Why do we use letters like x and y to represent numbers in algebra?
In: Mathematics
[Here’s](https://www.smithsonianmag.com/history/a-brief-history-of-the-letter-x-from-algebra-to-x-mas-to-elon-musk-180982647/) an article by the Smithsonian about it.
We use letters in math to show different quantities or unknown values in equations or formulas. It helps to keep things organized and avoid confusion when solving problems.
We want to describe the math as clearly as possible, right? So we need to use symbols that people aren’t going to confuse with math symbols. We also need a lot of them! So what are a bunch of symbols that everyone in Europe was familiar with? The alphabet! They also are very simple and quick to write. Bonus points, we can assign letters that make sense with the quantities we are using so they are easy to remember. Like for length, width, and height, we can use L, W, and H.
There are other reasons it persists, but that’s the core. We try not to use e thanks to Euler’s constant using that letter, but otherwise we get so many symbols to use. By my count, we get about 40 easily distinguishable symbols with lowercase and uppercase letters.
If you do math within a given specialty, you might find they prefer particular symbols instead of the alphabet. Like we use capital theta (a Greek letter) to show an angle. This is a tradition from when the Greeks did the same, and now most mathematicians know that it means angle. It seems silly, but since everyone knows what it means, it makes it a good symbol for something that shows up a LOT. Electrical formulas sometimes use capital omega for resistance, because the resistance unit (Ohm) has omega as its symbol (though mostly R is used instead, R for resistance!).
Why do sometimes use letters and sometimes other symbols? Letters are great when you tell someone that “y means this”, they know the letter and only need to remember the connection. Other symbols are used for things where everyone doing that math would know the symbol already, because it’s really important for that math.
Because it’s easier to write letters than drawing little squares, triangles and circles (children do algebra in 2nd/3rd grade when learning to add and substract) or drawing litte fruits or flowers (a lot of people do algebra in the form of “Only 2% guess the right answer!!!” challenges)
Fun fact: computer science has it’s own version of these, called [Metasyntactic variables](https://en.wikipedia.org/wiki/Metasyntactic_variable) which are used as stand-ins for “could be anything here”. Using x, y, etc. would be confusing because it might look like mathematics code. So you get `foo` and `bar` everywhere, then maybe `wibble` and `blarg` in New Zealand or different ones in other countries.
We started with ‘x’ the unknown and naturally went down the alphabet from there. x,y are used for 2D coordinates and x,y,z for 3D coordinates. We’re lucky we live in a 3 dimensional world because x,y,z,a is really ugly.
The magic of algebra is that we can “do math” using placeholders for different quantities in our equations and it actually still works!
We don’t *need* to use letters as the placeholders, we totally *could* do things like the following:
> 4×(Something)-8=12
> 4×🤔-8=12
However, writing out “(Something)” over and over again would get really tedious and make your hand cramp up, and drawing a little symbol or emoji over and over isn’t much better… on the other hand:
> 4×S-8=12
writing single letters like “S” is super easy.
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As for *”Why ‘x’ all the time?”* that’s a quirk of history.
Just like how in my example above I used a single letter “S” to abbreviate the “Something” placeholder, the original Algebra textbooks basically used “Thing” as their placeholder word… but in Arabic the word for “Thing” starts with “x” and so “Solve for x.” simply means “Solve for (Thing)”.
When Europeans started learning Algebra and making translations of the Arabic books, they kept using “x”. Maybe because they missed the fact that it was basically an abbreviation, or maybe just because x is just so nice and easy to write.
It’s a way to show the unknown number as an unknown number. It looks like none of the numbers 0-9 and is therefore hard to mistake for any of them.
In terms of constants, it can be viewed as redundant. Why not just use the actual number instead of a letter to represent it? In the case of large numbers, they help keep equations shorter. If I have a number like 0.0005, instead of writing that every time, I can represent it with a letter, and just write that letter instead. I know that every time I see that letter chosen, it represents 0.0005, saving me time.
But in terms of variables, and unknown quantities, they are simply placeholders for numbers not yet known. We could technically use any symbol we want to represent the unknown numbers, but we use letters as they are familiar.
Convenient unique label for unknown variables. That way you can keep track of them in the formula. If all unknowns were ? or X, we wouldn’t know which value it is.
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