Mathematical Notation

Mathematical notation is the language used by mathematicians to communicate unambiguously.

However, it can be very complicated to someone who doesn't know the language and even sometimes complicated to people who understand the concepts easily.

Introduction

Consider the following expression:

\[ \sum_{n=1}^{100}n \]

For someone who doesn't know any mathematical notation, the symbols and numbers can be overwhelming and confusing, especially . Once you learn the concepts involved, like any mathematician would, the concepts become simple.

It is in fact, so simple that even a first grader could understand what this expression is trying to convey. I'll try to explain it in very simple terms.

Imagine the following process of adding numbers:

• Start with the number 1
• Then, add 2
• Then, add 3
• Then, add 4
• ...
• Then, add 100

That's it? Yes! That's all that expression means.

Being able to parse the expression is the source of the complexity, and even seasoned mathematicians may not understand a mathematical statement or expression fully at first glance.

Information Density

I believe that one of the main issues with introducing mathematical notation to a layperson is that it is very information-dense. Changing any symbol could vastly change the meaning of any expression. In any other language, changing a single symbol or character usually doesn't completely alter the meaning of the sentence.

For example, if someone were to typo the following sentence:

\[ \text{"Hello, my name is Charles!"} \rightarrow \text{"Helo, my name is Charles!"} \]

The reader could easily infer that the sentence was misspelled, and in fact, probably reconstruct the original sentence. However, this is not the case for mathematics. the expression

\[ \sum_{n=1}^{100}n\rightarrow\sum_{n=11}^{100}n \]

are only off by 1 single character, yet they mean very different things. A reader, even a mathematician, even knowing that something was wrong with the expression, would not be able to reconstruct the original .

Assuming a human brain can only process a certain amount of information, they would necessarily have to read mathematical expressions more slowly than text written in any spoken language. For exmample, perhaps there is a person who is able to read 5 words per second. Even if they were a seasoned mathematician, the amount of time it would take them to read (and understand) the following expression

\[ f(x)=\int_3^5xe^{-x^2}\cos(2\pi y)dy \]

is probably more than a few seconds. Every single detail is important. From the 3 and 5 on the \(\int\) to the \(dy\) at the end, implying that the \(x\) and \(y\) need to be treated differently. (Side note: It turns out the equation evaluates to \(f(x)=0\))

So What?

The point is, most mathematical expressions aren't as complicated as you expect them to be when you're just a normal person looking at the math "calculating meme" (not included for copyright reasons). It's as if the creator of these "math things" designed them to be extremely esoteric and complex to a layperson. The good thing is, it's not impossible to learn. Just be patient, take it step by step, and most importantly try to understand the concept behind the statement rather than the statement itself.

Related

Mathematical Rigor