Mathematics is beautiful.
Mathematical notation, meanwhile, is a horrid mess second only to English itself for its arbitrariness.
For instance, basic arithmetic operators have three levels: Addition, Multiplication, and Exponentiation.
Addition notation is tidy: We add forward (a + b) or backward (a – b). We call these “addition” and “subtraction” for historical reasons, but “subtraction” should have faded away in favor of a more accurate term as soon as we accepted negative numbers. -4 – 3 is not a “subtraction”, it’s an addition on the opposite scale.
But I digress, since I’m ranting about notation.
Multiplication notation is… imperfect. Mathematically, we have two opposite operations, multiplication and division. Each needs to be complained about separately.
We have three different symbols in common use for multiplication: ✕ , *, and ·. Books for younger grades tend to use ✕ , but it has the obvious problem that it looks too much like an x when hand-written, so it’s largely abandoned by the time students take algebra. * is used for ease of typing, and is also used in computer programming for that reason… it’s not quite right, but it’s close enough. The symbol of choice at higher levels is · , when multiplication has to be explicitly marked at all.
Most of the time in secondary math and above, we imply multiplication. This creates its own problems, but it also gives primacy to multiplication over addition. Philosophically, I think this is the right thing to do, but it also means that we should be spending time explicitly reinforcing that, if there’s no operator where there ought to be one, we tend to imply multiplication.
When we get into explicitly written functions, we need to clarify “ought” again, so students realize that x(y) probably means x times y, while f(x) probably means the function f on x. “Probably”? Mathematics shouldn’t have “probably”!
And then, division. Ah, division. We have three ways of indication division: ÷, /, and the fraction bar (also called the vinculum, which disguises another truth).
The obelus (÷) is used in elementary school, but ought to be abandoned by high school. The overuse of the obelus and excessive focus on fractions as if they’re a separate mathematical concept seems to lead students to think of division and fractions as different operations.
While I understand why the vinculum is the symbol of choice for division at higher levels, it also doesn’t fit with the big four operators: The other three fit the pattern a⊕b. The inline option of a/b at least fits that pattern, but can contribute to confusion (is 4/2x equal to 2x or 2/x?).
So while Addition fits tidily and consistently in the a⊕b, frame, both multiplication and division violate it… but only sometimes. We have the option to write both multiplication and division consistently in that model, but it’s not the preferred method.
The Exponential level has three operations instead of two, so we need to be cautious about letting students think there are “opposites.” But I fear that by the time students get to logarithms, this is precisely what they’re doing.
One of these does have an a⊕b option, although it’s not the default. We generally write exponents as superscripts, but when we want to type in contexts that make superscripts difficult, we use ^: 4^3 = 64. I think it’s still fairly unusual to teach this notation for students handwriting mathematics, though.
I believe this is unfortunate, and something that we should be working to change. 4³ is too easy to confuse with 43; 4^3 is absolutely clear, it follows an established pattern, and given the widespread used of computers now to prepare documents, it ought to be comprehensible to any academic mathematician. The only downside that comes to mind is that 4-3-2 = -1 (not 3) because subtraction is evaluated from left to right, while 4^3^2 = 262144 (not 4096) because exponentiation is evaluated from right to left.
The first “opposite” operation that students see is radicals. We tend to teach square roots in terms of “what number squared equals 4?” so it’s the same sort of question as “what number plus 3 equals 7?” (subtraction) or “what number times 3 equals 12?” (division). At the same time, though, we can rewrite any radical into a rational exponent: √3 = 3^.5, for instance… something we can do with addition easily (7 – 3 = 7 + -3), but less easily with multiplication (12/3 = 12 * 1/3, but that seems circular from a notational standpoint).
The third operation involving exponents has absolutely terrible notation.
First, let’s talk about how notation COULD be. Consider 5^2 = 25. We have three values here in a relationship. If we want an analog to division, rather than radicals, we could use a new symbol for radicals: 25˅2 = 5, for instance. x˅2 = x^(1/2), just as 1/x = x^(-1). This could make for some nice parallels across the levels.
There are times that we want to know what our base is, though. This is how exponents seem to differ from our other two operators. Unlike addition and multiplication but like subtraction and division, exponentiation is not commutative. That is, 3 + 4 = 4 + 3 and 3 * 4 = 4 * 3, but 3 – 4 < 4 – 3, 3 / 4 < 4 / 3, and 3 ^ 4 > 4 ^ 3.
More importantly, the likelihood that the value that satisfies a ^ ▯ = b is irrational is far higher than with any other operator, so if want an exact value, we need a way of indicating that. Also, note that while a – ▯ = b is true when a – b = ▯ and a / ▯ = b is true when a / b = ▯ (and b is not zero), it’s not usually true that a ^ ▯ = b when a ^ b = ▯.
We could have a third symbol, say ♢. Just as 5^2 = 25 and 25˅2 = 5, 5♢25 = 2. a♢b would represent the value of the exponent x such that a^x = b. For these sample symbols, I’ve pulled existing Unicode symbols that already have different meanings in mathematics, but we could use anything available on the standard keyboard that’s not already reserved for something. My point is not to suggest new notation (although I wouldn’t say no, obviously), but to illustrate that we don’t have to use what we do.
But instead of this parallel notation, what we use is log_a b for a♢b, with no option for anything else.
Put those three next to each other and consider how it must look to someone learning mathematical notation.
5² = 25; √25 = 2; log₅ 25 = 2.
As standard practice, we don’t even mark the subscript as it is in LaTeX and other format-free documents, i.e., as log_5. We HAVE to write a subscript.
This is a mess, and it’s not necessary one. I see no inherent superiority to writing log₅ 25 instead of 5♢25. Meanwhile, although the habit of writing the square root without the 2 comes from the much higher usefulness of square roots over any other radical, it adds a level of confusion when students learn about the other roots.
Here’s a rule: When there’s no term where we need to add something, we put in 0. When there’s no coefficient or divisor where we need one, we put in 1. When there’s no marker on the radical, we use 2. On the one hand, there’s a nice parallelism here for the three levels of operators; on the other hand, where 0 and 1 seem logically motivated, 2 is nearly as arbitrary as it seems.
This said, ∛64 = 4, when handwritten, can easily be confused with 3√64, which equals 24, not 4. There is no such confusion with 64˅3.
Again, my aim here is not to promote new notation (although, again, I wouldn’t say no), but rather to illustrate how illogical the current notation might well seem to students first encountering it. If we are teachers, we should reflect on how our students see the mathematical world.