The evolution of mathematical symbols and their historical origins

From the equal sign to the Halmos box, how mathematical notation developed over centuries.

Illustration by Febrina Tiara

By Sarah Oktaviany and Laila Azzahra

The Language of Mathematics: The Stories behind the Symbols, by Raúl Rojas

Mathematical symbols play a crucial role in shaping how we understand and express numerical concepts. Among these, the equal sign is the most common, appearing in 94% of mathematical expressions. Parentheses are also widely used, appearing in nearly 60% of mathematical statements, while numbers themselves take up only 13% of written mathematical content.

The origins of these symbols often involve complex linguistic and historical interpretations. In The Language of Mathematics, Raúl Rojas provides a detailed survey of how mathematical notation developed, tracing symbols from ancient Greek arithmetic to the introduction of the concept of zero. His work explores how mathematical language evolved, sometimes through deliberate invention and sometimes by accident.

The inverted delta and the evolution of mathematical operators

One example of how mathematical symbols developed is the nabla, a mathematical operator resembling an inverted uppercase delta. The practice of inverting characters is a common method for creating new symbols. For instance, an inverted "A" signifies "for all," while an inverted "T" represents "false" as opposed to "T" for "true." Originally called "tilt," the nabla is now essential in equations describing electric and magnetic fields.

The role of Raúl Rojas in chronicling mathematical symbols

Raúl Rojas, a professor of mathematics and statistics at the University of Nevada, Reno, and an emeritus professor in computer science and mathematics at the Free University of Berlin, has spent decades compiling stories about mathematical symbols. While teaching a seminar on the history of mathematics, he gathered a collection of anecdotes, vignettes, and historical insights. His book, initially written in English and German, only took shape when he began writing in Spanish, his native language. Published in 2018, it was later translated into English by Eduardo Aparicio.

Rojas emphasizes that for centuries, mathematics was not expressed symbolically but rhetorically. This meant that mathematical ideas were written in full sentences rather than using symbols. As he explains, "In the 13th century, mathematicians had no standard symbols for addition, subtraction, multiplication, or even the equality symbol."

The invention of the equal sign

The equal sign, now a fundamental part of mathematics, was introduced by Welsh mathematician Robert Recorde in the 16th century. Before its invention, some mathematicians used a single line to separate equations, while René Descartes preferred a sideways squiggle that resembled an incomplete infinity symbol.

Recorde opted for two parallel lines to represent equality, famously explaining, "Noe 2 thynges can be moare equalle." Initially, this notation was primarily used in England, while continental European mathematicians continued using Descartes’s squiggle. However, by 1700, after Newton and Leibniz adopted Recorde's notation, the parallel lines became the standard equal sign.

Despite his contribution to mathematics, Recorde died in debt, failing to achieve financial "equilibrium," as Rojas wryly notes.

The introduction of parentheses

The use of parentheses in mathematical notation dates back to 1399, introduced by Italian scholar Coluccio Salutati. However, his version was more pointed than the rounded brackets used today. In 1470, a Venetian printer refined the design, introducing the curved brackets that are now a staple of mathematical and linguistic notation. Renaissance scholar Erasmus reportedly called them "little moons."

Over time, parentheses were occasionally used in reverse order, appearing as )this( rather than (this).

The mathematical arrow and its evolution

Another key mathematical symbol is the arrow, which first appeared in the mid-19th century when German mathematician Bernhard Riemann used it in his lectures to indicate transitions between solutions. However, at that time, the symbol was primarily used to mark steps in problem-solving rather than to define functions.

Today, the arrow is fundamental in mathematics. Expressions like f: A → B use it to indicate that a function f takes arguments from set A and produces results in set B. This typographic symbol has become a staple in modern mathematical notation, representing transitions between different mathematical domains.

The widespread adoption of the arrow occurred in the late 19th and early 20th centuries. It was first used by topologists like Felix Hausdorff, Witold Hurewicz, and Norman Steenrod before logicians such as Giuseppe Peano, Alfred North Whitehead, Bertrand Russell, and David Hilbert incorporated it into formal logic.

As Rojas colorfully summarizes, "Thousands of years ago, Homo sapiens crafted the first arrow for hunting. It was a small step for survival, but a giant leap for mathematics, which later expanded its symbolic toolbox with the arrows that fill textbooks today."

Unusual mathematical symbols and their hidden meanings

While many mathematical symbols have practical origins, some have more idiosyncratic histories.

One such example is the "dangerous bend" sign, first introduced in 1935 by the Bourbaki group, a collective of French mathematicians. This symbol, resembling a distorted "Z," appeared in the margins of textbooks to indicate particularly tricky sections where students were prone to making mistakes. The warning functioned similarly to a road sign marking a sharp turn.

A more recent adaptation of this idea comes from computer scientist Donald Knuth, who designed a modern version of the "dangerous bend" symbol. His version mimics an actual road sign, complete with a yellow cautionary background, to highlight sections of text containing obscure or advanced concepts.

Another striking symbol is the Halmos box, a notation used to signify the end of a mathematical proof. This symbol serves as an alternative to "Q.E.D." (quod erat demonstrandum, meaning "which was to be demonstrated"). Named after Hungarian-American mathematician Paul Halmos, the Halmos box is described by Rojas as "an oversized period, almost an exclamation point for the initiated—a triumphant mic drop at the end of a proof."

The enduring impact of mathematical symbols

Although mathematical notation may seem static, its evolution reflects centuries of intellectual progress. Symbols like the equal sign, parentheses, and arrows have transformed how mathematics is written and understood, allowing for greater precision and efficiency.

Even today, new mathematical symbols continue to emerge, adapting to the needs of modern mathematical research. As Rojas suggests, certain symbols—such as the iconic asterisk, which may date back to prehistoric cave paintings—remain subjects of ongoing exploration, providing material for future historical studies on mathematical notation.

Explore more in Books

Imani Perry explores the cultural and historical significance of blue, connecting it to Black identity and artistry in her latest work.
A semi-surreal debut novel where a mysterious blobfish-like creature sparks emotional growth and transformation.
Charmaine Wilkerson's novel Good Dirt delves into history, loss, and resilience across generations.