Try living one day without using anyone’s name. Depending on how many people you know and interact with each day, it might be a bit difficult. How would you get your friend’s attention halfway down the hall? “Hey you, there, with the red shirt and blue sneakers! No, no, not you, the other one… with the green purse!” Silly, yes? You would rather say “Hey, Janice!” Names serve a clear purpose: they allow you to easily reference things so that you can interact with them.
And if you think about it, there’s no way we could go through the day without using names for things. At home, you might say “would you mind giving me the cup on the kitchen table?” You would not want to say “would you mind giving me the liquid-holding thing that’s on the wood thing that’s on the floor, but not ON the floor, like on it but above it, you know what I mean, in the room where we cook?” Basic communication would be impossible without names like “cup” and “table” and “kitchen”. It’s really no different in mathematics.
Communication is incredibly important in mathematics. When you solve a problem or prove a theorem, you will want to communicate your accomplishments to others. In order to communicate clearly, you need names for things. Yes, you need names. If you want to ask your friend to graph a curve for you, would you say: “hey, draw that special thing we need to graph this curve, you know, with the two lines that are perpendicular to each other, and then you put numbers on the lines to make, like, a thing of all squares?” We hope not! You might say, “Hey pal, would you mind graphing this curve on the coordinate grid?” Assuming your friend knows what “coordinate grid” means, then you have just avoided many possibly awkward sentences trying to describe what it is! Having names for mathematical objects allows us to communicate mathematics to each other in clear, concise, and elegant ways.
So, we cook up special words to name mathematical objects, like real number and triangle. But we also use single letters to name things, which is called notation. In this week's problem, we didn’t feel like writing about our important group, “the group containing all the people who are simultaneously physicists and athletes” over and over again, so we called this group G. When we wanted to talk about that group, we simply referred to G. So nice and easy! Writing notation is wonderful for talking about mathematics, because it allows us to condense long, hard to read sentences into tiny little digestible snippets of information. It also makes everything we write easier to read, which is a must, since mathematics is hard enough as it is! By the way, notation doesn’t only involve letters, but also symbols of all shapes and sizes. We will see more of this in later problems.
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