Why is absolute value important?

Answer 1

There are many instances where what matters is only the absolute value and not the sign associated with the number.

Sometime functions, eg log(x), may be defined only for positive values of x and so, to ensure that it is valid, you need to take the positive value.

Some pseudo-functions, like square root, give two solutions: one positive and one negative. The principal value is the positive root or the absolute value of the root.

In statistics, one measure of spread is the distance from the mean. To summarise the result for a number of observations, though, you cannot simply add the deviations: that will always give an answer of zero. One option is to look at the absolute deviation instead. [The squared deviation is preferred because, if the deviations are independent, normally distributed, then the sum of the squares have a chi-square distribution which is well studied.