How Do Variables and Constants Differ in Mathematics?

[rahaverhma]

There are 2 things taught to us:variables(litersls ) and constants .
If we let a value 'a' as a constant ,meaning it would remain unchanged and also it is a set of real numbers .
If we take 'x' as a variable, meaning it can change and it is also a set of real nos.
For ex. Lim (x>a) x = a ,a:fixed value ,any value from end to end, but still it is not getting clear to me about 'x'?? If it does change then w.r.t to what ?

 

[jedishrfu]

X can be any arbitrary number that you care to try. When solving the limit, it is understood that x varies but that the constant a value remains at some specific but unspecified value.

Here's more discussion on it

http://www.school-for-champions.com/algebra/variables_constants.htm#.WMtSP5E8Kf0

 

[rahaverhma]

A constant can also be arbitrary where ever I take it.

 

[jedishrfu]

A constant can also be arbitrary where ever I take it.

Yes but for purposes of the problem it doesn't change.

It's like keeping y constant but varying x to produce a horizontal line on a graph.

 

[rahaverhma]

Can I say : a constant can be a variable but a variable can not be a constant.

 

[Stephen Tashi]

Can I say : a constant can be a variable but a variable can not be a constant.

It may be impossible to give a precise mathematical definition of a "variable" unless we are dealing with the abstractions of formal languages - such as the languages used to write computer programs. In writing about mathematics using English (or other natural languages) there are only informal and intuitive notions for the meaning of a "variable".

When we use symbols to represent "variables" , the variables have a "scope". For example, "x" on the top of a page in a textbook may not represent the same thing as "x" on the bottom of the page or as the "x" on a different page. If you have studied computer programming, you have seen that variables have a well defined "scope". A variable "X" in one function of a computer program may have a scope limited to the code in that function. It need have nothing to do with an "X" in some other code that is involved in the program. In writing mathematics, variables have a limited scope, but setting the boundaries for that scope is often done in an informal fashion.

Variables in computer programs can have "nested" scopes For example , the variable "X" might be introduced in one section of code and within that section there might be another section that introduces a variable "Y". The same situation can happen in writing mathematical discussions in English.

You are correct that there is a similarity in the way "constants" and "variables" are treated in writing mathematics. Technically, "constants" represented by symbols such as "C" can indeed be used as "variables".

An example of a mathematical claim is:

For each real number C {
For each real valued function f(x), if f(x) is the function defined by f(x)= Cx then f'(x) = C
}

However, the way this is typically expressed in English does not make the scope of the variables so clear. Typically we might write this as:

Let C be an arbitrary constant. If f(x) = Cx then f'(x) = C.

So we use the jargon "Let C be an arbitrary constant" to establish that we are introducing a variable "C" whose "scope" encompasses the use of the other variables "f(x)" and "x". Within that scope, we don't make other uses of the symbol "C" to represent something that changes its value as a function of "f(x)" or "x".

 

[jedishrfu]

Can I say : a constant can be a variable but a variable can not be a constant.

You're missing the operational phrase "for the purposes of the problem it doesn't change" meaning the 'a' constant doesn't change. It has a fixed value whatever it may be. The variable 'x' is the unknown in the problem which varies when working with a limit or may just be some unknown value you must discover using mathematical operations.

You would agree if 'a' was set to some value like 3 then 3 is a constant right? Well by using 'a' I can solve the same problem in a more general sense so that the 'x' answer I find is dependent on the 'a' value. The 'a' value represents all constants in the problem.

Hence: "For the purposes of the problem" the 'a' value is constant, it doesn't change.

 

[Mark44]

There are 2 things taught to us:variables(litersls ) and constants .

I assume you meant "literals". This is terminology that is used in programming languages, so I'm going to assume that's the context of your question.
Literals, in the programming and computer science sense are NOT variables. A literal is an expression with a fixed value, such as the character 'b', the numbers 5 and 3.2, the string of characters "Tuesday", and so on.
In the programming context, a variable is a named specific region in memory that can hold a value of some kind. In the mathematical sense, a variable is something that can hold a numeric value.

If we let a value 'a' as a constant ,meaning it would remain unchanged and also it is a set of real numbers .

No. A constant is not a set of real numbers. The value of the constant could come from the real numbers, but it would be only one real number, not a set of real numbers.

If we take 'x' as a variable, meaning it can change and it is also a set of real nos.

Again, no. x can take on a single value, but as a variable, its value is generally not known until we solve for it.

For ex. Lim (x>a) x = a ,a:fixed value ,any value from end to end, but still it is not getting clear to me about 'x'?? If it does change then w.r.t to what ?

It should be fairly obvious that $\lim_{x \to a} x = a$. In other words, if x takes on values ever closer to a, then x will get closer and closer to a.

A better example would be one that is not so obvious, such as $\lim_{x \to 0} \frac {\sin(x)} x$. The closer x is to 0, the closer the expression $\frac{\sin(x)} x$ is to 1.

Can I say : a constant can be a variable but a variable can not be a constant.

No.
A constant is fixed, at least for the purposes of the problem, as @jedishrfu said. The value of a variable can change.