Thursday, December 04, 2014

Blog: What is Interval Notation?

Another question for my students: What is Interval Notation?

Short Answer: It's a topic that's worth, at most, two points on the state exams, and that most of you aren't likely to see again. On the other hand, given the number of students I've known who have scored 63 or 64 on that same state exam, ignore this at your own peril.

Another Flippant Short Response: It's actually a very simple topic to grasp if you give your instructor, your facilitator, five minutes to explain and really use that time to understand, not just copy down a few rules in your notebook. Or you can try to figure it out for yourself from the worksheet and then read the textbook when you get home.

Seriously, it's a simple thing.

A couple of days ago, we were talking about Compound Inequalities. Interval notation is another way to express the range of data in the solution to an inequality. Why do we need another way to express this?

This is Algebra. We don't need no steenkin' reasons! However, I like to say that it's shorthand, even if it uses the same number of characters, with the exception of not requiring any underscores. For that matter, as I'm using a simplified form of a mark-up language, the less-than symbol has another meaning, and it actually takes me three keystrokes to ensure that you see this: <.

With Interval notation, there are no < symbols or variables, either. Just the boundaries.

For example, if we needed to represent the compound inequality -3 < x < 5, we could take the two endpoints and stick them in a pair of brackets, like this: [-3, 5].
This means the same thing and seems quicker to write, and doesn't require as much space (or, at least, doesn't seem to, depending on where you're writing it).

Well, that's all fine, but what if you have "less than" instead of "less than or equal to"? Simple, we don't use square brackets; we use parentheses, instead. That means -3 < x < 5 could be written as (-3,5).

Here is where you need to pay attention. I've said in the past that notion is important, and different symbols mean different things. This is one instance where notation can mean two different things. (-3,5) can be a inequality for one variable, such as x, or it can be a point on the co-ordinate plane, using two variables, (x,y). How do you know which? Context!

It's also important to note that you can (and the state will!) mix and match these: -1 < x < 4 would be written as [-1,4).

Finally (for now), what if it isn't a compound inequality. What is you only have, for example, x > 5? In this case, five is the lower boundary, but what is the upper boundary? The graph has an arrow going up to the right, continuing forever until it hits, as Buzz Lightyear might intone, Infinity, and Beyond! ... or infinity, at any rate. This would be written as (5, ∞).

Note that because infinity is not an actual number (that is, x cannot actually equal infinity), it will always have a parenthesis next to it, not a square bracket.

Finally, a proud moment because one of my students put themselves out there and took a chance. I asked where the answer goes to with x < 5. I got blank stares. "Well, think about where it goes off to the right." The student very cautiously hazarded the guess, "negative infinity?".

Absolutely. He had never heard of such a thing before and didn't know it could exist. Then he realized that he didn't know that it couldn't exist. I liked the way he thinks. I hope there's more of that because if you think of it, this isn't any more complicated than the compound inequalities they're representing.

Okay, maybe that's not saying as much as I'd like.

Postscript: For future reference, the infinity symbol (∞) is the ampersand (&), followed by "#8734" (no quotes).

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