What is a variable, really?

atran · 2013-12-21 07:25:26

I've understood it this way: a variable is a symbol representing an unknown number. It's called a variable since the same symbol can be used to represent other numbers. I recently read that a variable ranges over a given set: does it mean that a variable can stand for any member of that set? If so, how? Is there a solid definition of variable? In basic textbooks a variable is introduced as a symbol for an unknown, but more advanced books seem to assume that you know what a variable is and apply the "range" concept, etc...

Can you tell me what a variable exactly is, in all contexts?
I'm really frustrated, so I thank your assistance.

Last edited by atran (2013-12-21 07:25:41)

irspow · 2013-12-21 11:05:20

A variable is just that. It is a symbol to represent any value that is possible within the context it is used as long as mathematical rules are adhered to. You can and often do know the entire range of values possible for a given variable but use it as shorthand for all of those values. Can you imagine listing all the values instead of just x for an equation? In many cases you could not do so. For example:

y=2x perhaps the simplest of terms you will ever come across.

Well, x (and y for that matter) can take on any value. It would be impossible to list every possibility if you had an eternity. But what is powerful about variables is that an infinite number of terms can be expressed by the simple relationship demonstrated by the variables. And more times than not, that is what is really at the heart of all mathematics, logical symbolic relationships among abstract things.

I don't know if any of this helps, but a variable in simplest terms is a symbol that is used to look at mathematical relationships without having to actually assign a specific value at the time you are studying the relationship. It is treated mathematically like any other value, but one need not assign a specific value to it until necessary.

Nehushtan · 2013-12-21 12:13:45

Put simply, an independent variable is a member of the domain of a function while a dependent variable is a member of the range of the function.

atran · 2013-12-25 10:45:05

Hi,

Is the variable which ranges over a set called a bound variable?

As for ∈, does it literally mean "a member of" or "ranges over" when generally used with variables? For example I interpret ∈ in { x : x ∈ N} as "ranges over."

I also can't really understand how the solution is obtained to a system of equations. Assume I have two equations with two variables, x and y. Each equation has a solution set, but why is the solution to the system the intersection of equations' solution sets? I know this may sound stupid, but I can't understand it.

Thank you.

bobbym · 2013-12-25 11:54:30

I also can't really understand how the solution is obtained to a system of equations.

You do not know how to solve 2 equations in 2 unknowns when they are linear?

atran · 2013-12-25 12:01:19

bobbym wrote:

I also can't really understand how the solution is obtained to a system of equations.
You do not know how to solve 2 equations in 2 unknowns when they are linear?

I can solve linear systems but I'm feeling lost, meaning I can't understand the process even though I get right answers.

bobbym · 2013-12-25 12:05:28

When you graph 2x+5y = 10 do you understand what you are doing?

atran · 2013-12-25 12:10:11

I understand it this way: graphing an equation (with two variables) means graphing the solutions to the equation on a coordinate system.

bobbym · 2013-12-25 12:11:23

Do you see that the graph represents that there are an infinite number of solutions, that is for every x there is a y?

atran · 2013-12-26 04:46:29

Yes.

bobbym · 2013-12-26 04:57:36

And when you draw the line 3x - 5y = 7? What are you really doing?

atran · 2013-12-26 04:59:46

I graph the points which satisfy that equation.

bobbym · 2013-12-26 05:08:47

So, in your own words what does point A represent?

atran · 2013-12-28 06:19:18

Any equation (with variables) has infinite solutions, except a one-variable equation. A solution set is the set of solutions to a given equation. Now I have a system of equations:
2x + 5y = 10
3x - 5y = 7
Since the system is linear it must have infinitely-many, one, or no solutions. This system has a unique solution, which is x=3.4 and y=0.64.

The system has two variables, x and y. The purpose is to find one x and one y satisfying both equations. The first graph represents the solutions to the first equation, and the second graph is the solution to the second equation. Each solution is an ordered pair, so the first element of the ordered pair is substituted for x while the second for y. The point (3.4, 0.64) is the only solution to the system, so the graph of the system is only that point.

I'm honest, I still don't understand how the two equations are connected together and how the solution is obtained. I used to understand it before, but now, despite what I've just described, I'm still feeling lost. I don't know how to express my thoughts to make my problem visible. I also looked at many pages on internet but it didn't help...

Last edited by atran (2013-12-28 06:19:49)

bobbym · 2013-12-28 06:24:24

I still don't understand how the two equations are connected together and how the solution is obtained.

From your explanation above I would say you understand it about as well as I do, and as well as it is necessary to use it and to progress further.

Do not make the Babbage error of not moving on until you have total comprehension. Sometimes you have to keep moving and hoping it all comes together later on.

how the solution is obtained

Are you saying the process of solving is what you are unsure of?
This is nothing but rote algebra. It can be called an algorithm. It is so mechanical that it can be done by a computer. To learn it, you just have do some problems using it.

I don't know how to express my thoughts to make my problem visible.

If you can not express it then perhaps it is not really there. Take a look at my signature, the top line. It is true.

atran · 2013-12-28 09:40:26

Hi bobbym,
Thank you for the tip, that I should keep going despite not having total comprehension.

If I can, I will express my thoughts more clearly. Maybe it's the algorithm I've no "proof" of.
I will get back to this later, now I'm thinking to spend my time on something else.

bobbym · 2013-12-28 11:06:53

Hi;

You can proceed with the method knowing that it works and besides you can prove it worked when you plug in at the end.

atran · 2014-01-01 09:09:08

Hi. I have another question. Say x²=y², then what does follow? |x| = |y| or y=±x?

Last edited by atran (2014-01-01 09:09:18)

bobbym · 2014-01-01 09:19:36

Hi;

I like y = ± x.

anonimnystefy · 2014-01-01 11:43:39

Both arevcorrect, but y=+/- x is usually more useful.

atran · 2014-01-03 04:58:13

Hi;

Say I have the following equations,
y = x + 5
b = 2*a

The first equation has the solution sets, A1={(x, x+5) : x∈R} and A2={(x+5, x) : x∈R}.
The second equation has the solution sets, B1={(x, 2*x) : x∈R} and B2={(2*x, x) : x∈R}

Now if I want to find a point which satisfies both equations, I have first to decide whether x=a or x=b.
In the former case (i.e. x=a), the solution set is: A1 ∩ B1 = A2 ∩ B2 (This means the solution for 2 functions is the same as the solution for their inverses.)
In the latter case, the solution set is: A1 ∩ B2 = A2 ∩ B1

I'm not totally satisfied yet, but is my thinking OK? Am I complicating this?
Thanks for help.

bobbym · 2014-01-03 06:18:18

Hi;

I might be totally wrong on this so wait for some more answers.

If you say

y = x + 5
b = 2*a

how can we speak of a common solution? The variables are different. What it looks like you have done is to place both equations on the same plane as if you graphed them both on the xy plane. To do that they become:

y = x + 5
y = 2*x

now you would just solve for the intersection in the usual way. You would get (5,10) as the point of intersection. From my point of view there is no intersection of

y = x + 5
b = 2*a

Am I complicating this?

I would say yes.

Math Is Fun Forum

#1 2013-12-21 07:25:26

What is a variable, really?

#2 2013-12-21 11:05:20

Re: What is a variable, really?

#3 2013-12-21 12:13:45

Re: What is a variable, really?

#4 2013-12-25 10:45:05

Re: What is a variable, really?

#5 2013-12-25 11:54:30

Re: What is a variable, really?

#6 2013-12-25 12:01:19

Re: What is a variable, really?

#7 2013-12-25 12:05:28

Re: What is a variable, really?

#8 2013-12-25 12:10:11

Re: What is a variable, really?

#9 2013-12-25 12:11:23

Re: What is a variable, really?

#10 2013-12-26 04:46:29

Re: What is a variable, really?

#11 2013-12-26 04:57:36

Re: What is a variable, really?

#12 2013-12-26 04:59:46

Re: What is a variable, really?

#13 2013-12-26 05:08:47

Re: What is a variable, really?

#14 2013-12-28 06:19:18

Re: What is a variable, really?

#15 2013-12-28 06:24:24

Re: What is a variable, really?

#16 2013-12-28 09:40:26

Re: What is a variable, really?

#17 2013-12-28 11:06:53

Re: What is a variable, really?

#18 2014-01-01 09:09:08

Re: What is a variable, really?

#19 2014-01-01 09:19:36

Re: What is a variable, really?

#20 2014-01-01 11:43:39

Re: What is a variable, really?

#21 2014-01-03 04:58:13

Re: What is a variable, really?

#22 2014-01-03 06:18:18

Re: What is a variable, really?

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