This article is extracted from Joseph Mazur’s fascinating history of mathematical notation, Enlightening Symbols. For more Pi Day features from Princeton University Press, please click here.

When one sees *π* in an equation, the savvy reader automatically knows that something circular is lurking behind. So the symbol (a relatively modern one, of course) does not fool the mathematician who is familiar with its many disguises that unintentionally drag along in the mind to play into imagination long after the symbol was read.

Here is another disguise of *π*: Consider a river flowing in uniformly erodible sand under the influence of a gentle slope. Theory predicts that over time the river’s actual length divided by the straight-line distance between its beginning and end will tend toward *π*. If you guessed that the circle might be a cause, you would be right.

The physicist Eugene Wigner gives an apt story in his celebrated essay, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” A statistician tries to explain the meaning of the symbols in a reprint about population trends that used the Gaussian distribution. “And what is this symbol here?” the friend asked.

“Oh,” said the statistician. “This is pi.”

“What is that?”

“The ratio of the circumference of the circle to its diameter.”

“Well, now, surely the population has nothing to do with the circumference of the circle.”

Wigner’s point in telling this story is to show us that mathematical concepts turn up in surprisingly unexpected circumstances such as river lengths and population trends. Of course, he was more concerned with understanding the reasons for the unexpected connections between mathematics and the physical world, but his story also points to the question of why such concepts turn up in unexpected ways within pure mathematics itself.

# The Good Symbol

The first appearance of the symbol *π* came in 1706. William Jones (how many of us have ever heard of him?) used the Greek letter *π* to denote the ratio of the circumference to the diameter of a circle. How simple. “No lengthy introduction prepares the reader for the bringing upon the stage of mathematical history this distinguished visitor from the field of Greek letters. It simply came, unheralded.” But for the next thirty years, it was not used again until Euler used it in his correspondence with Stirling.

We could accuse *π* of not being a real symbol. It is, after all, just the first letter of the word “periphery.” True, but like *i*, it evokes notions that might not surface with symbols carrying too much baggage. Certain questions such as “what is *i ^{i}*?” might pass our thoughts without a contemplating pause. Pure mathematics asks such questions because it is not just engaged with symbolic definitions and rules, but with how far the boundaries can be pushed by asking questions that everyday words could ignore. You might think that

*i*makes no sense, that it’s nothing at all, or maybe a complex number. Surprise: it turns out to be a real number!

^{i}It seems that number has a far broader meaning than it once had when we first started counting sheep in the meadow. We have extended the idea to include collections of conceptual things that include the usual members of the number family that still obey the rules of numerical operations. Like many of the words we use, number has a far broader meaning than it once had.

Think only of the so-called imaginary quantities with which mathematicians long operated, and from which they even obtained important results ere they were in a position to assign to them a perfectly determinate and withal visualizable meaning.

It is not the job of mathematics to stick with earthly relevance. Yet the world seems to eventually pick up on mathematics abstractions and generalizations and apply them to something relevant to Earth’s existence. Almost a whole century passed with mathematicians using imaginary exponents while a new concept germinated. And then, from the symbol *i* that once stood for that one-time peculiar abhorrence √−1, there emerged a new notion: that magnitude, direction, rotation may be embodied in the symbol itself. It is as if symbols have some intelligence of their own.

What is good mathematical notation? As it is with most excellent questions, the answer is not so simple. Whatever a symbol is, it must function as a revealer of patterns, a pointer to generalizations. It must have an intelligence of its own, or at least it must support our own intelligence and help us think for ourselves. It must be an indicator of things to come, a signaler of fresh thoughts, a clarifier of puzzling concepts, a help to overcome the mental fatigues of confusion that would otherwise come from rhetoric or shorthand. It must be a guide to our own intelligence. Here is Mach again:

In algebra we perform, as far as possible, all numerical operations which are identical in form once for all, so that only a remnant of work is left for the individual case. The use of the signs of algebra and analysis, which are merely symbols of operations to be performed, is due to the observation that we can materially disburden the mind in this way and spare its powers for more important and more difficult duties, by imposing all mechanical operations upon the hand.

The student of mathematics often finds it hard to throw off the uncomfortable feeling that his science, in the person of his pencil, surpasses him in intelligence—an impression which the great Euler confessed he often could not get rid of.

A single symbol can tell a whole story.

There was no single moment when *x ^{n}* was first used to indicate the

*n*th power of

*x*. A half century separated Bombelli’s , from Descartes’s

*x*. It may seem like a clear-cut idea to us, but the idea of symbolically labeling the number of copies of

^{n}*x*in the product was a huge step forward. The reader no longer had to count the number of

*x*’s, which paused contemplation, interrupted the smoothness of reading, and hindered any broad insights of associations and similarities that could extend ideas. The laws

*x*=

^{n}x^{m}*x*and (

^{n+m}*x*)

^{n}*= x*

^{m}^{nm}, where

*n*and

*m*are integers, were almost immediately suggested from the indexing symbol. Not far behind was the idea to let

*x*denote √

^{½}*x*, inspired by extending the law

*x*to include fractions, so

^{n}x^{m}= x^{n+m}*x*=

^{½}x^{½}*x*.

^{1}Further speculation on what *n ^{x}* might be would surely have inspired questions such as what

*x*might be for a given

*y*in an equation such as

*y*= 10

*. Answer that and we would have a way of performing multiplication by addition. But Napier, the inventor of logarithms, already knew the answer long before mathematics had any symbols at all!*

^{x}Symbols acquire meanings that they originally didn’t have. But symbolic representation has, likewise, the disadvantage that the object represented is very easily lost sight of, and that operations are continued with the symbols to which frequently no object whatever corresponds.

Ernst Mach once again:

A symbolical representation of a method of calculation has the same significance for a mathematician as a model or a visualisable working hypothesis has for a physicist. The symbol, the model, the hypothesis runs parallel with the thing to be represented. But the parallelism may extend farther, or be extended farther, than was originally intended on the adoption of the symbol. Since the thing represented and the device representing are after all different, what would be concealed in the one is apparent in the other.