Search results
Results from the WOW.Com Content Network
The absolute value of a number may be thought of as its distance from zero. In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if is a positive number, and if is negative (in which case negating makes positive), and .
Ostrowski's theorem states that the nontrivial places of the rational numbers Q are the ordinary absolute value and the p-adic absolute value for each prime p. [3] For a given prime p , any rational number q can be written as p n ( a / b ), where a and b are integers not divisible by p and n is an integer.
The field of the rational numbers can be assigned one of a number of absolute value functions, including the trivial function | | =, when , the more usual | | =, and the -adic absolute value functions. By Ostrowski's theorem, every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or some -adic ...
A continuous function on the closed interval showing the absolute max (red) and the absolute min (blue). In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed and bounded interval , then must attain a maximum and a minimum, each at least once.
The first of these quadratic inequalities requires r to range in the region beyond the value of the positive root of the quadratic equation r 2 + r − 1 = 0, i.e. r > φ − 1 where φ is the golden ratio. The second quadratic inequality requires r to range between 0 and the positive root of the quadratic equation r 2 − r − 1 = 0, i.e. 0 ...
Absolute convergence. In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series is said to converge absolutely if for some real number Similarly, an improper integral of a function, is said to ...
This section gives two proofs of the following theorem: Cauchy–Schwarz inequality — Let u {\displaystyle \mathbf {u} } and v {\displaystyle \mathbf {v} } be arbitrary vectors in an inner product space over the scalar field F , {\displaystyle \mathbb {F} ,} where F {\displaystyle \mathbb {F} } is the field of real numbers R {\displaystyle ...
The graph of the absolute value function. If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold. Consider the absolute value function = | |, [,]. Then f (−1) = f (1), but there is no c between −1 and 1 for which the f ′(c) is zero.