One thing that many people dislike about math is all of its special symbols. Why does it need to be so specific? Why so many symbols? Why is writing things down in a specific format so important in math? Shouldn’t something be considered “correct” if it has the right ideas, regardless of how it is presented?
These are common questions, and often come in the form of a complaint, and understandably so. There is a huge problem with people being told in school they are wrong because they got to their answers in a way that, while correct, is not “what they wanted.” There are specific situations where it is reasonable to expect a student to learn a particular method for solving a problem. But in general a correct solution should be counted as such even if arrived at in an unexpected way.
But notation is completely different. One of the primary reasons that we use all of the symbols that we do is to make the mathematical content of what we write clear and unambiguous. This may sound strange to a lot of people. I’ve heard many, many people tell me things like “math is a foreign language to me. These people might think that all these variables and symbols are too confusing to understand. Ironically, this actually reveals why notation is so important – mathematics is a language, and languages need grammar.
If someone wrote a paper in English class with really good ideas but horrific grammar, they would not get a perfect score on their paper. There is a very good reason for this – the rules of grammar help us communicate. A good piece of writing should convey information clearly, and poor grammar prevents that from happening. Too many grammatical mistakes will severely impact the quality of the writing. Math is like this too. The reason we have the symbols we do is to make what we write down unambiguous. This is also the same reason that things like dictionaries exist – imagine trying to have a conversation with a person that defines every word of English differently than you do. It simply cannot be done. It is for exactly this reason that mathematics needs to be precise.
For example, the “equals sign” = serves a role similar to a punctuation mark. It shows us that an expression is ending, and tells us that the next statement is related to the one we just read. The “plus sign” + is sort of like the word “and.” It connects things together, just as the word “and” does in English. Other symbols have other meanings. Since the + symbol can only mean one thing, “add two numbers together,” this is certainly unambiguous. That is the point.
The other common complaint is that all of the symbols are too “abstract” and difficult to understand. Every so often, the school system will change the way it teaches math for reasons like this. This is worth considering, just as from time to time we update the rules of grammar to accommodate the ways that language has shifted over time, we should be cognizant of whether a similar shift has occurred in the ways we think about math.
But any change in the way we use our notation is necessarily only very minor. Even if we make larger changes to how we compute sums, the way we express our final answer will be the same. There is a very good reason we use the sorts of symbols we do, they are not arbitrary. There was a time in history where there was no standard way to write down mathematical statements, including numbers themselves. Today, we mostly use the number system developed in Arabia, but the Chinese had their own system, and so did the Romans. Many of us are semi-familiar with Roman numerals and can read them, but imagine being asked to multiply XIII by IV. How would one systematically go about that? There really isn’t any way to, because Roman numerals are written using patterns that make computation very hard, even if reading Roman numbers is not terribly difficult.
This is why Arabic numerals won out. They are both easy to read and easy to compute with. The major innovation in the number system we use is the positional notation – which basically means the location of a number actually matters. As a consequence of this, the short list of symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 can be combined to form any number. Consider for instance the numbers 1234 and 4321. Even though the same four symbols were written down, the fact they are in different positions makes the numbers different. The symbol “1” is being used to mean “one” in 4321 and “one thousand” in 1234. These facts make Arabic numerals very easy to work with, and so over the course of history this system of numbers won out, because it was more convenient.
The use of variables and equations happened much the same way. For a very, very long time, all mathematics had to be done using sentences. And there was no standard way to write your sentences about numbers. To make clear what this must have been like, I will state the famous Pythagorean theorem in two ways – one where I take advantage of symbols for variables, and one where I do not.
Without variables: For every right triangle, the area of the square standing on the longest side of that triangle is equal exactly to the combined areas of the squares standing on the smaller sides of the triangle.
With variables: If a right triangle has side lengths a, b, and c, with c the largest length, then
See how much quicker the second statement is? It’s a lot less writing, but conveys exactly the same information. Now, imagine if you had to solve the following problem:
Twice a number added to seven times a second number less three times the first is equal to the second number minus four times the first number. What is the relationship between the first and second numbers?
Now imagine if you had to write a solution in paragraph format. Imagine having to simplify this without symbols! You’d have to rewrite this long sentence four or five times. This is cumbersome. But using the notation we have today, we can write
See how much easier that is? The exact same information is presented in each of these situations, but the symbols and notation enable us to do this in a very efficient manner and also enables us to navigate towards solutions efficiently. When written in words, when you want to find an answer, you have to rewrite long, complicated statements every time you make even a tiny modification. Thus, over time people who do mathematics devised new ways of writing down their ideas that were easier. And as the peoples of the world interacted on a more global scale, the best ideas of different regions were adopted. The proof methodology we use now was developed most thoroughly in Europe, but the way we write numbers is Arabic in origin, the earliest usage of 0 as we use it today is Indian.
So, mathematics is written the way that it is because it makes math easier to do. This sometimes has the unfortunate side effect of making it harder to understand to someone who is not an expert. Even mathematicians still feel this way from time to time. But despite this, the many symbols, strange words and unusual definitions used in mathematics have reasons to be as they are. These abstractions are what have made mathematics as we know it possible.