Mathematics has been practiced for more than 2,000 years, yet most symbols used today are far younger. The plus, minus, multiplication and division signs entered common use only a few centuries ago. Behind each mark stands a specific person, place or practical need. Historians such as Kate Kitagawa and Raúl Rojas have documented how these signs emerged, changed and spread across continents. Their work fits into broader discussions in modern science about how knowledge systems evolve over time.
Rojas, a professor at the University of Nevada, Reno, has collected origin stories for nearly 30 years. He presents them in his book The Language of Mathematics The Stories Behind the Symbols. Kitagawa, now at La Trobe University in Melbourne, emphasizes that mathematical notation reflects personal decisions and historical context. Many symbols were adapted, altered or even erased over time. Readers interested in how ideas circulate across disciplines can find more here.
Table of contents
- Raúl Rojas and German merchants in 1489
- William Oughtred and Johann Rahn in England and Switzerland
- Al-Khwarizmi and algebra in the ninth century
- Archimedes, William Jones and the symbol π
- Karl Weierstrass and Giuseppe Peano in the nineteenth century
Raúl Rojas and German merchants in 1489
The rise of plus and minus in sea trade
The symbols + and – first appeared in a German mathematics text in 1489. Initially, they did not indicate addition or subtraction. They marked surplus and deficit in trade records. This shift coincided with the rapid growth of maritime commerce in the late 15th century.
Before symbolic notation, merchants wrote calculations entirely in words. A shipment description could require 234 characters. Using symbols reduced that to 83 characters, a decrease of 65 percent. Later shorthand could compress the same data to just 26 characters.
Key changes included
- replacing words with + and –
- grouping goods numerically
- simplifying repeated quantities
The adoption of symbols reduced writing time and simplified accounting for port officials and tax collectors.
William Oughtred and Johann Rahn in England and Switzerland
The spread of x and ÷ across Europe
The multiplication sign x is traced to William Oughtred. In 1631, he used it in a widely read textbook. He also introduced the colon as a division marker.
Earlier, the 12th-century Moroccan scholar al-Hassar used a horizontal bar to separate quantities, forming fractions. The modern ÷ symbol combines Oughtred’s colon with this dividing line. In 1659, Swiss mathematician Johann Rahn included ÷ in a book, though its exact origin remains uncertain.
Mathematician Sarah Hart notes that division notation reflects centuries of cross-cultural exchange. The evolution of ÷ demonstrates how mathematical ideas traveled from the Arab world to Europe.
Modern mathematics relies on a full system of symbols and notation that once did not exist - historians reveal how this unique language gradually evolved:
Film: YouTube / Kanal TED-Ed
Al-Khwarizmi and algebra in the ninth century
Practical mathematics for judges and merchants
The ninth-century scholar Al-Khwarizmi wrote a guide for judges on dividing inheritances. It contained no symbols. Everything was explained in words. His methods provided structured procedures for solving legal and commercial disputes.
Three centuries later, the text was translated into Latin. That translation carried algebra into Europe. Symbols such as +, –, x and ÷ appeared later, but the conceptual structure was already established. Historical reinterpretations of scientific ideas, similar to debates such as dinosaur speed questioned, show how even established knowledge can be reassessed.
Amir Alexander of the University of California, Los Angeles, points out that ancient Egyptians and Babylonians used mathematics to
- calculate taxes
- measure grain storage
- design stable buildings
Algebra developed from concrete legal and economic needs rather than abstract theory.
Archimedes, William Jones and the symbol π
From the Rhind papyrus to Srinivasa Ramanujan
The constant π represents the ratio between a circle’s circumference and diameter, approximately 3.14159. Around 3,600 years ago, Babylonians and Egyptians approximated it as 256 divided by 81, or 3.16 in decimal form. Problem 48 of the Rhind papyrus documents this early calculation. The manuscript, 5.2 meters long, is preserved in the British Museum in London.
More than 1,000 years later, Archimedes refined the value using geometry. In the early 1700s, Welsh mathematician William Jones introduced the symbol π, likely referencing the Greek word for perimeter.
In the 20th century, Srinivasa Ramanujan proposed formulas yielding the first nine exact digits of π. The symbol π represents a concept refined over more than three millennia.
Karl Weierstrass and Giuseppe Peano in the nineteenth century
Symbolic logic and absolute value
Karl Weierstrass, initially directed by his father to study law and finance, later became a mathematics professor at the University of Berlin at age 41. He is credited with introducing the absolute value symbol, written as two vertical bars enclosing a number. For example, |-4| = 4.
In 1858, Italian mathematician Giuseppe Peano began promoting symbolic logic. He attempted to express mathematics without words to remove language barriers. British mathematicians briefly adopted similar approaches in the early 1910s. However, fully symbolic writing proved difficult to interpret. Modern mathematics now combines words and notation.
Today’s mathematical language blends symbols and text, reflecting centuries of adaptation and exchange.
FAQ
When did the plus and minus signs first appear?
They first appeared in a German mathematics text in 1489.
What were the original meanings of the plus and minus symbols?
They were first used to denote surplus and deficit in trade records, not to add or subtract numbers.
Who is linked to the multiplication sign x?
William Oughtred is traced as the first known person to use x for multiplication, and he included it in a widely read textbook in 1631.
How did the modern division symbol ÷ develop?
It combines William Oughtred’s colon with the horizontal fraction bar credited to the 12th-century Moroccan scholar al-Hassar, and Johann Rahn used ÷ in a 1659 book.
Why was algebra written without symbols in Al-Khwarizmi’s work?
His ninth-century guide for judges explained procedures in words, focusing on practical legal tasks such as dividing inheritances.
Where is the Rhind papyrus kept today?
The 5.2-meter Rhind papyrus manuscript is housed in the British Museum in London.
Who first used the symbol π for the constant pi?
Welsh mathematician William Jones is thought to have first used π for the Archimedes constant in the early 1700s.
Which mathematician is credited with the absolute value symbol?
Karl Weierstrass is credited with developing the absolute value symbol, written as vertical bars enclosing a number.
Why did attempts to write mathematics using only symbols decline?
By the early 1910s, heavily symbolic writing proved hard to follow, so modern mathematics returned to blending symbols with explanatory words.
Source: Science News Explores