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Magnitude (mathematics)

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Magnitude (mathematics)

In mathematics, magnitude is the size of a mathematical object, a property by which the object can be compared as larger or smaller than other objects of the same kind. More formally, an object's magnitude is an ordering (or ranking) of the class (mathematics) of objects to which it belongs.

History

The 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 chiefly used in contexts in which zero is either the lowest size or less than all possible sizes.

Numbers

The magnitude of any number is usually called its "absolute value" or "modulus", denoted by |x|.

Real numbers

The absolute value of a real number r is defined by:[2]

\left| r \right| = r, \text{ if } r \text{ ≥ } 0
\left| r \right| = -r, \text{ if } r < 0 .

It may be thought of as the number's distance from zero on the real number line. For example, the absolute value of both 7 and −7 is 7.

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:[3]

\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 \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, where for any complex number z = a + bi, its complex conjugate is z = abi.

\left| z \right| = \sqrt{zz^* } = \sqrt{(a+bi)(a-bi)} = \sqrt{a^2 -abi + abi - b^2i^2} = \sqrt{a^2 + b^2 }

( recall i^2 = -1 )

Euclidean vectors

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 is most commonly defined as its Euclidean norm (or Euclidean length):[4]

\|\mathbf{x}\| := \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}.

For instance, in a 3-dimensional space, the magnitude of [4, 5, 6] is √(42 + 52 + 62) = √77 or about 8.775. This is equivalent to the square root of the dot product of the vector by itself:

\|\mathbf{x}\| := \sqrt{\mathbf{x} \cdot \mathbf{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:

  1. \left \| \mathbf{x} \right \|,
  2. \left | \mathbf{x} \right |.

A disadvantage to the second notation is that it is also used to denote the absolute value of scalars and the determinants of matrices and therefore its meaning can be ambiguous.

Normed vector spaces

By definition, all Euclidean vectors have a magnitude (see above). 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. A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space.[5] In high mathematics, not all vector spaces are normed.

Logarithmic magnitudes

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. Logarithmic magnitudes can be negative. It is usually not meaningful to simply add or subtract them.

"Order of magnitude"

In advanced mathematics, as well as colloquially in popular culture, especially geek culture, the phrase "order of magnitude" is used to denote a change in a numeric quantity, usually a measurement, by a factor of 10; that is, the moving of the decimal point in a number one way or the other, possibly with the addition of significant zeros.[6]

Occasionally the phrase "half an order of magnitude" is also used, generally in more informal contexts. Sometimes, this is used to denote a 5 to 1 change, or alternatively 101/2 to 1 (approximately 3.162 to 1).

See also

References

  1. ^  
  2. ^ Mendelson, Elliott, Schaum's Outline of Beginning Calculus, McGraw-Hill Professional, 2008. ISBN 978-0-07-148754-2, page 2
  3. ^ Ahlfors, Lars V.: Complex Analysis, Mc Graw Hill Kogakusha, Tokyo (1953)
  4. ^ Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International 
  5. ^ Golan, Johnathan S. (January 2007), The Linear Algebra a Beginning Graduate Student Ought to Know (2nd ed.), Springer,  
  6. ^ Brians, Paus. "Orders of Magnitude". Retrieved 5/9/2013. 
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