Mathematical notation is a language.
The study of mathematics is about finding patterns in our universe, but we need a method of communicating those patterns to other people. That’s where notation comes in.
Like the languages that we speak, this system of notation did not spring up all at once. It was not the result of a committee deciding what was best and voting on it. It developed over centuries, with one person using a notation and other people either adopting it or ignoring it. In many cases, there were duplicate ways of writing things, and a few of these duplicates still remain today.
And as with any other language, once you’ve learned it, you think of it as the most natural way to communicate. More efficient replacements will look strange and difficult to learn. So that leaves students of mathematics with the chore of learning this system, including all its imperfections, in order to be proficient in communicating mathematics.
By the time the typical student leaves high school, they’re aware of only a short list of symbols. The complexity is in learning an inefficient system of communication and in learning how to take apart that short list of symbols. In advanced mathematics (in universities and beyond), that list of symbols does grow, but most of them follow the same basic communication pattern established by the list learned by the end of high school.
Most of the high school list consists of four categories: Addition, multiplication, exponentiation, and trigonometry. There are also symbols for units, but we’re often lazy about those; except for trigonometry, we don’t need those to “do math”.
Let’s take a look at an example: \[5(8−3⋅2)=10\]
What does this say? It is a true statement, but what is it claiming? How do we read it aloud to make sense of it?
Mathematically, it says that if we multiply three times two, subtract that product from eight, and multiply the result by five, we’ll get a value of ten.
First, consider how I evaluated that: I didn’t go from left to right (as English and many other languages are read) or right to left (as Arabic and many other languages are read). Instead, I started in the middle and worked my way outward.
That’s not how all mathematical statements are read, that’s just how this one in particular works. That’s because part of the language of mathematical notation is the concept of operator precedence: Certain things have to be evaluated first.
This precedence is not dictated by any underlying laws of the science of mathematics; it is dictated by a consensus agreement based on the language of mathematics, so that everyone who knows the rules can read the statement the same way.
Next, note that there are two times that I multiplied when evaluating, but the statement only contains one symbol that explicit says to multiply.
An Alternate System
About a century ago, Jan Łukasiewicz used a system of notation for writing logic that was later adapted for mathematics in general. It is more efficient in that it doesn’t rely on everyone to agree on any operator precedence: The precedence is baked into the system. However, the system has been largely abandoned because it’s not what people (including mathematicians) are used to.
In his system, the same statement I made above might be written like this:\[=×5–8×3,2,10\]
Each time we get an operator (+,−,×,÷,∧,=,≠, etc.), it tells us to wait to interpret it until we have enough numerical values to do so. The first symbol says two things are equal, but doesn’t say what they are yet. The next one tells us we’re going to multiply two values. One of these is five, and the other is the result of a difference.
The difference is eight minus a product, and the product is three times two. Finally, we compare the first product to ten. That is, we calculate three times two, subtract the result from eight, multiply the result by five, then compare that product to ten.
The advantage of this system is that there is never any ambiguity. One disadvantage is that it can be harder to follow, especially since we’re not used to it. Another is that, while we’re reading from left to right, we have a lot of incomplete operations at any given time.
My point in describing this system, often called Polish Notation, is not to suggest it as an alternative, but rather to illustrate that our current system isn’t the only way we could write mathematical expressions. It’s just the way that we’ve collectively decided to write things.
Order Precedence
The standard way of writing mathematical expressions is superficially ambiguous. Without any rules regarding the order of evaluation, 3+4×2 can be interpreted as either fourteen (performing the addition first) or eleven (performing the multiplication first). We could use grouping symbols such as parentheses to make it clear, but this becomes cumbersome quickly.
So mathematicians agreed to perform operations in a specific order. On one level, this is an arbitrary decision, but it’s not without any reason: Multiplication of natural numbers has the same effect as repeated addition, so we perform any multiplication before any addition.
Likewise, the application of exponents within the natural numbers has the same effect as repeated multiplication, so we perform any exponents before any multiplication.
Multiplication and addition are both commutative. That is, it doesn’t matter what order we perform multiple multiplications in, we get the same product; it doesn’t matter what order we perform multiple additions, in, we get the same sum.
So having the rule that we do exponents, then multiplication, then addition, in that order, takes care of most cases of ambiguity. However, there are still four situations left to discuss, among these sorts of operations.
The first is that, if 3+4×2 is unambiguous, how do we express the case where we add first? We do this using parentheses: (3+4)×2. By only requiring parentheses to change the order in special cases, we can address these exceptions. It allows us to express more complicated situations without getting overloaded with parentheses.
The second is that subtraction, unlike addition, is not commutative. We can get around this by thinking of subtraction as addition of the opposite. Subtraction is indeed the inverse of addition. So if we think of 3–4+5 as 3+−4+5, order doesn’t matter.
Likewise, division, unlike multiplication, is not commutative. However, while subtraction has a consistent symbol, we write division several different ways. The way we get used to in early grades, using the ÷ sign, is mostly replaced in higher grades by / or the division bar. That is, while 3÷4=3/4=34, the use of the first form is frowned on in older grades.
However, if we use the obelus (÷) sign, we can think of it the same way as the minus sign, as being “attached” to the following number for the purposes of the order of operations. So 3÷4×8=3×8÷4.
The final issue is nested exponentiation, such as \(2^{3^4}\). Is this \((2^3)^4=4096\) or \(2^{(3^4)}=2^{81}≈2.42×10^{24}\)? There is no easy resolution here, so it was decided by a rule: Evaluate exponents from right to left, so \(2^{3^4}=2^{81}\). If you want \((2^3)^4\), you have to use parentheses.
So our order of operations can be understood this way:
- Unless directed otherwise, do exponentiation first, then multiplication (and division), then addition (and subtraction).
- Expressions inside grouping symbols (such as parentheses) should be evaluated first.
- Subtraction and division can be thought of as being “attached” to the following expression.
- Exponentiation is done right-to-left by default.
This order is often captured by an abbreviation like PEMDAS, BIDMAS, or GEMA, but rather than memorizing an acronym without understanding, it is always better to understand the motivation behind the decision to evaluate in the conventional order that we use.
This is not the only order we could use. The basic Microsoft Windows Calculator on “Standard” mode simply evaluates addition and multiplication from left-to-right, while squaring takes place immediately. That is, \(3+4^2×5=3+16×5=19×5=95\). Standard order of operations would instead interpret \(3+4^2×5=3+16×5=3+80=83\) (which is what we get by setting the Microsoft Windows Calculator on “Scientific” mode.
Concluding Remarks
Mathematical notation is a type of language and, as such, requires a careful study of its grammar and structure. Full fluency only comes after practice and use, through breaking down expressions and understanding the nuances.
The choices of convention that mathematicians have made are not always the best, or the most logical, conventions. Mathematicians balance detailed clarity with convenience and brevity, and sometimes the result errs to the side of brevity.
However, with patience, you can become adept at understanding the conventions used in mathematical notation.