The symbols we use every day have fascinating and surprisingly recent origins, explains author Joseph Mazur in his new book

A few years ago friends and I were talking about the origins of written music. When the conversation turned to the origins of math symbols, I was surprised to learn that few people knew that almost all maths was written rhetorically before the 16th century, often in metered poetry. Most people think symbols for addition, subtraction or equality had been around long before Euclid wrote his Elements in the first century BCE. No! The original Elements is rhetorical. There are no symbols in Euclid’s works, aside from the letters marking the ends of lines and corners of geometric objects. There are no symbols in any early Arab algebra books. Nor do we find any in early European printed algebra books.

Even our wonderful symbol for equality – you know, those two parallel lines – was not used in print before 1575, when the Welsh mathematician and physician Robert Recorde wrote an algebra book that he called the Whetstone of Witte. (We can only guess that the title is a pun on sharpening mathematical wit.) In it he wrote “is equal to” almost two hundred times for the first two hundred pages before finally declaring that he could easily “avoid the tedious repetition” of those three words by designing the symbol “=====” to represent them.

In that book we find + and – printed in English for the first time.



The first appearance of the equals sign in maths literature: "Howbeit, for easy alteration of equations, I will propound a few examples, because the extraction of their roots, may the more aptly be wrought. And to avoid the tedious repetition of these words: is equal to: I will set as I do often in work use, a pair of parallels, or Gemini lines of one length, thus =====, because no 2 things, can be more equal." The equation spells out 14 x(squared) + 15 = 71 Photograph: Wikipedia

Of course the Egyptians had their hieroglyphical indications of addition and subtraction in glyphs of men running toward or away from amounts to be respectively added or subtracted.



And from time to time, pre-16th century writers of mathematical texts had ventured into symbolic expression. So there were instances when writers experimented with graphic marks to represent words or even whole phrases. No doubt that some mathematicians had some private markings that they used to work things out. Recorde tells us that he did. But I searched in vain for any first hand manuscripts that contained private symbols for operating symbols.

The writer’s art is different from the mathematician’s. Writers have more freedom from myth and culture. They might use symbols to bring about emotions or to create states of mind. Emily Dickinson never uses the word “snake” in her poem “A Narrow Fellow in the Grass,” perhaps to avoid direct connections with evil, sneakiness, and danger. But the hint is there all the same. Joseph Conrad’s Heart of Darkness describes the Congo River as “an immense snake uncoiled, with its head in the sea.” That invokes all the connotations of slithering, sneaky evil.

But mathematicians create symbols in order to package complex information for better understanding.

Their symbols may seem different from those culturally flexible, emotional symbols found in music or those metaphorical symbols found in poems. Yet they also have subliminal connections that inescapably create states of mind through similarity, analogy and resemblance.

In reading an algebraic expression, the experienced mathematical mind leaps through an immense number of connections in relatively short neurotransmitter lag times, cutting to the chase of compact understanding. So, for instance, the relatively modern symbol π automatically invokes the mental shadow of something circular to the mathematician who is familiar with its many disguises.

Of course there are good symbols and bad ones. What could be better than Recorde’s Gemini symbol for equality? But math symbols cannot be simply squiggles that are not carefully thought out.

Mathematical lunacy: for a short time the symbol for positive and negative was a moon. The equation above is -4 + 6 = 2 Photograph: Joe Mazur

When negative numbers were first accepted as numbers, there was great debate about how to write them. As late as the 19th century, some writers suggested that negative numbers should be written as positive numbers flipped horizontally. Imagine the confusion over numbers containing a 0 or 8, or letters like b, d, p, q or w. For a short time negatives were written with a short line above them, and for another short time negative numbers were written with the waxing moon symbol , and positive numbers with the waning moon.

These days we have Maxwell’s equations: four interrelated equations that tell us how electric and magnetic fields relate to charge density and current density. They form a mathematical poem, written entirely in symbols. That poem holds the key to a great deal of creative thinking about electrodynamics, and, ultimately, to the most utilitarian needs of modern life. It can even tell us something about why our mobile phones work. Try to write that poem entirely in English. It can be done, but who then would understand it?

Joseph Mazur is the author of Enlightening Symbols: A Short History of Mathematical Notation and its Hidden Powers

