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The real and complex absolute values defined above are examples of absolute values for an arbitrary field. If v is an absolute value on F , then the function d on F × F , defined by d ( a , b ) = v ( a − b ) , is a metric and the following are equivalent:
By Ostrowski's theorem, every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or some -adic absolute value. The rational field is not complete with respect to non-trivial absolute values; with respect to the trivial absolute value, the rational field is a discrete topological space, so complete.
The standard absolute value on the integers. The standard absolute value on the complex numbers.; The p-adic absolute value on the rational numbers.; If R is the field of rational functions over a field F and () is a fixed irreducible polynomial over F, then the following defines an absolute value on R: for () in R define | | to be , where () = () and ((), ()) = = ((), ()).
For example, the real numbers form the real line, which is pictured as the horizontal axis of the complex plane, while real multiples of are the vertical axis. A complex number can also be defined by its geometric polar coordinates : the radius is called the absolute value of the complex number, while the angle from the positive real axis is ...
Alternatively, by defining the metric or distance function: using the absolute value function as (,) = | |, the real numbers become the prototypical example of a metric space. The topology induced by metric d {\displaystyle d} turns out to be identical to the standard topology induced by order < {\displaystyle <} .
The real value is the value expressed in terms of purchasing power in the base year. The index price divided by its base-year value / gives the growth factor of the price index. Real values can be found by dividing the nominal value by the growth factor of a price index. Using the price index growth factor as a divisor for converting a nominal ...
Logarithmic number systems (LNSs) represent a real number by the logarithm of its absolute value and a sign bit. The value distribution is similar to floating point, but the value-to-representation curve (i.e., the graph of the logarithm function) is smooth (except at 0). Conversely to floating-point arithmetic, in a logarithmic number system ...
The absolute value of a real number r is defined by: [4] | | =, | | =, < Absolute value may also be thought of as the number's distance from zero on the real number line. For example, the absolute value of both 70 and −70 is 70.