Mathematics is like a language.

Nowadays, it’s a language with the same symbols and meaning in every place it’s used. A multiplication sign is a multiplication sign everywhere, and everybody agrees on what a multiplication sign means.

But it wasn’t always so.

Because they’re so common and taken for granted, it can be strange to imagine that mathematical symbols were all invented and popularly agreed upon to have a common language of math.

Before the thirteenth or fourteenth centuries, for example, math equations were often written using words as part of what historians call the “rhetorical stage” of notation. The mathematician wrote out the equation like a sentence: one plus one equals two.

Numerical notation systems did exist as much as three thousand years ago, and mathematicians developed complex functions to work with numbers. But there were no agreed-upon symbols from one place to the next. Using words like “plus,” “minus,” “multiplied” and “equals” was the best way for mathematicians to communicate.

That did change, and slowly over time, mathematicians invented a system that everyone agreed upon. It was a shift to what historians call the “symbolic stage” of mathematical notation.

These symbols are the language of the Faculty of Mathematics at the University of Waterloo. It is the medium through which our researchers make groundbreaking discoveries and communicate with one another. And it’s a language our professors and students are helping to evolve all the time.

Here are 11 things you never knew about mathematical symbols.

- The humble plus sign is among the oldest agreed-upon mathematical symbols but only came into general use around 1360.
- The equals sign was invented in 1557 by a Scottish mathematician, Robert Recorde.
- From about 1500 to 1600, there were as many as ten rival notation systems used in Europe.
- In the late 13th century, Arab algebraists were the first to use only symbols to formulate equations. Historians often see this time as the turn to standard symbolic notation.
- Niccolò Tartaglia, an Italian mathematician, introduced parenthesis (or brackets) into general use around 1550.
- René Descartes, the famous mathematician and philosopher, introduced the modern system of using superscript notation for exponents.
- The symbols
**∴**(therefore) and**∵**(because) were used inconsistently by mathematicians and were a source of some controversy until they were formalized in the 19th century. - A somewhat more recently invented symbol in mathematical notation is the “end of proof sign,” also known as the “tombstone.” It was made popular in 1950 and looks like this: ∎
- The mathematical constant
*e*is also known as Euler’s number after its inventor, Leonhard Euler. He invented it around 1730. No one is entirely sure why Euler chose to use*e*, though some suggest it’s because*a*,*b*,*c*, and*d*were already commonly used for other constants. - Albert Einstein came up with his own system of notation, which has come to be called Einstein notation. This system preferred elegant and short summations of “notational brevity,” and is seen in Einstein’s famously brief formula
*E*= mc^{2}. - As mathematicians push into new territory and make discoveries, they create new symbols and notation. Exciting work in quantum computing, string theory, and machine learning all require a constantly updating and evolving language of mathematics.

*By the Numbers is a weekly series that reflects on the lighter side of student life, research and innovation in the Faculty of Mathematics at the University of Waterloo. Stay tuned to this space for the next instalment.*