Mathematical Journeys: The Three Types of Symbols

We're going back to basics today with a Math Journey covering the three broad categories of symbols. I've found this concept very handy when introducing Algebra to middle school students. So let's go!

Math is a language, and I find it often helps to think of it as such right from the beginning. Just as there are different parts of speech in a language, so there are different 'parts of speech' in math. Where a spoken language includes parts of speech such as nouns, verbs, and adjectives, math has three major types of symbols: constants, operators, and variables. Let's go over each one in detail.

These would be the equivalent of your nouns. A Constant is a number – it has a single, discrete place on the number line. Even if the number itself is ugly – a non-terminating decimal, for example – it still does exist in a specific spot somewhere on the number line. In addition to the obvious constants, math frequently uses what I refer to as 'special constants' or 'named constants' – ugly numbers that are important enough for some reason or other that mathematicians have given them special names and symbols. Pi is a good example of this; mathematicians figured out that performing a specific calculation on a circle always yields the same number, regardless of which circle is used, and figured that that number was special enough to warrant a name. In much the same way, other constants such as e and i have been given names and special symbols to represent them due to their importance for certain calculations. But the important thing to remember here is that all of these named constants do have specific spots on the number line – they don't change value depending on the situation. Pi will always be approximately 3.14159, no matter what you do to the rest of the problem.

But that's not always the case.

Variables also represent constants, however in this case the actual value of the constant is unknown. The variable does have a specific spot on the number line, but we don't know where it is. Its location on the number line can vary from problem to problem, but within a single problem it is always consistent. We generally refer to variables using lowercase letters, traditionally starting with x, y, and z, and then moving to others if necessary. In practice, a variable behaves just like a constant, since it does actually represent a constant. It can be manipulated the same way you would a constant, except of course you don't know the value so you'll have to leave some calculations unfinished until you get to a point where you can identify the mystery number. Funnily enough, many elementary math programs use the concept of variables, but they don't define them as such. If you've ever seen a basic math worksheet with a question mark in a problem, you've seen a variable. All algebra does is change over from using a question mark to a series of lowercase letters.

3 + ? = 7
3 + x = 7

All the constants and variables in the world won't help us without an operator. Remember how your grammar teacher was always going on about how every sentence needs a verb? Well, every mathematical sentence or phrase needs an operator. An operator is a symbol that performs an action on a constant or set of constants. Plus signs, minus signs, multiplication and division symbols are all operators, but so are square root bars and fraction bars. In fact, as you may have read in one of my earlier Math Journeys, a fraction is just an indication of the top constant being divided by the bottom constant. The equals sign is also an operator of sorts, though it doesn't perform an action on the constants so much as declare a relationship between them. The greek letter Sigma is an operator as well, used to represent taking the sum of a series.

And then there's the special operator known as 'a function.' We've talked about functions multiple times before in my blog, and I usually introduce it as a machine that turns one number into another by applying a set rule. That sounds like an operator to me! The key here is that a function is kind of a general operator, one that you can define within a given problem any way you want or need it to be. Want to indicate a specific sequence of operations performed on a number repeatedly over the course of a single problem? Use a function and define it appropriately!

Breaking down the world of math symbols in this way helps to clear up some of the confusion that often results from the particulars of traditional naming conventions. Consider, for instance, the following six symbols:

e         ∏
x         ?
f()       ∑

All three in the first column are lowercase letters, and all three in the second column are greek symbols. However, their usage in math is better represented by the horizontal rows. The first row are constants, the second variables, and the third operators. And the way they behave differs accordingly. So the next time you're confused, take a look at which type of symbol you're working with!


This is awesome except that pi's not ugly!
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