Absolute Value: Keeping a Positive Attitude

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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.

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:

absolute value definition
Formalized notation defining the representation of absolute values based on sign

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:

properties of absolute values
Common properties of absolute values.

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:

absolute value notation
The vertical bars on either side of the -5 indicate its absolute value is being represented resulting in a positive value

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.

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Zαck West
Entrepreneur, programmer, designer, and lifelong learner. Can be found taking notes from Mother Nature when not hammering away at the keyboard.