Why do we use letters like x and y to represent numbers in algebra?
In: Mathematics
Others have answered the question, “why do we use the specific letters ‘x,y, etc.’ in algebra?” I’m instead going to try to answer, “why do we use letters at all in algebra?”
If you’re in school and wondering why suddenly your math teacher is introducing these weird letters into math, a subject based around numbers, it’s because you’ve shifted from studying one part of mathematics to another, specifically shifting from arithmetic to algebra. Arithmetic, what you’ve probably been doing since starting school, is about executing a series of operations on some numbers to get to a final number (e.g., 2+3*5=17). Algebra on the other hand, is about using arithmetic to solve for the unknown (x, y, etc.). Arithmetic assumes everything you need is already known; algebra assumes there is something unknown and we’re using math to figure out what it is.
I’ll use this for another question:
In 2D you have x and y axis
In 3D you have x, y and z axis
What happens if you tried to project something in 4D? What would the axis be called?
Descartes kindly anticipated that using Roman letters for variables would make it easy to type simple equations on the internet, whereas using Greek letters or another special symbol would require some kind of annoying workaround outside of typesetting software (where they’re still more annoying than simple letters).
I’m not sure if this is “the reason” per se, but it is a good reason that I can think of. Symbols used in physics and engineering equations are often associated with their name – e.g. “G” is the gravitational constant, “c” often just means “constant”, τ is the time constant, “i” is an imaginary number, etc. “x” and “y” are not the first letters of many words, so they fit nicely when the variable does not represent anything specific. x + 5 = 10, for example, just means “some variable plus five equals ten”. x and y do not stand for anything, so they are nice to use as variables where they do not represent anything in the real world.
tl;dr – x and y just mean “some generic variable” because they do not correspond to a specific term
Like you’re 5; it wouldn’t make sense to use numbers because we don’t know the number, so we use a letter to show it’s a number that changes, we call that a “variable”.
We use letters because they are symbols most of us know, we can all picture “y” or “x”, that letter could represent anything from dogs to coins to the length of rope.
If we made up our own symbols it would technically still work, but it would get quite confusing fast, you could even use pictures, but the equation would get crowded, easiest thing is to use letters.
In a lot of programming, we’d mostly use whole (or parts of…) words instead of just letters to help visualize and reference these variables as clear as we can bothered to make it i.e. “carrot_Thickness”, “carrot_Length”, “carrot_Colour”.
Other people have answered why we use those symbols specifically. But the reason we use symbols at all is because it makes it easier to understand. I know that might be surprising, especially if you don’t already know how the notation works.
But consider these two mathematical expressions:
>The square of the sum of two quantities is equal to the sum of the square of the first quantity, the square of the second quantity, and twice the product of the two quantities.
and
> (x + y)^2 = x^2 + 2xy + y^2
Those two things mean exactly the same thing. It takes a little bit of getting used to, but once you’ve learned how it works, working with symbols is a lot easier than words.
Stuff You Should Know podcast did an episode on this. It basically confirms the other responses given here but it’s still a good listen.
a few months ago, a user going by u/phiwong explained it like this and I really like it!
“Descartes was the mathematician who proposed and formalized the coordinate system that was subsequently named after him (cartesian).
He used the first letters of the alphabet to signify known quantities and the last few letters to signify unknowns. The tradition started from there. (this might be apocryphal since x and y were probably already used before his time)
Very generally speaking, this developed into an informal ‘system’.
a,b,c are used for scalar (or real number) constants. (e isn’t typically used because it refers to euler’s constant. In physics, c, is usually reserved for the speed of light)
f,g,h are used for function names typically. In kinematics physics, g is usually reserved for the acceleration due to earth’s gravity
i,j (and sometimes k) is used as subscripts, indices or with a “hat” to denote unit vectors.
k,m,n, (mostly m and n) are used to denote integer constants or integer unknowns.
p,q,r are also commonly used to represent vector unknowns. (a,b,c are usually scalar constants or unknowns)
s,v and t are used in physics typically to denote displacement, velocity and time. (In kinematics physics, u is used for initial velocity)
u,v,w are also quite often used for vector variables.
x, y, z are used for variables. z is commonly used to represent complex number variables. (so if a problem states z^3 =1, you should assume that z refers to complex number solutions and responding with z = 1 is incomplete or incorrect.)
l and o are rarely used because it too easy to mistake it for “1” and “0”
“
I remember one year in elementary school, where the unknown was always listed as “n”. No idea why as every year afterwards was x and y. I do remember wondering why they switched from n, when we were used to it. This was back in the “60s, so I have no idea if that is still a thing.
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.
Latest Answers