Absolute Value Function

An Absolute Value Function of any real number is always its positive value

• AKA: Absolute Value, Modulus.
• Context:
  • It can be a defined as

[math]\displaystyle{ |x| =\left\{\begin{array}{ll}x \quad\textrm{for} \quad x \geq 0\\-x \quad\textrm{for}\quad x \lt 0\end{array}\right. }[/math]

• The absolute value of vector is its distance from zero, i.e. vector norm
• The absolute value of complex number is complex number modulus
• It can range from being a 0 to being positive number.

• Example(s):
  • |-3|=3
  • [math]\displaystyle{ |(3,0,4)|=\sqrt{3^2+0^2+4^2}=5 }[/math]
  • [math]\displaystyle{ |3+i4|=\sqrt{3^2+4^2}=5 }[/math]
• Counter-Example(s):
  • Positive Value.
  • Negative Value

See: Numeric Function, Positive Value, Negative Value, Complex Number Modulus.

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References

2016

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Absolute_value Retrieved 2016-07-10
  • In mathematics, the absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a positive x, |x| = −x for a negative x (in which case −x is positive), and |0| = 0. 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. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

• (Eric W. Wesstein, 2016) ⇒ Weisstein, Eric W. (1999-2016)"Absolute Value.” From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/AbsoluteValue.html Retrieved 2016-07-10
  • The absolute value of a real number x is denoted |x| and defined as the "unsigned" portion of x,

[math]\displaystyle{ |x|=xsgn(x) }[/math]

where sgn(x) is the sign function. The absolute value is therefore always greater than or equal to 0. (...) The absolute value of a complex number [math]\displaystyle{ z=x+iy }[/math], also called the complex modulus,

2008

• (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised.” Oxford University Press. ISBN:0199541450

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