Absolute Value – Distance from Zero

The absolute value of a number is its distance from zero on the number line – always a positive number (or zero), regardless of whether the original number was positive or negative. It is written with two vertical bars: |n|. Since distance can never be negative, |5| = 5 and |-5| = 5 as well – both are exactly 5 units from zero.

The vertical bar notation for absolute value was introduced by the German mathematician Karl Weierstrass in 1841, as part of his broader effort to make 19th-century mathematics more rigorous and precisely defined. Before Weierstrass's notation became standard, mathematicians had various ad hoc ways of describing “the size of a number, ignoring its sign” – his simple two-bar symbol proved so clear and useful that it spread throughout mathematics and is still used exactly the same way today.

Evaluating Absolute Value

|n| always equals n if n is positive or zero, and equals -n (which flips a negative to positive) if n is negative.

Find |-17|.

-17 is 17 units from zero, so |-17| = 17.

Comparing Absolute Values

Which has the greater absolute value: -12 or 8?

|-12| = 12 and |8| = 8. 12 is greater than 8, so -12 has the greater absolute value – even though -12 is the smaller number.

Real-Life Application

Speed: the magnitude of velocity, ignoring direction.
Error margins: how far a measurement is off, regardless of over or under.
Distance travelled: total ground covered, regardless of direction.

Key Takeaways

|n| is the distance of n from zero on the number line, and is always positive or zero.
Karl Weierstrass introduced the |n| notation in 1841.
A number's absolute value can be larger than another number even if the number itself is smaller (e.g. -12 vs 8).

Practice: Absolute Value

Evaluate the Absolute Value