The Basic Operators

By the time most students graduate from high school in the United States, they have seen the following operators*: Addition, subtraction, negation, multiplication, division, reciprocation, exponentiation, radicals, logarithms, sine, cosine, and tangent.

There is a certain symmetry in this list: They are clearly grouped in threes, and I’ve listed them so the traditionally dominant one in each set is first. Also, the groups are in order of when its first member is taught.

In my experience, it’s unusual for the distinction between subtraction and negation to be specifically highlighted and clarified. Negation is seen either as a form of subtraction or as “flipping the sign”. Likewise, reciprocation is either a form of division or “flipping the fraction”.

As a thought experiment, I’ve been working on how to characterize these twelve operators as minimally as possible.

With regards to trigonometry, everything can be expressed in terms of sine (since \(\cos\theta = \sin(\theta + \pi/2)\), and so \(\tan\theta = \sin(\theta)/\sin(\theta + \pi/2)\)).

We can replace subtraction and division with negation and reciprocation, respectively. That is, \(a – b = a + -b\) and \(a / b = a \times 1/b\).

But we could go a step further through inverse functions. That is, if \(A(x, a)\) is \(x + a\), then \(AI(y, a) = y – a\). Negative numbers are \(AI(0, x)\).

Likewise, if \(M(x, a)\) is \(x \times a\), then \(MI(y, a)\) is \(y \div a\) and the reciprocal is \(MI(1, x)\).

An advantage is that it allows us to characterize subtraction/negation and division/reciprocation as inverses rather than functions in their own right.

So what to do with exponentiation? Unlike addition and multiplication, exponential forms are not commutative. That is to say, \(a^b \ne b^a\) for most values.

There are at least two ways to see the radical, but one is simpler in this scheme: Radicals are convenient stand-ins for rational exponents. Hence, \(\sqrt[n] x = x^{MI(1, n)}\). The “true” inverse of an exponential function is the logarithm. Hence, if \(E(x, b) = b^x\), then \(E(MI(1, n), b) = \sqrt[n] b\) and \(EI(y, b) = \log_b y\).

One struggle I have here is that our default notation is conceptually backwards between subtraction/division on the one hand and exponentiation/logarithms on the other. With subtraction and division, we write the result we’re inverting first, and then the parameter we’re using. In other words, the dependent variable is first. With exponentiation and logarithms, the variable is second, while the parameter is first.

If we’re devising a new notation to be more consistent, binary** functions should have a consistent order. Using the order of subtraction and division, we have \(FI\)(dependent variable, parameter). For the sake of consistency, the general pattern for binary functions is (variable, parameter).

Revisiting the trig functions with this notation gives us \(\sin\theta = S(\theta), \cos\theta = S(A(\theta, M(\pi, MI(1, 2)))),\) and (the stuff of nightmares!) \(\tan\theta = MI(S(\theta), S(A(\theta, M(\pi, MI(1, 2)))))\).

A separate discussion, by the way, is proper function notation. I’m using the current standard of \(F(x, b)\), but I think something like \(x:\rightarrow F[b]\) would be better. But, as I say, that’s a separate discussion.

* The last four of these are usually characterized as functions, while it’s not even clear what to call the radical sign.

** I’m being loose with the term “binary” here for the sake of discussion. In the case of \(y = F(x, b)\), \(x\) is the independent variable, \(b\) is a parametric (constant) variable, and \(y\) is the dependent variable. Of course, \(b\) could also be a true variable.

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