Absolute Difference
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An Absolute Difference between two numbers is the Absolute Value of the subtraction between the two, i.e. [math]\displaystyle{ |A-B| }[/math] .
- Example(s):
- [math]\displaystyle{ |3-5|= |-5|=5 }[/math]
- Counter-Example(s):
See: Absolute Value, Number, Mean Deviation.
References
2016
- (Wikipedia, 2016) ⇒ http://www.wikiwand.com/en/Absolute_difference Retrieved 2016-07-10
- The absolute difference of two real numbers x, y is given by |x − y|, the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y. It is a special case of the Lp distance for all 1 ≤ p ≤ ∞ and is the standard metric used for both the set of rational numbers Q and their completion, the set of real numbers R.
As with any metric, the metric properties hold:
- The absolute difference of two real numbers x, y is given by |x − y|, the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y. It is a special case of the Lp distance for all 1 ≤ p ≤ ∞ and is the standard metric used for both the set of rational numbers Q and their completion, the set of real numbers R.
- |x − y| ≥ 0, since absolute value is always non-negative.
- |x − y| = 0 if and only if x = y.
- |x − y| = |y − x| (symmetry or commutativity).
- |x − z| ≤ |x − y| + |y − z| (triangle inequality); in the case of the absolute ::difference, equality holds if and only if x ≤ y ≤ z.
- By contrast, simple subtraction is not non-negative or commutative, but it does obey the second and fourth properties above, since x − y = 0 if and only if x = y, and x − z = (x − y) + (y − z).
- The absolute difference is used to define other quantities including the relative difference, the L1 norm used in taxicab geometry, and graceful labelings in graph theory.
When it is desirable to avoid the absolute value function – for example because it is expensive to compute, or because its derivative is not continuous – it can sometimes be eliminated by the identity
- |x − y| < |z − w| if and only if (x − y)2 < (z − w)2.
- This follows since |x − y|2 = (x − y)2 and squaring is monotonic on the nonnegative reals.
- (Eric W. Weisstein, 2016) ⇒ Weisstein, Eric W. "Absolute Difference." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/AbsoluteDifference.html Retrieved 2016-07-10
- The absolute difference of two numbers [math]\displaystyle{ n_1 }[/math] and [math]\displaystyle{ n_2 }[/math] is [math]\displaystyle{ |n_1-n_2| }[/math], where the minus sign denotes subtraction and |x| denotes the absolute value.
2008
- (Upton & Cook, 2008) ⇒ Graham Upton, and Ian Cook. (2008). “A Dictionary of Statistics, 2nd edition revised." Oxford University Press. ISBN:0199541450