## What do the less-than and greater-than symbols mean?

My love read a thread about the < and > signs, and mnemonics people had learned to tell which was which. And my love wondered, is a mnemonic needed? The symbol is wider on the side with the larger quantity; that’s what it means, right? Why imagine an alligator that’s already swallowed the smaller and is ready to eat the larger? In my elementary school it was goldfish, not alligators. Much easier to draw them in.

All right, but just because an interpretation seems obvious doesn’t mean it is. The questions are, who introduced the < and > symbols to mathematics, and what were they thinking?

And here we get complications. The symbols first appear, meaning what they do today, in Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas (“The Analytical Art by which Algebraic Equations can be Resolved”). This is a book, by Thomas Harriot, published in 1631. Thomas Harriot was one of the great English mathematicians of the late 16th and early 17th centuries. He worked on the longitude problem, on optics, on astronomy. Harriot’s observations are our first record of sunspots. He almost observed what we now call Halley’s Comet, with records used to work out its orbit. And he worked on how to solve equations, in ways that look at least recognizably close to what we do today.

The thing is that Harriot died in 1621. The Analytical Art was was assembled from his papers. Harriot’s biography at St Andrews’s mathematics page reports that there’s no manuscripts by Harriot with the < or > symbols.

There is a tradition that holds Harriot drew these symbols from the arm markings on a Native American. Harriot did sail to the New World at least once. He was on Walter Raleigh’s 1585-86 expedition to Virginia and observed the solar eclipse of April 1585. This was a rare chance to calculate the longitude of a ship at sea. So that’s possible. But there is also an argument that Harriot (or editor) drew from the example of the equals sign.

The = sign we first see in the mid-16th century, written by Robert Recorde, another of the great English mathematicians. Recorde did write, in The Whetstone of Witte (1557) that he used parallel lines of a common length because no two things could be more equal. Good mnemonic there. It seems Harriot (or editor) interpreted the common distance between the lines in the equals sign as the thing kept equal. So, on the side of the symbol with the greater number, make the distance between lines greater. On the lower-number’s side, make the distance between lines smaller. Which is another useful mnemonic for the symbol, if you need one.

It’s not an inevitable scheme. William Oughtred also had symbols for less-than and greater-than. Oughtred’s another vaguely familiar name in mathematics symbols. He gave us the $\times$ symbol for multiplication, and $sin$ and $cos$ for the trig functions. He also pioneered slide rules. Oughtred’s symbols look like a block-letter U set on its side, with the upper leg longer than the lower. The vertical stroke and the shorter horizontal stroke would be on the left, to represent the left being greater than the right. The vertical stroke and shorter horizontal stroke would be on the right, for the left being less than the right. That is, the “open” side would face the smaller of the numbers, opposite to what we do with < and >.

And that seems to be as much as can be definitely said. If I’m reading right, we don’t have Harriot’s (or editor’s) statement of what inspired these symbols. We have guesses that seem reasonable, but that might only seem reasonable because we’ve brought our own interpretations to it. I’d love to know if there’s better information available.

## Hopefully, Saying Something True

I wanted to talk about drawing graphs that represent something, and to get there have to say what kinds of things I mean to represent. The quick and expected answer is that I mean to represent some kind of equation, such as “y = 3*x – 2” or “x2 + y2 = 4”, and that probably does come up the most often. We might also be interested in representing an inequality, something like “x2 – 2 y2 ≤ 1”. On occasion we’re interested just in the region where something is not true, saying something like “y ≠ 3 – x”. (I’ve used nice small counting numbers here not out of any interest in these numbers, or because larger ones or non-whole numbers or even irrational numbers don’t work, but because there is something pleasantly reassuring about seeing a “1” or a “2” in an equation. We strongly believe we know what we mean by “1”.)

Anyway, what we’ve written down is something describing a relationship which we are willing to suppose is true. We might not know what x or y are, and we might not care, but at least for the length of the problem we will suppose that the number represented by y must be equal to three times whatever number is represented by x and minus two. There might be only a single value of x we find interesting; there might be several; there might be infinitely many such values. There’ll be a corresponding number of y’s, at least, so long as the equation is true.

Sometimes we’ll turn the description in terms of an equation into a description in terms of a graph right away. Some of these descriptions are like as those of a line — the “y = 3*x – 2” equation — or a simple shape — “x2 + y2 = 4” is a circle — in that we can turn them into graphs right away without having to process them, at least not once we’re familiar and comfortable with the idea of graphing. Some of these descriptions are going to be in awkward forms. “x + 2 = – y2 / x + 2 y /x” is really just an awkward way to describe a circle (more or less), but that shape is hidden in the writing.