absolute value
The distance between a real number and zero on a number line.
The absolute value of a real number x, denoted by |x|, is defined as the distance between x and 0 on the number line. More formally, we define:
| x | = {
x, if x >= 0
-x, if x < 0
}
In other words, the absolute value of a positive number is the number itself, and the absolute value of a negative number is its opposite (i.e., the same magnitude but with the opposite sign). The absolute value of zero is also defined to be zero, since zero is equidistant from both positive and negative numbers on the number line.
The absolute value function is continuous for all real numbers, meaning that small changes in x result in small changes in the absolute value of x. It is also symmetric around the origin, since |x| = |-x| for any real number x.
The absolute value has many useful properties in mathematics and is commonly used in various fields such as calculus, linear algebra, and number theory. For example, the absolute value can be used to define the distance between two points in a plane or to measure the error between an approximation and a true value. It is also used to define the norms of vectors in linear algebra, which are important tools in the study of vector spaces and matrices.
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