomorphism (Greek: isos "equal", and morphe "shape") is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e., structure-preserving mappings.
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In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.
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A number line, invented by John Wallis, is a one-dimensional picture in which the integers are shown as specially-marked points evenly spaced on a line. Although this image only shows the integers from -9 to 9, the line includes all real numbers, continuing "forever" in each
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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.
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In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3.
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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.
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In mathematics, the Pythagorean theorem (AmE) or Pythagoras' theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and
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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 . It is denoted by Re or , where is a capital R in the Fraktur typeface.
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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 . It is denoted by Im or , where is a capital I in the Fraktur typeface.
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spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. A vector can be thought of as an arrow in Euclidean space, drawn from an initial point A pointing to a terminal point B.
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Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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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.
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In mathematics, a square root of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x.
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dot product, also known as the scalar product, is an operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. It is the standard inner product of the Euclidean space.
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Rn. It turns out that the following properties of "vector length" are the crucial ones.
  1. The zero vector, 0, has zero length; every other vector has a positive length.

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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.
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logarithm (to base b) of a number x is the exponent y that satisfies x = by. It is written logb(x) or, if the base is implicit, as log(x).
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Loudness is the quality of a sound that is the primary psychological correlate of physical strength (amplitude).

Loudness, a subjective measure, is often confused with objective measures of sound pressure such as decibels or intensity.
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Sound is a disturbance of mechanical energy that propagates through matter as a wave (through fluids as a compression wave, and through solids as both compression and shear waves).
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The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power) relative to a specified or implied reference level.
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Brightness is an attribute of visual perception in which a source appears to emit a given amount of light. In other words, brightness is the perception elicited by the luminance of a visual target. This is a subjective attribute/property of an object being observed.
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STAR is an acronym for:

Organizations:
  • Society for Telescopy, Astronomy, and Radio, a non-profit astronomy club in New Jersey
  • Special Tasks and Rescue or Special Tactics and Response, synonyms for SWAT

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