The Evolution & History of Famous Mathematical Symbols

Today, mathematical symbols appear so natural and ubiquitous that we rarely pause to question where they came from. When we write $e^{i\pi} + 1 = 0$ or $\int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2}$, we take for granted a compact, universal language shared across every continent, culture, and computational device.

Yet for thousands of years, mathematics possessed virtually no symbols at all. Ancient Babylonian, Egyptian, Greek, and medieval Islamic mathematicians solved sophisticated quadratic equations, geometric theorems, and astronomical orbits entirely using prose sentences. This guide traces the dramatic evolutionary history of mathematical symbolism and the intellectual pioneers who freed mathematics from the shackles of verbal prose.

1. From Words to Symbols: The 3 Stages of Algebra

Historian of mathematics G.H.F. Nesselmann categorized algebraic evolution into three great historical epochs:

1. Rhetorical Algebra (Pre-250 CE): All calculations and arguments were written out in ordinary spoken language. To express $x + 5 = 12$, a Mesopotamian or Roman scribe wrote: "A certain quantity added to five units equals twelve units."
2. Syncopated Algebra (ca. 250 – 1591 CE): Pioneered by Diophantus of Alexandria and later Islamic and Italian Renaissance mathematicians (like Luca Pacioli). Mathematicians used standard words interspersed with abbreviations (such as p for plus and m for minus).
3. Symbolic Algebra (1591 CE – Present): Initiated by French jurist and mathematician François Viète in 1591, who introduced vowels for unknown variables and consonants for known constants. This was completed by René Descartes in 1637, who introduced the modern convention of $x, y, z$ for unknowns and $a, b, c$ for constants.

2. The Plus (+) and Minus (-) Signs

Before the plus and minus signs, Roman scribes wrote the Latin word et (meaning "and") to indicate addition. Over centuries of hurried handwriting, the cursive ligature for et was compressed until only the cross remained ($+$).

The first printed appearance of $+$ and $-$ in a mathematical text occurred in 1489 in Leipzig, in Johannes Widmann's arithmetic treatise Behende und hüpsche Rechenung auff allen Kauffmanschafft. However, Widmann did not use them as operational signs; he used them as warehouse inventory marks! A barrel stamped with $+$ indicated surplus weight over the standard measure, while a barrel marked with $-$ indicated a deficit.

Within decades, European algebraists recognized their utility as operational arithmetic operators, replacing the clumsy Italian abbreviations p̄ (più) and m̄ (meno).

3. The Equals Sign (=): Robert Recorde's Gemowe Lines

For centuries, mathematicians wrote phrases like aequales, gleich, or is equal to between expressions.

In 1557, Welsh scholar Robert Recorde grew exhausted of repeating these words while writing his algebra textbook, The Whetstone of Witte. He wrote:

"And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe [twin] lines of one lengthe, thus: ======, bicause noe 2 thynges, can be moare equalle."

Recorde's original equals sign was significantly longer than our modern symbol. It took over a century to achieve universal adoption; rival mathematicians in France and the Netherlands continued using two vertical bars ($||$) or a crescent shape ($\propto$) well into the 1700s.

4. Multiplication (×, ·) and Division (÷, /)

The Multiplication Cross vs Centered Dot

English mathematician William Oughtred (inventor of the slide rule) introduced the St. Andrew's cross ($\times$) for multiplication in his 1631 work Clavis Mathematicae.

However, Gottfried Wilhelm Leibniz violently objected. Writing in a letter to Henry Oldenburg in 1698:

"I do not like $\times$ as a symbol for multiplication, as it is easily confused with $x$."

Leibniz instead championed the centered interpunct dot ($\cdot$), which became the standard in scientific and tensor algebra.

The Division Obelus (÷)

The division sign with dots above and below a bar ($\div$) is called an obelus. In ancient Alexandrian manuscripts, the symbol was used by editorial critics to mark corrupt or spurious lines of text.

Swiss mathematician Johann Rahn first used $\div$ to denote arithmetic division in his 1659 algebra book Teutsche Algebra. While popular in primary schools in the English-speaking world, higher mathematics and European universities almost exclusively favor horizontal fraction bars $\frac{a}{b}$ (introduced by Islamic scholars and Fibonacci) or the solidus forward slash $a/b$ (popularized by De Morgan in 1845).

5. The Square Root (√) and Descartes' Vinculum

In medieval Latin texts, the square root was written as Radix (Latin for "root" or "branch"). German mathematician Christoph Rudolff simplified this in 1525 in his book Coss by using a stylized lowercase letter r with an elongated downward stroke: $\sqrt{}$.

Originally, Rudolff's symbol did not extend over the numbers. To take the square root of multiple terms, mathematicians had to write dots. In 1637, René Descartes combined Rudolff's radical sign with a horizontal bar called a vinculum, creating the unified symbol we use today: $\sqrt{x^2 + y^2}$.

6. The Symbol Pi (π): From William Jones to Euler

The circle ratio—circumference divided by diameter—was calculated by Archimedes as lying between $3\frac{10}{71}$ and $3\frac{1}{7}$, but neither the Greeks nor the Romans had a single character for it.

In 1706, self-taught Welsh mathematics tutor William Jones published Synopsis Palmariorum Matheseos, where he selected the Greek letter $\pi$ as an abbreviation for $\pi\epsilon\rho\iota\phi\epsilon\rho\epsilon\iota\alpha$ (peripheria, the Greek word for circumference).

Despite Jones's publication, the symbol languished in obscurity until Leonhard Euler adopted it in his landmark 1737 treatise Mechanica. Because Euler was the most prolific and influential mathematician in Europe, his adoption made $\pi$ the universal global constant.

7. Infinity (∞): John Wallis's Mysterious Lemniscate

The sideways figure-eight symbol for infinity ($\infty$) is known mathematically as a lemniscate (from the Latin lemniscus, meaning "ribbon").

English mathematician John Wallis introduced it in 1655 in his treatise on conic sections, De Sectionibus Conicis. Wallis never explained exactly why he chose the figure, leading to several intriguing historical theories:

• It represents the ancient Roman numeral for 1,000 (often written as $\text{CI}\supset$ or $\infty$), which Romans used colloquially to mean "an unimaginably large number".
• It represents the Greek uppercase letter Omega ($\omega$ or $\Omega$), the final letter of the Greek alphabet, symbolizing the ultimate bound.
• It derives from the mythical Ouroboros—the ancient serpent swallowing its own tail, symbolizing endless cyclical eternity.

8. Leonhard Euler: The Father of Modern Notation

More than any other single human being in history, Swiss polymath Leonhard Euler (1707–1783) standardized the mathematical notation we teach in schools today. Euler's inventions include:

• $f(x)$: Function notation (1734)
• $e$: The base of natural logarithms ($2.71828\dots$, 1727)
• $i$: The imaginary unit $\sqrt{-1}$ (1777)
• $\sum$: Capital Greek Sigma for summation (1755)
• $a, b, c$ and $A, B, C$: Sides and opposite angles of triangles

Euler combined five of these fundamental constants into what physicists and mathematicians consider the most beautiful theorem in all of human thought:

9. Chronological Timeline of Mathematical Notation

Modern Symbol Typing Tools

Generations of mathematicians spent centuries inventing these symbols. Today, you can access every single one instantly. Visit our Symbols Keyboard to type and copy all historical and modern mathematical characters right in your browser.