X: The Notation You Didn't Know Was Broken

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You're sitting in algebra class. Your teacher writes on the board: 2x + 3(x + 4) = 5x

You don't even blink at it. But imagine if your teacher had written: 2 × x + 3 × (x + 4) = 5 × x

That looks horrible, right? Your brain gets confused. Is that the letter x? Is that multiplication? What's going on?

This is exactly the problem mathematicians ran into about 400 years ago. And their solution was radical: they basically just stopped using the multiplication symbol in certain situations.

But here's the weird part: this change didn't happen because someone made a grand decision in a math conference. It happened slowly, painfully, and grudgingly, as mathematicians realized they had created a notational disaster. The story of why we write 2x instead of 2 × x is actually a fascinating look at how symbols gain meaning, how mistakes spread, and how it takes centuries to fix them.

The Beginning: When × Was New and Cool

To understand the problem, we need to go back to the 17th century. For most of human history, multiplication didn't have a symbol. If you wanted to multiply 5 by 3, you either wrote it out in words ("five times three") or you just put the numbers next to each other and hoped people understood what you meant.

Then, in the 1600s, an English mathematician named William Oughtred got tired of this ambiguity. In 1631, he invented the multiplication symbol we all know: × This symbol became popular because it worked great. It was clear. It was unambiguous. When you saw ×, you knew someone was multiplying. It was a shorthand. It saved time and ink.

For about 50 years, everything was fine. The × symbol spread across mathematics. Teachers taught it. Textbooks used it. Everyone was happy. Then, in the late 1600s, something changed that would make the × symbol a problem forever.

The Plot Twist: René Descartes Invents the Variable

Around the same time that the × symbol was becoming standard, a French mathematician named René Descartes was developing something revolutionary: algebraic notation. Before Descartes, algebra was mostly written out in words. If you wanted to solve a problem, you might write something like "A number, when multiplied by three and then increased by five, equals twenty. Find the number." Descartes thought this was inefficient. He wanted a way to write mathematics using symbols instead of sentences. And crucially, he needed a symbol to represent unknown numbers.

He could have chosen any letter. He could have used A for "unknown." He could have used U. He could have used Q.

But Descartes chose x. Why x? According to historical sources, the choice came from a translation issue. When Spanish scholars were translating Arabic mathematical texts, they kept coming across the Arabic word "shay-un," which means "thing." In mathematics, an unknown value is "a thing"—something we don't know yet.

Spanish doesn't have the "sh" sound, so translators substituted the Greek letter chi, which looks like X. When these Spanish texts were later translated into Latin and other European languages, the X stuck around. By the time Descartes was developing his notation, X was already becoming the standard symbol for "unknown."

So Descartes used x for his unknowns. And suddenly, the problem was created.

The Collision: When Symbols Started Fighting

Here's what happened:

In elementary mathematics, you had: 3 × 4 = 12 (the × symbol means multiply)

In algebra, you had: 3 × x = 15 (wait... is that × multiplication, or is that the letter x, or both?)

The problem sounds silly when you say it out loud, but try reading an equation with both symbols mixed in: 5 × x + 3 × y = 20

Now read that again carefully. Did your brain trip over which x's were the multiplication symbol and which were the variable x? Did you have to read it multiple times to parse it correctly? Imagine you're a 17th-century mathematician trying to read dozens of these equations every day. Your brain is constantly getting tripped up. The symbol for multiplication and the symbol for an unknown variable are identical (or at least very similar). It's a notational nightmare.

The Historical Problem: Two Symbols, One Shape

To make matters worse, the two symbols weren't even written exactly the same way. The multiplication symbol × was supposed to be a centered cross, symmetrical and precise. The letter x was just a letter, written naturally in whatever handwriting you used. But in handwritten mathematics, they look almost identical. In printed text, they still look very similar. Your brain has to do extra work to figure out which is which based on context. And the problem got worse as algebra became more common. As more mathematicians started using x as a variable, more equations had both symbols appearing together. By the late 1700s and 1800s, mathematicians were really struggling. The notation was confusing. Students were making mistakes. Equations were hard to read. Something had to give.

The Solution Emerges: Three Alternatives

Rather than change the × symbol or stop using x as a variable (both of which would require convincing every mathematician in the world to change their habits), mathematicians gradually adopted three alternatives:

Alternative 1: The Dot Notation

A German mathematician named Leibniz suggested using a small dot (·) to indicate multiplication instead of ×.

Instead of writing 3 × x, you'd write 3 · x. The dot is smaller, less visually prominent, and doesn't look like the letter x. It's easy to see that 3 · x is clearly multiplication, and x is clearly the variable. This solution worked reasonably well, and it's still used in some contexts today, especially in higher mathematics and in some European countries.

Alternative 2: Parentheses

Another option is using parentheses to make multiplication explicit without needing a symbol at all.

Instead of 3 × x, you write 3(x). Or instead of (a + b) × (c + d), you write (a + b)(c + d). The parentheses make it clear that you're multiplying the two expressions. And it works perfectly fine. In fact, as notation evolved, parentheses became the standard in most American textbooks.

Alternative 3: Juxtaposition (Just Put Them Next to Each Other)

This is the most radical solution, and it's the one that became most common in modern algebra: just write the variables next to each other with no symbol at all. Instead of 3 × x, write 3x. Instead of x × y, write xy. Instead of 2 × (a + b), write 2(a + b). This works because mathematicians reasoned: "In this context, we're working with variables, not numbers. Everyone knows that when you see a number next to a variable with nothing between them, it means multiplication."

But here's the rule: this only works with variables, not with numbers. You can write 3x to mean "3 times x," but you can't write 34 to mean "3 times 4" because 34 is already the number thirty-four.

So the rule is:

3x means 3 times x (no symbol needed)
3(4) means 3 times 4 (symbol or parentheses needed)
3 · 4 means 3 times 4 (dot symbol)
3 × 4 means 3 times 4 (but we try to avoid this in algebra)

How the Change Actually Happened

Here's an interesting thing: mathematicians didn't declare a universal law saying "Stop using × in algebra." Instead, the change happened gradually, textbook by textbook, teacher by teacher. By the 1900s, most higher-level mathematics textbooks had stopped using × in algebraic contexts. The dots, parentheses, and juxtaposition became standard. But the change was slow and inconsistent. Even today, some countries and some textbooks use different conventions. British mathematics education uses certain symbols differently than American education. Different mathematical traditions have slightly different preferences. However, by the time students reach algebra class in most modern American schools, the × symbol has essentially disappeared. Teachers don't explicitly say "We're stopping using ×," but students notice it anyway. Suddenly, multiplication is written completely differently.

Why This Matters: How Notation Shapes Math

Here's something important: the symbol you use isn't just a cosmetic choice. It actually affects how you think about mathematics.

When you write 3 × x, you're emphasizing the operation: "multiply 3 and x."

When you write 3x, you're emphasizing the result: "a quantity called three-x."

These are subtle differences, but they affect how your brain processes the mathematics. The different notations literally change the way you conceptualize what's happening. Also, the notation affects what you can do next. Once you start treating 3x as a single unit (which you kind of do when you write it without a symbol), you can do algebraic manipulations more naturally.

For example, if you have: 3x + 5x = ?

With the juxtaposition notation, it's clear that you're adding two quantities that are both some number of "x's." You can combine them to get 8x. With the original notation, it would look like: 3 × x + 5 × x = ? That's much harder to parse. You have to work through the order of operations. You have to remind yourself that you're multiplying first, then adding. The notation makes the math harder to see. By changing the notation, mathematicians actually made algebra easier to understand and work with.

The Bigger Picture: Why Notation Matters in Science

This story about multiplication symbols is actually about something much bigger: notation in science and mathematics.

Symbols are how scientists communicate complex ideas efficiently. A single symbol can represent volumes of meaning. The equation E = mc² is incredibly powerful because it's so compact. But symbols can also confuse if they're not well-designed. A symbol that looks like two different things will cause problems. A notation that's ambiguous will lead to mistakes. Over the history of science, many notations have been created, improved, abandoned, or redesigned. The × symbol in algebra is just one example.

Other examples include:

Calculus notation has changed multiple times. Isaac Newton used one notation, Leibniz used a different one, and the notation has evolved again since then. Today, the Leibniz notation (dy/dx) is standard, but Newton's dot notation (ẏ) is still used in some contexts.

Chemistry notation had to evolve because early attempts to represent molecules were confusing. Today, chemical formulas are standardized in ways that make it immediately clear what elements are present.

Physics notation uses different conventions in different countries. An American physicist and a German physicist might use slightly different symbols for the same thing.

The point is: notation isn't fixed. It evolves as people figure out what works best. And sometimes, that evolution takes centuries.

Why You Should Care: Notation Is Design

Understanding why we changed the multiplication notation teaches you something important about how knowledge actually works. Math isn't a fixed, eternal thing that mathematicians discovered in heaven and then taught us. It's something humans created, and like all human creations, it's imperfect. It has design flaws. It needs to be improved over time.

When you encounter confusing notation in any field, you can ask: "Why was this designed this way? Is there a historical reason? Could it be improved?" This kind of thinking applies far beyond math. Computer scientists design programming languages, and sometimes old programming languages get replaced because their notation is confusing. Doctors use medical abbreviations, and sometimes those abbreviations cause dangerous confusion (which is why there are official lists of abbreviations to avoid). The story of the × symbol and x variable collision is really a story about design, communication, and how humans improve their systems over time.

The Modern Situation: It's Still a Bit Messy

Interestingly, even though we've largely abandoned × in algebraic contexts, it hasn't completely disappeared.

In elementary school, you still learn with ×. Teachers use it. Textbooks use it. And it makes sense at that level because you don't have variables yet, so there's no confusion. But then you get to algebra, and it suddenly stops. Some teachers explain the reason. Many don't. Students just notice it's gone.

In some countries and some mathematical traditions, the × symbol is still used more frequently in algebra. In computer science, we often use * (asterisk) for multiplication because computers use that symbol. In some advanced mathematics, you might see different conventions altogether.

There's also the issue of digital text. How do you type a dot for multiplication? It's not always obvious on a keyboard. So in typed mathematics, you might see other conventions.

The bottom line: there's no single, universal standard. Different places use different notations. But the general trend in modern algebra is to avoid × and use parentheses, dots, or juxtaposition instead.

The Lesson: How Small Decisions Echo Through History

One last interesting thing: this entire problem came about because of a translation. Spanish scholars translating Arabic texts substituted X for SH. This choice spread through mathematics over centuries. And then it created a notational crisis that took hundreds of years to resolve. All because of one translation choice.

This shows how small decisions in one moment can have massive consequences that ripple through time. A translator trying to solve a problem (how do I represent the "sh" sound in Spanish) created a problem that hundreds of future mathematicians would have to deal with.

It's a good reminder that when you're designing anything (notation, symbols, words, computer systems), you should think ahead about how your choices might cause problems later.

Sources

Oughtred, William. "Clavis Mathematicae." Oxford University Press, 1631. (Historical text establishing the × symbol)
History of Algebra. Wikipedia Encyclopedia. Last accessed May 2026.
"Why Is 'X' Used to Represent the Unknown?" NBC News Science. nbcnews.com
Krista King Math. "Different Symbols for Multiplication." Online Math Education Resource, October 25, 2020.
Everyday Mathematics. "Multiplication Symbols." University of Chicago School Mathematics Project. everydaymath.uchicago.edu
"The Order of Operations and Variables: Introduction to Variables." Brigham Young University - Idaho. content.byui.edu
Goodreads Author Blog. "Mathematical Symbols: Origins and History." Author Profile Pages, 2020.
"Why Aren't We Using the Multiplication Sign?" Khan Academy Video Lecture. khanacademy.org
"Rhetorical Algebra and the Development of Symbolic Notation." Mathematics Education Research. scholarpedia.org
Descartes, René. "La Géométrie" (The Geometry). Appendix to Discourse on Method, 1637. (Historical text establishing x as a variable)