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Absolute Value

April 8th, 2009 |

Author: Merrin

Absolute value is related to the distance between two points on a number line. We want to measure the distance as a positive quantity. Suppose we have two values x and y. Then the distance is y-x if y > x and the distance is x-y if x > y. Absolute values allow us to bunch these two cases into one system. We can write

$\left| x-y \right| = x-y \quad x>y$

$\left| x-y \right| = -x+y \quad x<y$

Absolute value can occur in other contexts besides distance on the real line. Suppose we take

$\sqrt{x^2}$

This will have give us the result |x|. This is because the square root is always defined to be positive. Even though you can input a negative x the result has to be positive. Working with absolute values is usually simplified by breaking the expression down into two equations for each case of the absolute value. Expressions such as

$(x^4)^{1/2}=x^2$