Concept
Absolute value
Overview
In mathematics, the absolute value or modulus of a real number
x
{\displaystyle x}
, denoted
|
x
|
{\displaystyle |x|}
, is the (non-negative) magnitude of
x
{\displaystyle x}
measured without regard to its sign. Namely,
|
x
|
=
x
{\displaystyle |x|=x}
if
x
{\displaystyle x}
is a positive number, and
|
x
|
=
−
x
{\displaystyle |x|=-x}
if
x
{\displaystyle x}
is negative (in which case
−
x
{\displaystyle -x}
is positive), and
|
0
|
=
0
{\displaystyle |0|=0}
. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.
Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
The terms 'two cubits long', 'three cubits long', and so on indicate quantity, the terms 'great' and 'small' indicate relation, for they have reference to an external standard. It is, therefore, plain that these are to be classed as relative.
Aristotle, The Categories
States what Absolute value is.
The notion of continuity depends upon that of _order_, since continuity is merely a particular type of order. Mathematics has, in modern times, brought order into greater and greater prominence. In former days, it was supposed (and philosophers are still apt to suppose) that quantity was the fundamental notion of mathematics. But nowadays, quantity is banished altogether, except from one little corner of Geometry…
Bertrand Russell, Mysticism and Logic and Other Essays
States what Absolute value is.
So that when the scale is moved over the line by any continuous series of shiftings of one kind, there are two points which no numbers on the scale can ever reach, except the numbers fixed there. This pair of points, thus unattainable in measurement, is called the Absolute. These two points may be distinct and real, or they may coincide, or they may be both imaginary.
Charles S. Peirce, Chance, Love, and Logic
States what Absolute value is.
Related ideas
- Mathematics
- Real number
- Oxford English Dictionary
- Pythagorean theorem
- 0
- Complex number
- Continuous function
- Cardinality
- Identity of indiscernibles
- Element of a set
- Karl Weierstrass
- Inverse function
- Maximum and minimum
Place in the tradition
Part of Universe.