Interval Notation

Have you ever written something, and accidentally added or omitted a comma or other bit of punctuation? If so, you likely noticed how that changed the meaning of what you’d written. The classic example involving Grandma demonstrates the tremendous effect of the tiniest of punctuation:

“Let’s eat, Grandma!”

versus

“Let’s eat Grandma!”

Improper punctuation results in altered meaning and mistakes.

Math, too, has a unique system of punctuation. It also is a system in which accuracy is of the utmost importance. In math, a misplaced decimal might represent the difference between bankruptcy and abundance! As with language, the improper use of math’s punctuation leads to mistakes.

If the grammar of math is its formulas, equations, and the like, its punctuation is its notation. Just as the period is the first bit of written punctuation one learns, interval notation is the foundation of clear mathematical meaning. Interval notation precisely communicates a specific range of mathematical possibilities.

Every numerical range has starting and ending points. The numbers between them, which may or may not include these points, are called a set. An interval is a set representing the real numbers between the first and last numbers.

Perhaps you wish to notate the set of numbers in the current month leading up to and including your birthday. You might wish to indicate all numbers greater than 5. You could desire to write down the numbers less than or equal to 20. No matter the numerical content of a given set, it can be communicated with interval notation.

The easiest way to visualize an interval is with a number line.

To indicate the interval from -4 and 3, color the space above the line that stretches between -4 and 3. You must also communicate whether -4 and 3 are part of the indicated interval. Do this by drawing circles on the notches indicating the relevant numbers. An open circle indicates the number is outside the interval. A closed (colored) circle indicates inclusion. The interval is considered closed when both endpoints are members of it.

A number line isn’t the only way to mathematically depict this interval, however. There are quite a few ways to say the same thing. If the number line were depicting the set of values possible for the variable x, you could achieve the same effect by writing:

-4 ≤ x ≤ 3

An alternate way to express this interval is:

[-4, 3]

The brackets indicate that the number closest to each bracket is within the interval. A parenthsis instead of a bracket would indicate exclusion.

Another way to notate the interval is:

{x ε R | -4 ≤ x ≤ 3}

Translated, this says, “The set of x’s that are members of the set of real numbers such that -4 is less than or equal to x is less than or equal to 3.”

Again, one could write:

X ε R | x ε [-4, 3]

Notate your interval any way you like, but keep in mind one thing: if you get even one small part of your mathematic punctuation wrong, you’ll likely change the meaning of the whole thing!