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'The '''magnitude''' of a mathematical object is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an [[ordering]] (or ranking) of the [[class (mathematics)|class]] of objects to which it belongs. The Greeks distinguished between several types of magnitude, including: *(positive) [[fractions]] *[[line segment]]s (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. == 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'' | = &minus;''x'', if ''x'' &lt; 0. This gives the number's distance from zero on the real [[number line]]. For example, the modulus of &minus;5 is 5. == Complex numbers == Similarly, the magnitude of a [[complex number]], called the '''[[absolute value|modulus]]''', gives the distance from zero in the [[Argand diagram]]. The formula for the modulus is the same as that for [[Pythagoras' theorem]]. :<math> \left| z \right| = \sqrt{\Re(z)^2 + \Im(z)^2 }</math> where &real;(''z'') and &image;(''z'') are the respectively [[real part]] and [[imaginary part]] of ''z''. For instance, the modulus of &minus;3 + 4<var>''i''</var> is 5. poooooooooooooooooooooooooooop == Euclidean vectors == The magnitude of a [[Vector (geometric)|vector]] '''x''' of real numbers in a [[Euclidean space|Euclidean <var>n</var>-space]] is most often the [[Norm (mathematics)#Euclidean norm|Euclidean norm]], derived from [[Euclidean distance]]: the [[square root]] of the [[dot product]] of the vector with itself: :<math>\|\mathbf{x}\| := \sqrt{x_1^2 + \cdots + x_n^2}.</math> where '''x''' = [''x''₁, ''x''₂, ..., ''x''_''n'']. For instance, the magnitude of [4, 5, 6] is √(4² + 5² + 6²) = √77 or about 8.775. Two similar notations are used for the magnitude or Euclidean norm of a vector '''x''': #<math>\left \| \mathbf{x} \right \|,</math> #<math>\left | \mathbf{x} \right |.</math> However, the second notation is generally discouraged, because it is also used to denote the [[absolute value]] of scalars and the [[determinant]]s of matrices. == General vector spaces == By definition, all Euclidean vectors have a magnitude (see above). More generally, 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 (mathematics)|norm]]. A [[vector space]] endowed with a norm, such as the Euclidean space, is called a [[normed vector space]]. In high mathematics, not all vector spaces are normed. == Practical math == A magnitude is never negative. When comparing magnitudes, it is often helpful to use a [[logarithm]]ic scale. Real-world examples include the [[loudness]] of a [[sound]] ([[decibel]]), the [[brightness]] of a [[star]], or the [[Richter magnitude scale|Richter scale]] of earthquake intensity. To put it another way, often it is not meaningful to simply [[addition|add]] and [[subtract]] magnitudes. ==See also== * [[Number sense]] ==References== {{reflist}} {{Unreferenced|date=August 2009}} [[Category:Elementary mathematics]] [[ar:مقدار (رياضيات)]] [[es:Magnitud (matemática)]] [[ru:Величина]] [[simple:Magnitude (mathematics)]] [[th:ขนาด (คณิตศาสตร์)]] [[zh:量]]'