The absolute value of a number can be defined as the measure of that number’s distance to zero. This makes consideration for sign (positive vs. negative) redundant in that absolute value is a positive measure only. A measure of negative five is still *five* measures away from zero—regardless of direction.

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The first formal mention of absolute value is found in an 1857 mathematics book where it is used to distinguish *absolute* values from *relative* values. Absolute value is sometimes referred to as the *numerical *value or the *magnitude* depending on context.

Absolute value measures are useful in any application where the magnitude of change is measured—physics, frequency analysis, and even securities analysis. Absolute value have some complex implications but knowing the basics of its use stands to benefit all.

## Formula

The formal definition of absolute value represents the range of numbers (negatives) of which will have their value inversed and the range of numbers (positives) that will retain sign. This is expressed in the following notation:

## Properties

Absolute value bears many implications. These echo into the addition, subtraction, multiplication, and even *integration* of absolute values. This definition of absolute value is not intended to be canonical so I’ll simply leave you with the following:

I suggest referencing the Math is Fun definition of absolute value for a deeper dive. I find their animations, graphics, and simplistic explanations extremely helpful.

## Example

Absolute value is denoted by the use of two vertical bars surrounding the value of concern:

Simple yet true.

## Final Thoughts

The absolute value is an incredibly *simple yet useful* function to measure a value’s distance from zero. It makes relative comparisons of change a breeze, can be useful in analyzing security volatility and helps wrangle spatial problems involving vector calculations.