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The real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist. The subdifferential of | x | at x = 0 is the interval [−1, 1]. [14] The complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann ...
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 ((), ()) = = ((), ()).
The real numbers form a metric space: the distance between x and y is defined as the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology ; the topology arising from the metric and the one arising from the order are identical, but yield different presentations for the topology—in the order ...
Restricting the standard absolute value on to gives an archimedean absolute value on ; such an absolute value is also referred to as a real place of . On the other hand, the roots of factors of degree two are pairs of conjugate complex numbers, which allows for two conjugate embeddings into C {\displaystyle \mathbb {C} } .
When the ordered abelian group Γ is the additive group of the integers, the associated valuation is equivalent to an absolute value, and hence induces a metric on the field K. If K is complete with respect to this metric, then it is called a complete valued field .
A multiplication by a negative number can be seen as a change of direction of the vector of magnitude equal to the absolute value of the product of the factors. When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The sign of the product is determined by the following rules:
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