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 of objects to which it belongs.
The Greeks distinguished between several types of magnitude, 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.
The magnitude of any number is usually called its "absolute value" or "modulus", denoted by |x|.
The absolute value of a real number r is defined by:
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.
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:
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 . 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∗ = a − bi.
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