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X marks the unknown in algebra – but X’s origins are a math mystery

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X marks the unknown in algebra – but X’s origins are a math mystery

The use of the letter x as a mathematical unknown is a relatively modern convention. Algebra has been around for a lot longer.
Daryl Benson/Stockbyte via Getty Images

Peter Schumer, Middlebury

Even though x is one of the least-used letters in the English alphabet, it appears throughout American culture – from Stan Lee’s X-Men superheroes to “The X-Files” TV series. The letter x often symbolizes something unknown, with an of mystery that can be appealing – just look at Elon Musk with SpaceX, Tesla’s Model X, and now X as a new name for Twitter.

You might be most familiar with x from math class. Many algebra problems use x as a variable, to stand in for an unknown quantity. But why is x the letter chosen for this role? When and where did this convention begin?

There are a few different explanations that math enthusiasts have put forward – some citing translation, others pointing to a more typographic origin. Each theory has some merit, but historians of mathematics, like me, know that it’s difficult to say for sure how x got its role in modern algebra.

Ancient unknowns

Algebra is a branch of math in which abstract symbols are manipulated, using arithmetic, to solve different kinds of equations. But many ancient societies had well-developed mathematical systems and knowledge with no symbolic notation.

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All ancient algebra was rhetorical. Mathematical problems and were completely written out in words as part of a little story, much like the word problems you might see in elementary school.

A piece of papyrus in a wooden frame, with black and red writing.
A portion of the Rhind Mathematical Papyrus, dated circa 1650 B.C.E.
The Trustees of the British Museum, CC BY-NC-SA

Ancient Egyptian mathematicians, who are perhaps best known for their geometric advances, were skilled in solving simple algebraic problems. In the Rhind papyrus, the scribe Ahmes uses the hieroglyphics referred to as “aha” to denote the unknown quantity in his algebraic problems. For example, problem 24 asks for the value of aha if aha plus one-seventh of aha equals 19. “Aha” means something like “mass” or “heap.”

The ancient Babylonians of Mesopotamia used many different words for unknowns in their algebraic system – typically words meaning length, width, area or volume, even if the problem itself was not geometric in nature. One ancient problem involved two unknowns termed the “first silver thing” and the “second silver thing.”

Mathematical know-how developed somewhat independently in many lands and in many languages. Limitations in communication prevented any immediate standardization of notation. However, over time some abbreviations crept in.

In a transitional syncopated phase, authors used some symbolic notation, but algebraic ideas were still presented mainly rhetorically. Diophantus of Alexandria used a syncopated algebra in his great work Arithmetica. He called the unknown “arithmos” and used an archaic Greek letter similar to s for the unknown.

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Indian mathematicians made additional algebraic discoveries and developed what are essentially the modern symbols for each of the decimal digits. One especially influential Indian mathematician was Brahmagupta, whose algebraic techniques could handle any quadratic equation. Brahmagupta’s name for the unknown variable was yãvattâvat. When additional variables were required, he instead used the initial syllable of color names, like kâ from kâlaka (black), ya from yavat tava (yellow), ni from nilaka (blue), and so on.

Two pages of a book with writing in Arabic.
In the treatise Al-jabr wa’l muqabalah, the words al-jabr and muqabalah roughly mean ‘restoration’ and ‘reduction,’ respectively.
Al-Khwarizmi via Wikimedia Commons

Islamic scholars translated and preserved a great deal of both Greek and Indian scholarship that has contributed immensely to the world’s mathematical, scientific and technical knowledge. The most famous Islamic mathematician was al-Khowarizmi, whose foundational book Al-jabr wa’l muqabalah is at the root of the modern word “algebra.”

So what about x?

One theory of the genesis of x as the unknown in modern algebra points to these Islamic roots. The theory contends that the Arabic word used for the quantity being sought was al-shayun, meaning “something,” which was shortened to the symbol for its first “sh” sound. When Spanish scholars translated the Arabic mathematical treatises, they lacked a letter for the “sh” sound and instead chose the “k” sound. They represented this sound by the Greek letter χ, which later became the Latin x.

It’s not unusual for a mathematical expression to about through convoluted translations – the trigonometric word “sine” started as a Hindu word for a half-chord but, through a series of translations, ended up coming from the Latin word “sinus,” meaning bay. However, there is some evidence that casts doubt upon the theory that using x as an unknown is an artifact of Spanish translation.

The Spanish alphabet includes the letter x, and early Catalonian involved several pronunciations of it depending on context, a pronunciation akin to the modern sh. Although the sound changed pronunciation over time, there are still vestiges of the sh sound for x in Portuguese, as well as in Mexican Spanish and its use in native place names. By this reasoning, Spanish translators conceivably could have used x without needing to resort first to the Greek χ and then to the Latin x.

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Moreover, although the letter x may have been used in mathematics during the Middle Ages sporadically, there is no consistent use of it dating back that far. Western mathematical texts over the next several centuries still used a variety of words, abbreviations and letters to represent the unknown.

For instance, a typical problem in the algebra book “Sumario Compendioso of Juan Diez,” published in Mexico in 1556, uses the word “cosa” – meaning “stuff” or “thing” – to stand in for the unknown.

A portrait of a man with a mustache. He has shoulder-length brown hair and is wearing a black robe with a white collar.
The French mathematician, scientist and philosopher René Descartes.
Painting by Frans Hals via Wikimedia Commons

I think that the most plausible explanation is to credit the influential French scholar René Descartes for the modern use of x. In an appendix to his major work “Discourse” in the 17th century, Descartes introduced a version of analytic geometry – in which algebra is used to solve geometric problems. For unspecified constants he chose the first few letters of the alphabet, and for variables he chose the last letters in reverse order.

Although scholars may never know for sure, some theorize that Descartes may have chosen the letter x to appear often since the printer had a large cache of x’s because of its scarcity in the French language. Whatever his reasons for choosing x, Descartes greatly influenced the of mathematics, and his mathematical writings were widely circulated.

Xtending beyond algebra

Even if the origins of x in algebra are uncertain, there are some instances in which historians do know why x is used. The X in Xmas as an abbreviation for Christmas definitely does come from the Greek letter χ. The Greek word for Christ is Christos, written χριστοσ and meaning “anointed.” The χ monogram was used as a shorthand for Christ in both Roman Catholic and Eastern Orthodox writings dating back as far as the 16th century.

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There are also some contexts in which x was chosen specifically to indicate something unknown or extra, such as when the German physicist Wilhelm Roentgen accidentally discovered X-rays in 1895 while experimenting with cathode rays and glass.

But there are other cases in which scholars can only guess about the origins of x’s role, such as the phrase “X marks the spot.” And there are other contexts – such as Elon Musk’s affinity for the letter – that may just be a matter of personal .The Conversation

Peter Schumer, Professor of Mathematics and Natural Philosophy, Middlebury

This article is republished from The Conversation under a Creative Commons license. Read the original article.

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Will your phone one day let you smell as well as see and hear what’s on the other end of a call?

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theconversation.com – Jian Liu, Assistant Professor of Electrical Engineering and Computer Science, of Tennessee – 2024-09-16 07:27:05

Phones that transmit odors seem like a great idea, but careful what you wish for!

Teo Mahatmana/iStock via Getty Images

Jian Liu, University of Tennessee

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Curious Kids is a for children of all ages. If you have a question you’d like an expert to answer, send it to curiouskidsus@theconversation.com.


Is it possible to make a phone through which we can smell, like we can hear and see? – Muneeba K., age 10, Pakistan


Imagine this: You pick up your phone for a call with a friend. Not only can you see their face and hear their voice, but you can also smell the cookies they just baked. It sounds like something out of a science fiction , but could it actually happen?

I’m a computer scientist who studies how machines sense the world.

What phones do now

When you listen to music or talk to someone on your phone, you can hear the sound through the built-in speakers. These speakers convert digital signals into physical vibrations using a tiny component called a diaphragm. Your ears sense those vibrations as sound waves.

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Your phone also has a screen that displays images and . The screen uses tiny dots known as pixels that consist of three primary colors: red, green and blue. By mixing these colors in different ways, your phone can show you everything from beautiful beach scenes to cute puppies.

Smelling with phones

Now how about the sense of smell? Smells are created by tiny particles called molecules that float through the and reach your nose. Your nose then sends signals to your brain, which identifies the smell.

So, could your phone send these smell molecules to you? Scientists are working on it. Think about how your phone screen works. It doesn’t have every color in the world stored inside it. Instead, it uses just three colors to create millions of different hues and shades.

How your sense of smell works.

Now imagine something similar for smells. Scientists are developing digital scent technology that uses a small number of different cartridges, each containing a specific scent. Just like how pixels mix three colors to create images, these scent cartridges could mix to create different smells.

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Just like images on your phone are made of digital codes that represent combinations of pixels, smells produced by a future phone could be created using digital codes. Each smell could have a specific recipe made up of different amounts of the ingredients in the cartridges.

When you a digital scent code, your phone could mix tiny amounts of the different scents from the cartridges to create the desired smell. This mix would then be released through a small vent on the phone, allowing you to smell it. With just a few cartridges, your phone could potentially create a huge variety of smells, much like how red, green and blue pixels can create countless colors.

Researchers and companies are already working on digital odor makers like this.

The challenges to making smell phones

Creating a phone that can produce smells involves several challenges. One is designing a system that can produce thousands of different smells using only a few cartridges. Another is how to control how strong a scent should be and how long a phone should emit it. And phones will also need to sense odors near them and convert those to digital codes so your friends’ phones can send smells to you.

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The cartridges should also be easy to refill, and the chemicals in them be safe to breathe. These hurdles make it a tricky but exciting area of research.

An odiferous future

Even though we’re not there yet, scientists and engineers are working hard to make smell phones a reality. Maybe one day you’ll be able to not only see and hear your friend’s birthday party over the phone, but also smell the candles they blew out!


Hello, curious kids! Do you have a question you’d like an expert to answer? Ask an adult to send your question to CuriousKidsUS@theconversation.com. Please tell us your name, age and the where you .

And since curiosity has no age limit – adults, let us know what you’re wondering, too. We won’t be able to answer every question, but we will do our best.The Conversation

Jian Liu, Assistant Professor of Electrical Engineering and Computer Science, University of Tennessee

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a double shot of US history

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theconversation.com – Kyle G. Volk, Professor of History, University of Montana – 2024-09-16 07:28:46

a beer in Raceland, La.

Russell Lee for Farm Security Administration/WPA

Kyle G. Volk, University of Montana

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Text saying: Uncommon Courses, from The Conversation

Uncommon Courses is an occasional from U.S. highlighting unconventional approaches to teaching.

Title of course:

“Intoxication Nation: Alcohol in American History”

What prompted the idea for the course?

I wanted to get students about studying the past by learning about something that is very much a part of their own lives.

Alcohol – somewhat surprisingly to me at first – featured prominently in my own research on minority rights and U.S. democracy in the mid-19th century. As a result, I knew quite a bit about the temperance movement and conflicts over prohibition during that period. Designing this course me to broaden my expertise.

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What does the course explore?

Prohibition is a must-do subject. Students expect it. But I several hundred years of history: from the 17th-century invention of rum – as a byproduct of sugar produced by enslaved people – to the rise of craft beer and craft spirits in the 21st century.

A faded poster with an illustration of a person about to smash a huge bottle of alcohol, and the message 'Close the saloons' at the top.

A temperance poster from the World War I era.

Office of Naval Records and Library via National Archives Catalog

Along the way, I’m thrilled when students get excited about details that allow them to a more complicated historical cocktail. For example, they learn why white women’s production of hard cider was crucial to the survival of colonial Virginia. The short answer: Potable water was in short supply, alcoholic drinks were far healthier, and white – and their indentured and enslaved workforce – were busy raising tobacco. It fell to women to turn fruit into salvation.

Why is this course relevant now?

Alcohol remains a big and almost inescapable part of American society. But of late, Americans have been drinking differently – and thinking about drinking differently.

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Examples abound. Alcohol producers, we learn, now face competition from legalized weed. Drinking l evels rose during the COVID-19 pandemic, yet interest is declining among Gen Zers. The “wine mom” culture that brought some mothers together now faces mounting criticism.

And, of course, there’s the never-ending debate about the health benefits and risks of alcohol. Of late, the risks seem to be dominating headlines.

What’s a critical lesson from the course?

Alcohol has been a highly controversial, central aspect of the American experience, shaping virtually all sectors of our society – political and constitutional, business and economic, social and cultural.

What materials does the course feature?

What will the course prepare students to do?

Like any history course, this one aims to develop student’s analytical, written, research and verbal skills. In lots of ways, the topic is just a tool to get students to grow their brains. But I also seek to grow students’ critical awareness of the place of alcohol in their own lives. The course has also informed students’ paths after graduation – including some who wound up working in the alcohol industry or recovery .The Conversation

Kyle G. Volk, Professor of History, University of Montana

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Sunflowers make small moves to maximize their Sun exposure − physicists can model them to predict how they grow

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theconversation.com – Chantal Nguyen, Postdoctoral Associate at the BioFrontiers Institute, of Colorado Boulder – 2024-09-13 07:31:40

Sunflowers use tiny movements to follow the Sun’s path throughout the day.

AP Photo/Charlie Riedel

Chantal Nguyen, University of Colorado Boulder

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Most of us aren’t spending our days watching our houseplants grow. We see their signs of only occasionally – a new leaf unfurled, a stem leaning toward the window.

But in the summer of 1863, Charles Darwin lay ill in bed, with nothing to do but watch his plants so closely that he could detect their small movements to and fro. The tendrils from his cucumber plants swept in circles until they encountered a stick, which they proceeded to twine around.

“I am getting very much amused by my tendrils,” he wrote.

This amusement blossomed into a decadeslong fascination with the little-noticed world of plant movements. He compiled his detailed observations and experiments in a 1880 book called “The Power of Movement in Plants.”

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A zig-zagging line showing the movement of a leaf.

A diagram tracking the circumnutation of a leaf over three days.

Charles Darwin

In one study, he traced the motion of a carnation leaf every few hours over the course of three days, revealing an irregular looping, jagged path. The swoops of cucumber tendrils and the zags of carnation leaves are examples of inherent, ubiquitous plant movements called circumnutations – from the Latin circum, meaning circle, and nutare, meaning to nod.

Circumnutations vary in size, regularity and timescale across plant species. But their exact function remains unclear.

I’m a physicist interested in understanding collective behavior in living . Like Darwin, I’m captivated by circumnutations, since they may underlie more complex phenomena in groups of plants.

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Sunflower patterns

A 2017 study revealed a fascinating observation that got my colleagues and me wondering about the role circumnutations could play in plant growth patterns. In this study, researchers found that sunflowers grown in a dense row naturally formed a near-perfect zigzag pattern, with each plant leaning away from the row in alternating directions.

This pattern the plants to avoid shade from their neighbors and maximize their exposure to sunlight. These sunflowers flourished.

Researchers then planted some plants at the same density but constrained them so that they could grow only upright without leaning. These constrained plants produced less oil than the plants that could lean and get the maximum amount of sun.

While farmers can’t grow their sunflowers quite this close together due to the potential for disease spread, in the future they may be able to use these patterns to up with new planting strategies.

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Self-organization and randomness

This spontaneous pattern formation is a neat example of self-organization in nature. Self-organization refers to when initially disordered systems, such as a jungle of plants or a swarm of bees, achieve order without anything controlling them. Order emerges from the interactions between individual members of the system and their interactions with the .

Somewhat counterintuitively, noise – also called randomness – facilitates self-organization. Consider a colony of ants.

Ants secrete pheromones behind them as they crawl toward a food source. Other ants find this food source by the pheromone trails, and they further reinforce the trail they took by secreting their own pheromones in turn. Over time, the ants converge on the best path to the food, and a single trail prevails.

But if a shorter path were to become possible, the ants would not necessarily find this path just by following the existing trail.

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If a few ants were to randomly deviate from the trail, though, they might stumble onto the shorter path and create a new trail. So this randomness injects a spontaneous change into the ants’ system that allows them to explore alternative scenarios.

Eventually, more ants would follow the new trail, and soon the shorter path would prevail. This randomness helps the ants adapt to changes in the environment, as a few ants spontaneously seek out more direct ways to their food source.

A group of honeybees spread out standing on honeycomb.

Beehives are an example of self-organization in nature.

Martin Ruegner/Stone via Getty Images

In biology, self-organized systems can be found at a range of scales, from the patterns of proteins inside cells to the socially complex colonies of honeybees that collectively build nests and forage for nectar.

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Randomness in sunflower self-organization

So, could random, irregular circumnutations underpin the sunflowers’ self-organization?

My colleagues and I set out to explore this question by following the growth of young sunflowers we planted in the lab. Using cameras that imaged the plants every five minutes, we tracked the movement of the plants to see their circumnutatory paths.

We saw some loops and spirals, and lots of jagged movements. These ultimately appeared largely random, much like Darwin’s carnation. But when we placed the plants together in rows, they began to move away from one another, forming the same zigzag configurations that we’d seen in the previous study.

Five plants and a diagram showing loops and jagged lines that represent small movements made by the plants.

Tracking the circumnutations made by young sunflower plants.

Chantal Nguyen

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We analyzed the plants’ circumnutations and found that at any given time, the direction of the plant’s motion appeared completely independent of how it was moving about half an hour earlier. If you measured a plant’s motion once every 30 minutes, it would appear to be moving in a completely random way.

We also measured how much the plant’s leaves grew over the course of two weeks. By putting all of these results together, we sketched a picture of how a plant moved and grew on its own. This information allowed us to computationally model a sunflower and simulate how it behaves over the course of its growth.

A sunflower model

We modeled each plant simply as a circular crown on a stem, with the crown expanding according to the growth rate we measured experimentally. The simulated plant moved in a completely random way, taking a “step” every half hour.

We created the model sunflowers with circumnutations of lower or higher intensity by tweaking the step sizes. At one end of the spectrum, sunflowers were much more likely to take tiny steps than big ones, leading to slow, minimal movement on average. At the other end were sunflowers that are equally as likely to take large steps as small steps, resulting in highly irregular movement. The real sunflowers we observed in our experiment were somewhere in the middle.

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Plants require light to grow and have evolved the ability to detect shade and alter the direction of their growth in response.

We wanted our model sunflowers to do the same thing. So, we made it so that two plants that get too close to each other’s shade begin to lean away in opposite directions.

Finally, we wanted to see whether we could replicate the zigzag pattern we’d observed with the real sunflowers in our model.

First, we set the model sunflowers to make small circumnutations. Their shade avoidance responses pushed them away from each other, but that wasn’t enough to produce the zigzag – the model plants stayed stuck in a line. In physics, we would call this a “frustrated” system.

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Then, we set the plants to make large circumnutations. The plants started moving in random patterns that often brought the plants closer together rather than farther apart. Again, no zigzag pattern like we’d seen in the field.

But when we set the model plants to make moderately large movements, similar to our experimental measurements, the plants could self-organize into a zigzag pattern that gave each sunflower optimal exposure to light.

So, we showed that these random, irregular movements helped the plants explore their surroundings to find desirable arrangements that benefited their growth.

Plants are much more dynamic than people give them credit for. By taking the time to follow them, scientists and farmers can unlock their secrets and use plants’ movement to their advantage.The Conversation

Chantal Nguyen, Postdoctoral Associate at the BioFrontiers Institute, University of Colorado Boulder

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