🇺🇸▼
Have you ever paused to consider the origins of the mathematical symbols you use daily? Everyone who engages with mathematics utilizes signs such as the plus (+) and minus (-). However, these symbols are not ancient artifacts; they are relatively modern inventions. In fact, they have been in consistent use for less than two hundred years. Many individuals find these symbols to be a source of stress. Nevertheless, experts suggest that we should view them not as obstacles, but as helpful allies. Each of these allies possesses an odd, entertaining, or colorful history waiting to be discovered.
The primary goal of mathematics is to uncover clear, undeniable truths. Yet, the symbols we rely on today emerged from the personal decisions of a select few influential figures. "Every math symbol carries with it a unique and often complex history," states Kate Kitagawa. She serves as a historian of mathematics at La Trobe University in Australia. She further adds, "Their stories are rarely straightforward."
Raúl Rojas knows these narratives intimately. For nearly thirty years, he has been collecting these historical accounts. Currently, he teaches mathematics at the University of Nevada, Reno. Rojas believes that understanding the history behind these symbols can significantly increase student appreciation for the subject. Unfortunately, these backstories are seldom taught in modern classrooms. Rojas is determined to change this reality. He shares these stories with his students and has compiled them into a book titled The Language of Mathematics.
It may be difficult to imagine a time before the plus and minus signs existed. However, these symbols were not utilized until 1489, when they appeared in a German mathematics book. Initially, people used them to mark goods that were extra or missing. They were not originally intended for adding or subtracting numbers. So, what prompted this shift?
In the late 1400s, sea trade expanded rapidly. Merchants in port cities had to meticulously track goods across many different ships. They recorded every single item in words, including the quantities. This process was incredibly slow and laborious.
Consider a simple shipment scenario: Ship one carried three crates of apples, with forty apples in each crate. It also transported two hundred fish: fifty flounder, seventy-five bream, twenty-seven sharks, and forty-eight anchovies. This totals three hundred and twenty items. Writing this narrative out required 234 letters and spaces. In contrast, using mathematical symbols, we could write: Ship 1 (320 items) = (50 flounder) + (75 bream) + (27 sharks) + (48 anchovies) + (3 crates of 40 apples). This concise version uses only 83 characters. Over time, this notation could be shortened even further.
A simple example: Ship one brought in three crates of apples, with each crate equal to forty apples. It also carried two hundred fish: Fifty were flounder, seventy-five were bream, twenty-seven were sharks and forty-eight were anchovies. That equals a shipment of three hundred and twenty items.
Merchants and tax collectors required a faster method. They began using symbols to save time and prevent the hand cramps associated with writing so extensively.
The symbols for multiplication and division took even longer to emerge. Rojas discovered that the "x" for multiplication was first used by William Oughtred. He was an English mathematics expert in the 1600s. Oughtred later employed a colon ":" to signify "divided by." His symbols gained popularity due to a textbook he published in 1631.
Long before Oughtred, Arab societies utilized a line to divide two numbers and create fractions. In the 1100s, a mathematics expert named al-Hassar from Morocco is credited with creating the horizontal fraction bar. Today, we use "÷" for division. Mathematician Sarah Hart explains that it is a combination of Oughtred's colon and al-Hassar's line. The Swiss mathematician Johann Rahn first used "÷" in a 1659 book.
"The story of mathematical symbols," Hart says, "is also the story of how mathematical ideas have spread around the world." These wordless symbols for mathematical concepts have evolved over centuries.
For thousands of years, mathematics has served a very practical purpose. Ancient Egyptians and Babylonians used it to calculate taxes and grain supplies. They utilized it to construct stable buildings.
However, today, mathematics often feels disconnected from practical application. "The way people are taught to think about math is that it is separate from our world," says math historian Amir Alexander. "Most people feel that, beyond pretty elementary math, it's sort of irrelevant for them." But he insists that this perception is incorrect.
Consider algebra, where symbols represent unknown numbers. In a classroom, you might encounter the equation 7 + a = 10 and solve for 'a'. This may not appear useful. Yet, algebra began as a method to solve significant legal and business problems.
A ninth-century Arabic scholar named al-Khwarizmi wrote a book about this concept. It was not a mathematics book in the traditional sense. It was a guide for judges. It explained how to fairly divide inheritances into specific shares. Think of it as a recipe. Once you understand how to solve one problem, you can adjust the steps to solve similar ones.
This knowledge was valuable for lawyers and merchants. Three hundred years later, the book was translated into Latin. This brought algebra from the Middle East to Europe. The symbols we now use for algebra, like +, -, x, and ÷, appeared later.
Consider the constant pi, represented by the letter π. It is the ratio of a circle's circumference to its diameter. Its value is approximately 3.14159. Today, π is used in complex calculations for astronomy and engineering. However, the history of this symbol demonstrates that it is more than just a number to memorize.
The story begins about 3,600 years ago with the Babylonians and Egyptians. They needed to survey land, which required finding the area of circular fields. They realized that the ratio of a circle's diameter to its circumference is always roughly the same. Their estimate was remarkably close to the true value of pi.
More than a thousand years later, the Greek scientist Archimedes used geometry to calculate pi more accurately. In his honor, pi was nicknamed the "Archimedes constant."
Later mathematicians worked to find its value with greater precision. The Indian genius Srinivasa Ramanujan, who died in 1920, discovered a formula for the first nine digits of pi. He claimed a Hindu goddess revealed the value to him in a dream.
The symbol π was first used in the early 1700s by the Welsh mathematician William Jones. He likely chose π because it is the first letter of the Greek word for perimeter.
"These tangential stories are a great way to bring [math] alive," says math writer Alex Bellos.
While studying, Rojas found many stories with surprising twists. Consider Karl Weierstrass, who lived in the 1800s. His father sent him to study law and finance. However, the young man enjoyed partying and was deeply interested in science.
He dropped out of law school, much to his father's displeasure. He moved to another university to study mathematics. This degree did not lead to a lucrative job, so he became a high school mathematics teacher.
"He had no money," says Rojas, "but he had a lot of good ideas." Weierstrass began solving extremely difficult mathematical problems. He even developed a new theory. When a major journal published his work, famous mathematicians took notice. At age 41, he secured a position as a professor in Germany.
Weierstrass is credited with creating the absolute value symbol: two vertical lines around a number, like |-4| = 4. Behind that simple symbol is a story of a late bloomer. Knowing people's stories can make mathematics feel less intimidating. "We should treat math as part of human heritage, not just science," says Bellos.
Once mathematical symbols became common, an interesting phenomenon occurred. Mathematicians began using fewer words. An Italian named Giuseppe Peano, born in 1858, attempted to write mathematics using only symbols. He wanted to break down language barriers and make mathematics accessible to everyone.
Other mathematicians tried this approach as well. However, a problem arose. Even experts found it difficult to understand work that consisted solely of symbols. The symbols could obscure meaning, functioning like a foreign language.
Nowadays, mathematicians blend symbols with words to explain their ideas. The symbols we possess are valuable tools. Rojas hopes that if people understand their origins, mathematics will seem less abstract.
"I find it deeply fascinating," says Kitagawa, "to see how what once began as culturally specific practices has become a universal language we call mathematics. Yet the journey is far from complete."
Mathematical symbols will continue to evolve as mathematicians share ideas and tackle great mysteries that may take centuries to solve.