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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.
The absolute value is a norm for the real line; as required, the absolute value satisfies the triangle inequality for any real numbers u and v. If u and v have the same sign or either of them is zero, then | + | = | | + | |. If u and v have opposite signs, then without loss of generality assume | | > | |.
where , is the inner product.Examples of inner products include the real and complex dot product; see the examples in inner product.Every inner product gives rise to a Euclidean norm, called the canonical or induced norm, where the norm of a vector is denoted and defined by ‖ ‖:= , , where , is always a non-negative real number (even if the inner product is complex-valued).
When u and v are real numbers, they can be viewed as vectors in , and the triangle inequality expresses a relationship between absolute values. Pythagorean theorem : It states that the area of the square whose side is the hypotenuse (the side opposite the right angle ) is equal to the sum of the areas of the squares on the other two sides.
Bennett's inequality, an upper bound on the probability that the sum of independent random variables deviates from its expected value by more than any specified amount Bhatia–Davis inequality , an upper bound on the variance of any bounded probability distribution
A power inequality is an inequality containing terms of the form a b, where a and b are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises. Examples: For any real x, +.
These order-theoretic properties lead to a number of fundamental results in real analysis, such as the monotone convergence theorem, the intermediate value theorem and the mean value theorem. However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects.
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 ((), ()) = = ((), ()).