Instant Floppy Math - Interval Notation

This article created 13 December 2019 by Deven Blake


Interval notation is a way to express intervals in terms of numbers.[1]

Take, for example, the inequality x < 5, where x can be any number lower than five. This inequality also represents an interval; x exists within the interval between the lowest number possible and five, excluding negative-infinity (because that unfortunately is not a number) and excluding five (because x has to be less than five). Mathemeticians try to express things like intervals with symbols rather than language, because not all mathematicians speak English (and I can tell you from personal experience that many of my peers exclusively speak English) - we do that by putting intervals in this interval notation.

So, in `x < 5`, the lower exclusive bound is -infinity, and the upper exclusive bound is five. Interval notation dictates exclusive bounds to be marked with parentheses- keep in mind that these parentheses are not grouping symbols, so be careful to not read interval notation like you would a math equation. The bounds face into the equation, so this `x < 5` becomes (-∞, five) - with a comma to separate the bounds.

But what if we had `x <= 5`? How do we differentiate the now inclusive boundary at five from the exclusive boundary? In interval notation brackets are used rather than parentheses for inclusive boundaries, but only for inclusive boundaries. A right bracket may not have a left bracket, or vice versa! This is something worth taking note of; if you're used to writing mathematical equations you may get the itch to pair that parenthesis with another parenthesis. Don't. In sum, this `x <= 5` becomes `(-∞, 5]`, because the lower bound (exclusive) is at negative infinity and the upper bound (inclusive this time) is five.


[1]: Mathwords - Interval Notation