# Three coins of unknown denomination **Teaches:** `species-of-the-unknown` · **Assumes:** nothing · **~20 min** --- ## The refusal > **Add.** Every pair. Write *refused* where you cannot. | $+$ | $x^2$ | $x$ | $7$ | |---|---|---|---| | $3x^2$ | | | | | $5x$ | | | | | $2$ | | | | > **Now multiply.** Same nine pairs. | $\times$ | $x^2$ | $x$ | $7$ | |---|---|---|---| | $3x^2$ | | | | | $5x$ | | | | | $2$ | | | | Count the refusals in each grid. Write the two numbers down before you turn the page. --- One grid refused most of its cells and the other refused none of them. Nothing about the six quantities changed between the grids — only what you were asked to do to them. So the refusal is not a property of $x^2$ and $7$. It is a property of *addition* applied to $x^2$ and $7$, and multiplication does not share it. Any account of what is going on here has to explain that asymmetry, and "you can't add unlike terms" does not: it restates the first grid and says nothing about the second. ## What a number was We think of $3$ as a thing. It sits on a line, it can be added to any other thing on that line, and it does not care what it is counting. That is a recent idea. Ben Syversen puts the older view better than I can, so here it is in his words: > Before the 17th century, numbers were thought of differently from the way we think of > numbers today. At the time, a number was an adjective describing the quantity of some > specific type of thing. Numbers came in what's called quantity species pairs. It's > like how the number *dozen* doesn't really exist by itself. Unlike the number three, > the word *dozen* is only meaningful if it's used to describe how many eggs or bagels > or whatever you have. It's describing a tangible thing rather than an abstract > concept. So while today the expression $3x$ tells us to multiply an abstract number > three by some other abstract unknown number $x$, the pre[m]odern version of this, > *three things*, means that you're taking three of some unknown thing. There's no > multiplication involved. And you can think of those unknown things as being like > **three coins of unknown denomination.** > > — Ben Syversen, *Why Did it Take 1,900 Years to Invent the Exponent?*, 9:55–10:45. > [youtube.com/watch?v=dgMvkb9Waco](https://www.youtube.com/watch?v=dgMvkb9Waco) Sit with *three coins of unknown denomination*. It is the best one-line summary of pre-modern algebra in circulation. $3x$ is not an instruction to multiply. It is a description of what you are holding. You have three of them. You do not know what they are worth. Nothing about that description involves multiplication, and nothing about it lets you combine your three coins with the seven apples in your other hand. ## Why $x^2$ was a different species, not a bigger $x$ Syversen continues, and this is the sentence the rest of this library is built on: > back then each power of the unknown was seen as a different type or a different > species. So saying *two squares, five things and four numbers* is kind of like saying > 3 hours 21 minutes and 13 seconds. Each part has a different unit to it which can't > be combined without doing some kind of conversion. > > — *ibid.*, 10:53–11:12 Read your refusal table again. You refused $3x^2 + 7$ for exactly the reason you would refuse to write "3 hours and 13 seconds" as a single number. Not because a rule forbids it. Because there is no single number there to write. And the older names make the species visible in a way our notation hides. A **thing** (*res*, *shay'*, *cosa*) is a length. A **square** (*census*, *māl*) is an area. A **cube** is a solid. These are not three sizes of the same object. They are a line, a patch of ground, and a block of stone. Adding them is a category error you can see. > **Do this.** Rewrite $2x^2 + 5x + 4$ in species: *two squares, five things and four > numbers.* Now rewrite $x^2 + 6x + 9$ and $3x^2 - 12$ the same way. Then say out loud > what $x^4$ would have to be called if a square is a square and a cube is a cube. You will not have a good answer to the last one. Neither did they. The sixteenth-century German answer was *zenzizenzic* — "the square of squares" — and the notation for the eighth power, *zenzizenzizenzic*, holds the record for the most Zs of any word ever admitted to the Oxford English Dictionary. That is not a fun fact. It is a symptom: a naming system that has to invent a new word for every power is a system in which powers are kinds of things rather than counts of a repetition. ## The one thing the refusal buys you A modern student experiences the refusal as an obstacle. It is worth seeing it once as a tool, because for the next several lessons it is going to do real work. Consider these two ways of writing the same quantity: $$x^2(500-x)^2 \qquad\text{and}\qquad x^4 - 1000x^3 + 250000x^2$$ Multiply out the first and you get the second; they are equal for every $x$. But say them in species. The first is *a square, times a square* — an area multiplied by an area. Whatever it is, it is one kind of thing, and it is that kind of thing because it is **something squared**. The second is a square-of-squares, minus a thousand cubes, plus two hundred fifty thousand squares: three species, no one of which you can hold, combined by an operation the language does not license. Both are correct. Only one is *readable*. A mathematician in 800 or 1200 or 1500 would not expand the first into the second, and not merely because it was laborious. Doing so would have destroyed the only description that meant anything. That is the habit this library is trying to install, and the history is the argument for it: **do not expand a product until you know what the expansion is going to tell you.** Nine times out of ten the factored form already answered the question. > **Do this, and keep it.** On a card, write $x^2(500-x)^2$ on one side and its expansion > on the other. Under each, write down every fact you can read off *without doing any > further work*. Keep the card. You will be asked for it again in > `quad-04-do-not-expand`, and the two lists are not the same length. ## Where the refusal finally lifts It lifts in 1637, and it lifts because René Descartes does something that looks like a triviality and is not: he picks a segment and calls it **one**. Once there is a unit length, $x^2$ is no longer an area. It is the length you get from $x$ and $1$ by a construction, and it can be marked on the same line as $x$. The species collapse into a single kind, and $x^2 + x$ becomes something you can write, and then something you can *draw*. That is the last lesson in this arc, not the first, and deliberately so. You are going to prove a large number of things about quadratics before you are allowed to see a graph, for the same reason you would not hand someone a calculator during a lesson on what division means. When the picture finally arrives it should be a relief and a recognition, not a method. --- ## Exit 1. Explain to somebody, without using the phrase "like terms," why $3x^2 + 5x$ cannot be written as $8x^3$ or $8x^2$ or anything else with one term. 2. *Two squares, five things and four numbers*, and separately, *three things and six numbers*. Add them, in species. What can you combine, and what has to stay apart? 3. In the "3 hours 21 minutes 13 seconds" analogy, hours and seconds *can* be combined — there is a conversion. Why is there no conversion between things and squares? Answers and commentary: [`ANSWERS.md`](ANSWERS.md). ## Related - Next in the historical arc: `quad-02-parts-of-500` (uses `species-of-the-unknown`). - The payoff for the refusal: `quad-04-do-not-expand`. - Where the refusal lifts: `quad-06-the-turn`. - Source and citation notes, including a caveat about how this transcript was obtained: [`../../SOURCES.md`](../../SOURCES.md).