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Magnitude (mathematics)

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	Topics Magnitude (mathematics) Topic Home 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: (positive) fractions line segmentLine segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line betw... s (ordered by lengthLength Length is the long dimension of any object.... ) Plane figures (ordered by areaArea Area is a physical quantity expressing the size of a part of a surface.... ) Solids (ordered by volumeVolume *'Volume', also called capacity, is a quantification of how much space a certain region occupies.... ) 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 valueAbsolute value Overview In mathematics, the absolute value of a real number is its numerical value without regard to its sign.... 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 lineNumber line A number line is a one-dimensional picture in which the integers are shown as specially-marked points evenly spaced on a lin... . For example, the modulus of −5 is 5. Complex numbers Similarly, the magnitude of a complex numberFacts About Complex number In mathematics, a complex number is a number of the form ... , called the modulusAbsolute value In mathematics, the absolute value of a real number is its numerical value without regard to its sign.... , 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 respectively real partReal part In mathematics, the real part of a complex number , is the first element of the ordered pair of real numbers representing , ... and imaginary partImaginary part In mathematics, the imaginary part of a complex number , is the second element of the ordered pair of real numbers represent... of z. For instance, the modulus of −3 + 4`i` is 5. Euclidean vectors The magnitude of a vectorVector (spatial) In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a... x of real numbers in a Euclidean `n`-spaceEuclidean space Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "plane Euclidean geometry", wh... is most often the Euclidean normNorm (mathematics) In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive len... , derived from Euclidean distanceFacts About Euclidean distance In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" distance between the two points that one ... : the square rootSquare root In mathematics, a square root of a number x is a number whose square is x.... of the dot productDot product In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over ... of the vector with itself: where x = [x₁, x₂, ..., x_n]. The notation \|x\| is also used for the norm. For instance, the magnitude of [4, 5, 6] is v(4² + 5² + 6²) = v77 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 spaceNormed vector space Summary In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and c... . The function that maps objects to their magnitudes is called a normNorm (mathematics) In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive len... . Practical math A magnitude is never negative. When comparing magnitudes, it is often helpful to use a logarithmFacts About Logarithm The logarithm is the mathematical operation that is the inverse of exponentiation .... ic scale. Real-world examples include the loudnessLoudness Summary Loudness is the quality of a sound that is the primary psychological correlate of physical intensity.... of a soundSound Sound is a disturbance of mechanical energy that propagates through matter as a wave.... , the brightnessBrightness Brightness is an attribute of visual perception in which a source appears to emit a given amount of light.... of a starStar A star is a massive, compact body of plasma in outer space that is held together by its own gravity and, unlike a planet, is... , or the Richter scaleRichter magnitude scale Richter magnitude test scale assigns a single number to quantify the size of an earthquake.... of earthquake intensity. To put it another way, often it is not meaningful to simply addAddition Addition is the mathematical operation of increasing one amount by another.... and subtract magnitudes.