**1.**

For example, an absolute value is defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.

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For example, an absolute value is defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces.

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The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

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The term absolute value has been used in this sense from at least 1806 in French and 1857 in English.

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Absolute value is thus always either a positive number or zero, but never negative.

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Absolute value has the following four fundamental properties, that are used for generalization of this notion to other domains:.

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The real absolute value function is a piecewise linear, convex function.

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Real absolute value function has a derivative for every, but is not differentiable at.

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Real absolute value function is an example of a continuous function that achieves a global minimum where the derivative does not exist.

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Complex absolute value function is continuous everywhere but complex differentiable nowhere because it violates the Cauchy–Riemann equations.

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Definition of absolute value given for real numbers above can be extended to any ordered ring.

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