# What is the difference between variable, argument and parameter?

I'm sure that these terms should be different since there exists a difference between parameter and argument in computer science but I'm not sure about their differences in math.

• A function has argument-places, like the "sum" function, that has two (binary function). In order to synbolize them, we use variables : $f(x,y)$. – Mauro ALLEGRANZA Jan 25 '17 at 10:22
• In principle we may write $f(-,-)$ with place-holders, but then we are in trouble because we cannot make a difference between $f(x,x)$ and $f(x,y)$. – Mauro ALLEGRANZA Jan 25 '17 at 10:22
• @MauroALLEGRANZA So, argument is a place and variable is the stuff that fills that place? – ankit Jan 25 '17 at 10:23
• In a mathematical expression, like e.g. $ax+by$ we call $a,b$ parameters in order to convey the fact that - in the context of the "discourse" about that expression - we will consider them constant while $x,y$ are variables. But the "discourse" will holds generally, irrespective of the specific values of $a,b$, and this is why we use letetrs instead of (individual) numbers. – Mauro ALLEGRANZA Jan 25 '17 at 10:25
• Not exactly: variables are symbols and they are the place-holders. We assign values (input) to variables and "compute" the resulting value (output) of the function (or expression) for those input values. Basically, it is the same as in computer science: we use math formulae in the same way as the computer uses FORTRAN or C code. – Mauro ALLEGRANZA Jan 25 '17 at 10:26

Consider family of functions $\{f_a : a\in [0,1]\}$, where $f_a:\mathbb{R}\rightarrow \mathbb{R}$ is given by $$f_a(x)=x^2 - a.$$ When you take some $x\in \mathbb{R}$, this is a variable. This variable becomes an argument for function $f_1$ if you begin to consider $f_1(x)$. In this expression, $1$ is a parameter.
Remember, that these terms, though diffeerent, are sometimes used interchangeably. For example, you can consider a function of two variables $f(x,y)$ as a function of one variable $y$ with parameter $x$.