Why do we use letters like x and y to represent numbers in algebra?
In: Mathematics
Part of it is just tradition. Renee Descartes, for whom the Cartesian plot is name, used x, y, z for unknown variables and a, b, c, for known variables. His notation carried over to Newton, Leibniz, and Euler, with added conventions introduced as they were needed.
Ultimately since the convention is used while learning it just carries forward. Also, an x and y usually don’t look like numbers, which helps with legibility for homework.
Essentially, Descartes used them in a very influential book and they caught on. See [https://hsm.stackexchange.com/questions/8109/why-are-x-and-y-commonly-used-as-mathematical-placeholders](https://hsm.stackexchange.com/questions/8109/why-are-x-and-y-commonly-used-as-mathematical-placeholders) . At the end of the day, it’s just convention. It’s useful that when you’re looking at something new, it’s probably true that x is an unknown, n is a natural number, p is a prime, etc.
If your looking for the historic reason why we picked them, I can’t help there. Probably because it is an easy way to write it out.
But, for the math reason, it’s because they represent either unknowns or variables.
A unknown is a number we don’t know yet. Think of it like a question mark. We use these in early algebra to get used to the idea. Sometimes there’s more than one, so we use x, y, z, and more letters.
But algebra isn’t that useful if it’s just looking for an unknown. That’s where variables come in.
Variables are numbers that can change. That means we can choose a number, any number, and do the math.
Take X + 1 = Y.
X can be any number, and Y will always be one more than that. We don’t know what we will choose every time we do the math, so we use variables to describe how the math will behave, instead of solving one specific problem.
Normally, we say that X means something, like time or distance, and Y is something else similar.
A classic example is figuring out how tall a tree is by measuring its shadow and your own shadow.
Its really hard to take a tape measure to a tree, but not so bad to do that with the shadow, so you measure the shadow, and your own shadow, and then you can figure out how tall the tree is.
Let’s say that your shadow is half as tall as you are. That gives us an equations.
You are twice your shadow.
You are Y.
The word Are means equals.
Twice, so 2.
Your shadow is X.
That means we can turn the sentence ‘You are twice your shadow’ into the equation Y = 2X.
Well, say the tree shadow is 10 feet tall, now we can do the math and know the tree is 20 feet tall.
Essentially, think of the letters in algebra as the empty boxes you had in maths questions in school. Stuff like
2+ [ ] = 5,
where you’re supposed to fill in the missing number to make the maths add up. When the equations get more complicated and you need to start moving stuff around, it is very useful for that box, aka the missing part, or the “unknown”, if you like, to be represented by a symbol that is easy to write, as well as easy to say out loud. So we use letters, which we are already used to writing and saying out loud. Mathematicians are lazy, and this is the most convenient solution.
Why those letters specifically? Tradition, could be anything really.
Why letters (symbols) and not numbers? It’s either a number we don’t know yet, or not even just one specific number, but basically “any number”. But then somewhere later we need to refer back to it as “whatever it was back there, same here.”
Maybe more answering your question — variables are useful for solving math problems, by setting the variables up in an equation and then reconfiguring that equation to get a solution. Making a difficult / confusing problem much easier to solve.
When you’re representing an unknown number, you have to use a symbol other than a number. Letters work well, because we already recognize them and can tell them apart, so it’s easy to follow them through the equations. Also, the familiarity allows you to use a letter that’s related to what it represents, such as a for acceleration.
Classic sudoku is similar. You could use any nine symbols and it would work the same, but it’s much easier to see which one is missing from the row when it’s nine digits you’re familiar with
I heard somewhere that it comes from the arabic word شيء which means “something” . When transliterated it can be written x and read chi (like the Spanish x)
If you have this question, it might help you to understand if you use other names occasionally. When helping my kids with math, I like to use variable names like “poop” or “butt flavored pizza”. It makes it clear that it’s not anything fancy, it’s just a name for a number you don’t know yet, and can sometimes help further, if for example, you need to cut your butt-flavored pizza into slices for fractions or something like that.
Because without letters, you have to invent other symbols to represent numbers. We already have letters.
Latest Answers