Difference between a variable and a constant?

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I know this is a very elementary question, but I suddenly realized in calculus that I don't really know precisely what the definition of a variable and constant was.

I know what people tend to call constants and variables in something like:

ax + by = c, where you'd call x and y a variable and a,b,c constants.

...But aren't a and b subject to change just as much as x and y?
And x and y just represent a SINGLE VALUE, not many values! They don't "vary".

So isn't everything a constant?
x is supposed to represent some number, or in other words, some CONSTANT.


Also, why is it that in ∫ f(x)dx = F(x) + C, C is called the constant while x is a variable?
 

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  • #2
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I know this is a very elementary question, but I suddenly realized in calculus that I don't really know precisely what the definition of a variable and constant was.

I know what people tend to call constants and variables in something like:

ax + by = c, where you'd call x and y a variable and a,b,c constants.

...But aren't a and b subject to change just as much as x and y?
No, not in the usual contexts. Variables are placeholders into which we can insert whatever values are appropriate. Although we don't know the values of a, b, and c, they should be treated is fixed constants, albeit ones whose values aren't specified. Some people call these parameters.


And x and y just represent a SINGLE VALUE, not many values! They don't "vary".
No, not true. The equation ax + by = c, with a, b, and c fixed (i.e., constants), has a graph that is a straight line. Every pair of numbers (x, y) that is on this line is also a solution to this equation. There are an infinite number of points (x, y) on the line, which means that x and y can take on an infinite number of values. Of course, with a, b, and c being fixed, if you know the value of y, then there is only one value of x for which (x, y) satisfies the equation. The point is, though, that there are many, many possible values for x or y.
So isn't everything a constant?
x is supposed to represent some number, or in other words, some CONSTANT.
No, as explained above.
Also, why is it that in ∫f(x)dx = F(x) + C, C is called the constant while x is a variable?
Here's a specific example: ∫x2 dx = (1/3)x3 + C
This equation says that all antiderivatives of the function f(x) = x2 are of the form (1/3)x2 plus some constant. The opposite statement is that the derivative of (1/3)x3 + C is x2.
Here we have two functions, x → x2 and x→(1/3)x3. The output of each function depends on what went in as an input value. If you put in two different x values (one at a time), you get two different output values.

In contrast, a constant's value doesn't depend on some variable. Its value remains unchanged, even when its value is not explicitly stated.

A formula that comes to mind is the one that gives the gravitational force between two objects.
$$F = G\frac{M_1 * M_2}{r^2}$$

I think I am remembering this formula correctly...
Here G is the constant of gravitation, and M1 and M2 are the masses of the two objects. r is the distance between the centers of the two objects.

For any two given objects, M1 and M2 would be constants, but we can calculate the force due to gravitational attraction for various values of r, so r would be the variable in this scenario. If we wanted to calculate the force between a given object of mass M1 and an arbitrary mass (M2) at an arbitrary distance, M2 and r would be the variables.
 
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Here's a specific example: ∫x2 dx = (1/3)x3 + C
This equation says that all antiderivatives of the function f(x) = x2 are of the form (1/3)x2 plus some constant. The opposite statement is that the derivative of (1/3)x3 + C is x2.
Here we have two functions, x → x2 and x→(1/3)x3. The output of each function depends on what went in as an input value. If you put in two different x values (one at a time), you get two different output values.

In contrast, a constant's value doesn't depend on some variable. Its value remains unchanged, even when its value is not explicitly stated.

A formula that comes to mind is the one that gives the gravitational force between two objects.
$$F = G\frac{M_1 * M_2}{r^2}$$

I think I am remembering this formula correctly...
Here G is the constant of gravitation, and M1 and M2 are the masses of the two objects. r is the distance between the centers of the two objects.

For any two given objects, M1 and M2 would be constants, but we can calculate the force due to gravitational attraction for various values of r, so r would be the variable in this scenario. If we wanted to calculate the force between a given object of mass M1 and an arbitrary mass (M2) at an arbitrary distance, M2 and r would be the variables.

Thanks, I think that cleared it up for me a lot more! I've been thinking a lot more about this after your response...

From what I understand, it seems that what we call a variable is really case-dependent, as you showed in your gravity example.
So a constant is a constant only in respect to some other variable.


I think I can understand it in terms of a "tree" of implications:
Given some relation R(A,x,y), where we call A a "constant" and x and y "variables"...
If A = some number,
then:
x = this number
OR
x = that number
OR
x = another number
OR
x = yet another number
OR
x = some other number in its domain
OR
....
x = n

I *think* what confused me here was that the "OR" makes it so that only ONE x can be true, which made it semantically seem like x assumed only one value, much like a constant. But although it represented a single value, it was able to ASSUME several others in different cases, so I can now see the difference. x is a variable in relation to A, and A is a constant in relation to x.
Now that I've written it like this, it makes a lot more sense.


I guess I could extend it:
And then y would HAVE to be a certain number (assuming this is a function), if this were a relation with no other variables:
If x = some number,
then y = cool number,
so we have x = some number AND y = cool number
written as (x,y) or (some number, cool number)

If x = n,
then y = m,
So we have (x = n) → (y = m), so x AND y.
So we can write it as (x,y) ⇔ (n,m)

So a graph, in a sense, is a "splaying out" of all POSSIBILITIES of solutions, or all points (x,y).
 
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