While all of us regularly use basic math symbols such as those for plus, minus, and equals, few of us know that many of these symbols weren't available before the sixteenth century. What did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? In *Enlightening Symbols*, popular math writer Joseph Mazur explains the fascinating history behind the development of our mathematical notation system. He shows how symbols were used initially, how one symbol replaced another over time, and how written math was conveyed before and after symbols became widely adopted.

Traversing mathematical history and the foundations of numerals in different cultures, Mazur looks at how historians have disagreed over the origins of the numerical system for the past two centuries. He follows the transfigurations of algebra from a rhetorical style to a symbolic one, demonstrating that most algebra before the sixteenth century was written in prose or in verse employing the written names of numerals. Mazur also investigates the subconscious and psychological effects that mathematical symbols have had on mathematical thought, moods, meaning, communication, and comprehension. He considers how these symbols influence us (through similarity, association, identity, resemblance, and repeated imagery), how they lead to new ideas by subconscious associations, how they make connections between experience and the unknown, and how they contribute to the communication of basic mathematics.

From words to abbreviations to symbols, this book shows how math evolved to the familiar forms we use today.

extremely difficult to algebraically manage. It is a cumbersome notation. Even Diophantus tells us, “You will think it hard before you get thoroughly acquainted with it.”31 Since there was no sign for addition, it was necessary to group all the negative terms together after the sign for subtraction. Moreover, his notation gave no signal to the mind that x and x2 are of the same number species. We might say that Diophantus’s notation is terribly awkward and acutely difficult to process compared

same achievement? At least a huge dedicated math populace. Italy wasted no time in cultivating the seeds of algebra that drifted to Europe after the Arabs brought that art to Spain. Unfortunately, except for the works of Fibonacci, almost nothing is known about European works of algebra before 1300. The earliest works of that period were those of Fibonacci, Paolo de l’Abacco, and Belmondo de Padua. By the end of the fifteenth century, algebra went no further than quadratic equations with just

these operations are constructible with straightedge and compass, and therefore provable from Euclid’s axioms. And any problem that can be expressed through a geometric construction that uses straightedge and compass alone can also be expressed by a polynomial equation of degree one or two. The Cartesian coordinate system is more than just an orienting system, more than just a way to get from here to there. It is a way to see geometry through the lens of algebra. Descartes (and Fermat too) gave

it may afterwards furnish the grammar of a science. The proficient in a symbolic calculus would naturally demand a supply of meaning.11 Chapter 18 The Symbol Master A seemingly modest change of notation may suggest a radical shift in viewpoint. Any new notation may ask new questions. —Barry Mazur Gottfried Leibniz, a man “of middle size and slim figure, with brown hair, and small but dark and penetrating eyes,” was the genius of symbol creation.1 Alert to the advantages of proper

expressed only in words, would that association have come so quickly from a verbal description of the hint that came from my fade-in-and-out equation? Asking a question such as “What goes through your mind when…?” the way I did reminds us of how social science experiments were performed back in the mid-twentieth century, when there were few mechanisms in place for measuring responses. My sample size was so small, there was no real way to tally the frequencies of the answers. Moreover, even if