Logs, Roots, and Fractions

How notation gets in the way of understanding

The other day I tweeted this:

Objectively, I realize that \(\sqrt2\), \(\log6\), and \(\frac57\) are all specific numbers and that they’re the simplest way to write those specific numbers.

But I struggle with convincing my brain of that. And if I struggle, I don’t at all wonder that my students do as well.

It’s true that these three numbers belong to different subsets of the real numbers. \(\frac57\) is a rational number, meaning that it can (and is, in this case) written as a ratio of an integer and a positive integer. Rational numbers can be split into those whose decimals eventually terminate (such as 5 and \(\frac52 = 3.5\)) and those whose decimals follow a repeating pattern (as is the case with \(\frac57 = 0.\overline{714285}\)).

Since we can never write all the digits of the second group, and since it’s often difficult to typeset the vinculum (overbar), \(\frac57\) is typically seen as the “best” way to write that specific numeric value. For that matter, it’s not unusual to prefer fractional forms for rational numbers that do have terminating forms, such as \(\frac5{64} = 0.078125\).

(Sidebar: The decimal form of a rational number terminates if and only if its denominator can be written in the form \(2^m5^n\), where \(m\) and \(n\) are nonnegative integers.)

While \(\frac57\) is a rational number, \(\sqrt{2}\) is not, but it’s an algebraic number, meaning that it’s a root of a polynomial (specifically, the polynomial \(x^2 – 2\), which is why it’s called a “square root”). All numbers of the form \(\sqrt[a]{b}\), where \(a\) and \(b\) are nonnegative integers, are either integers themselves (\(\sqrt{9}\)) or irrational but algebraic (being roots of \(x^a – b\)).

Because the decimal form of irrational numbers never reach a point of repeating a pattern, it is not possible to write their exact form. Every decimal form will be an approximation. So \(\sqrt{2}\) has no “exact” decimal form, and \(\sqrt{2}\) is the simplest way to write that specific irrational number.

All rational numbers are algebraic, but not all algebraic numbers are rational. Real numbers which are not algebraic are transcendental. Most logarithmic numbers, including \(\log 6\), are transcendental. The circle constants \(\pi\) and \(\tau\), as well as the natural base \(e\), are also transcendental, as are most values of \(\sin a\).

Incidentally, given \(a\) and \(b\) as integers greater than 1, \(\log_a b\) can be whole, rational, or transcendental, but it can’t be an irrational algebraic number, according to the Gelfond-Schneider theorem (thanks to David Radcliffe for this information).

All this said, I don’t think it’s the nature of the numbers that get in the way of me accepting them as numbers. I certainly objectively know that \(\log 2\) and \(\sin 1\) are ways of writing an exact value that can’t be written exactly in decimal form, the same way that \(\sqrt[5]{7}\) and \(\frac89\) are.

I think it’s the notation that’s creating the biggest obstacle for my brain. I was trained to think of division as an operator and logarithm as a function. The radical sign sits somewhere in between.

This is linguistic distinction, though, with no immediately discernible mathematical difference. We could just have easily created \(\text{add}_4 5\) for addition, or created an operator such as \(4 \downarrow 3\) for logarithms. We didn’t, but there’s nothing in our notational system preventing us from having done so, or from doing so now.

The mathematically relevant reality, which I’m struggling to get my brain to accept, is that \(\log\), \(\sin\), \(\sqrt{ }\), and the fraction bar vinculum are all operators/functions, and that more often than not they’re used to exactly identify a real number whose full decimal form can’t be written finitely.

By using distinctly different forms of notation, we’ve gotten in the way of that realization.

This struggle is further compounded by the fact that \(\log\) and \(\sqrt{ }\) have default values for one of their two arguments. We have another such operator: The minus sign. When we are subtracting from zero, taking the logarithm in base 10 (or base \(e\) in some sources), or finding the square root, we don’t generally write the first argument.

In the case of \(-1\), students seem to understand that we’re pointing to a specific numeric value. But in general they don’t seem to think, “Oh, that’s a value that’s one less than zero, and since we don’t have a specific symbol for that number, we have to apply the subtraction operator to one.”

If they did, they wouldn’t struggle with \(-1^2 = -1\) vs. \((-1)^2 = 1\). This struggle comes from thinking that \(-1\) is a single symbol for a value, not an operator attached to a value.

Meanwhile, we usually introduce the radical sign without an index, calling it the square root sign, well before talking about other radicals, and so students struggle with understanding both why there’s an index in the first place, and why the default index is two.

As for logarithms, they tend to be introduced all at once. So even though students have already experienced two cases of a default parameter, as well as the notion that \(a + 0\), \(a \times 1\), and \(a ^ 1\) are all \(a\), I find students wondering why I’m sometimes writing \(\log\) without a subscripted value.

Once again, I find myself nattering at our odd and often frustrating notation. It could be fixed, but we continue to choose to leave it as is. As such, then, I must continue to try to convince my brain that \(\sqrt2\), \(\log6\), and \(\frac57\) are all representations of specific numbers than can’t be written in decimal exactly.

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