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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.

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  • $\begingroup$ A function has argument-places, like the "sum" function, that has two (binary function). In order to synbolize them, we use variables : $f(x,y)$. $\endgroup$ – Mauro ALLEGRANZA Jan 25 '17 at 10:22
  • $\begingroup$ 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 $f(x,y)$. $\endgroup$ – Mauro ALLEGRANZA Jan 25 '17 at 10:22
  • $\begingroup$ @MauroALLEGRANZA So, argument is a place and variable is the stuff that fills that place? $\endgroup$ – ankit Jan 25 '17 at 10:23
  • $\begingroup$ 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. $\endgroup$ – Mauro ALLEGRANZA Jan 25 '17 at 10:25
  • $\begingroup$ 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. $\endgroup$ – Mauro ALLEGRANZA Jan 25 '17 at 10:26
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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.

More generally, parameter is a "selector" from a family of "similar" functions. Variable is basically any element from any set. If you take variable from domain of some function to consider value of this function at given point, then variable becomes an argument for the function.

I hope it made this topic more understandable.

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 $x$.

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