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It can be defined as follows: For any real number a, the absolute value of a, denoted |a|, is equal to a if a ≥ 0, and to −a, if a < 0 (see also: inequality). |a| is never negative, as absolute values are always either positive or zero. Put another way, |a| < 0 has no solution for a.
The absolute value can be regarded as the distance of a number from zero; indeed the notion of distance in mathematics is a generalisation of the properties of the absolute value. When real numbers are considered as one-dimensional vectors, the absolute value is the magnitude, and the p-norm for any p. Up to a constant factor, every norm in R1 is equal to the absolute value: ||x||=||1||.|x|
The absolute value has the following properties:
The last two properties are often used in solving inequalities; for example:
For real arguments, the absolute value function f(x) = |x| is continuous everywhere and differentiable everywhere except for x = 0. For complex arguments, the function is continuous everywhere but differentiable nowhere (One way to see this is to show that it does not obey the Cauchy-Riemann equationsPartial differential equations Complex analysis In complex analysis, the Cauchy-Riemann differential equations are two partial differential equations which provide a necessary and sufficient condition for a function to be holomorphic. Let f u + iv be a fu).
For a complex number z = a + ib, one defines the absolute value or modulus to be |z| = √(a2 + b2) = √ (z z*) (see square rootIn mathematics, the square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is. For example, since. This example suggests how square roots can arise whe and complex conjugateIn mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. Thus, the conjugate of the complex number z a + ib (where a and b are real numbers) is defined to be z a − ib''. It is also often denoted). This notion of absolute value shares the properties 1-6 from above. If one interprets z as a point in the plane, then |z| is the distance of z to the origin.
It is useful to think of the expression |x − y| as the distance between the two numbers x and y (on the real number line if x and y are real, and in the complex plane if x and y are complex). By using this notion of distance, both the set of real numbers and the set of complex numbers become metric spacesIn mathematics, a metric space is a set (or "space") where a distance between points is defined. History Maurice Frechet introduced metric spaces in his work Sur quelques points du calcul fonctionnel Rendic. Palermo 22(1906) 1-74. Formal definition Formal.
The function is not invertible, because a negative and a positive number have the same absolute value.
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