In algebra what does x represent?

[laymanhobbist]

The letter "x" is often used in algebra to mean a value that is not yet known. It is called a "variable" or sometimes an "unknown". In x + 2 = 7, x is a variable, but we can work out its value if we try!

Can i call "x" an unknown object that varies ?

 

[fresh_42]

Can i call "x" an unknown object that varies ?

Why do you want to? It is unknown, yes, but neither an object, because it is a number, nor does it vary, because it is the fixed number $5$.

 

[laymanhobbist]

So X denotes an unknown something , and that's it ?
Is this letter X called a variable name of an unknown something ?

 

[fresh_42]

So X denotes an unknown something , and that's it ?
Is this letter X called a variable name of an unknown something ?

In your example it is an unknown number, which turns out to represent $x=5$.

If you had written $y=x+2$ then it would have been a variable, since we can substitute any number for $x$ and get an $y$. Thus the set $\{(x,y)\, : \,y=x+2\}$ is a straight in the $( x,y)$ coordinate system. One point of this straight is $(x_0,y_0)=(5,7)$, but along the straight, the now variable $x$ varies.

 

[laymanhobbist]

So the unknown variable name X can either be fixed or it varies depending on the equation ?
i think that is why i was trying to narrow it down like this ,
"x" as an unknown object that is fixed or varies ?

Is that more correct ?

 

[fresh_42]

So the unknown variable name X can either be fixed or it varies depending on the equation ?
i think that is why i was trying to narrow it down like this ,
"x" as an unknown object that is fixed or varies ?

Is that more correct ?

Mostly. Unknown variable name is still a bit strange. We know the name, we called it $x$. It is unknown as long as we haven't figured out for which number it stands, and it is a variable if it can stand for a bunch of numbers of a set which we call the domain. Unknown, if it stands for $x=5$ as in your example and we haven't done the math yet, variable if it can stand for any real number $x\in \mathbb{R}$ as in $y=x+2$.

However, if it is the case that you have a second equation, say for example $y=3x$, then we have two straights which intersect at a certain point. $x$ and $y$ are then the coordinates of this point, which turns them into unknowns.

You can roughly say: if it is part of a function, then it is a variable, and if it can be calculated - at least theoretically - then it is an unknown. Purist will write $x_0,y_0,\ldots$ for unknowns instead of $x,y,\ldots$ which are reserved for variables. See my example in post #4. Specific values of a varible should carry an index to distinguish its roles as variable or unknown.

 

[laymanhobbist]

@ fresh_42
After trying to learn some computer programming i lost some basic concepts of both mathematics and computer science

Thanks a lot for all those explanations .

 

[fresh_42]

@ fresh_42
After trying to learn some computer programming i lost some basic concepts of both mathematics and computer science

Thanks a lot for all those explanations .

If we write $y=x+2$ and $y=3x$ then we have two straights, each function represents one.

If we want to know, where those straights intersect, then we are looking for a point $P=(x_0,y_0)$ such that $y_0=x_0+2$ and $y_0=3x_0$ hold at the same time. These are unknowns and not variables anymore. They changed their role. The index ${}_0$ is usually omitted, but that comes to a price: It is confusing if the roles are important, which is definitely the case when we differentiate functions. The lack of the index is the main source of confusion in formulas with derivatives.

There is a similar situation in C++ programming. A pointer and a reference address the same thing, but in different ways.

 

[laymanhobbist]

@ fresh_42

I was also trying to prepare for a government test and it has lot of mathematics and quantitative aptitude tests .
So all these things came to my mind , so i thought i would ask here
I will ask here if i have more doubts

Thanks for the all the important points .

 

[jbriggs444]

@ fresh_42
After trying to learn some computer programming i lost some basic concepts of both mathematics and computer science

Thanks a lot for all those explanations .

The mathematical notion of a variable and the computer programming notion are, in some ways, diametric opposites.

In mathematics, we assign a name to a value.
In computer programming, we assign a value to a name.

However, sometimes they behave similarly. The notion of the scope of a variable name should be familiar.

For instance: "For all $x$ in $\mathbb{R}$, $x^2$ $\ge$ 0". In this construct, the variable name "$x$" is only within scope inside the "for all" quantifier. It makes no sense to ask what "$x$" is outside that scope.

 

[laymanhobbist]

@ jbriggs444

Thanks for the reply .

In computer science , it looks like this from my notes ...

This is a declaration.
Declare an Integer line called x
int x;

This is an initialization.
Initialize Integer line called x
int x = 0;

For mathematics i wrote it down like this in my notes
In an equation ,
If x is unknown , Find the value of x

In a function ,
The unknown x varies , so its a variable

I hope that is correct

 

[jbriggs444]

Whether you call it a "variable" or a "value" does not change much. It is how you use it that counts.

But let me give a specific example where computer and math usage coincide nicely.

Suppose in computers you have the declaration of a subroutine:

real f ( x )
real x
f = x**2
return

! Main line code
integer t
t = 3
print f(t)

Here, "x" is a dummy variable. It does not matter what variable name you use. The caller does not care.

Or in mathematics:

Let $f()$, $\mathbb{R} \to \mathbb{R}$ be defined so that $f(x) = x^2$.
Let $t$ = 3
What is $f(t)$?

Here, "x" is a dummy variable. It does not matter what variable name you use. The function is what it is. One might call $t$ a "constant", a "named constant" or even a "variable". I wouldn't expect eyebrows to raise at any of those choices.

 

[laymanhobbist]

@ jbriggs444

OK , i think i understand

Thanks a lot for all those examples

 

[symbolipoint]

Can i call "x" an unknown object that varies ?

This is not mysterious and is not a difficult idea.

Any letter of the alphabet, depending on reason, can play as a mask for a number. We have at least one common sense rule: A variable must represent the same number throughout any single example in which this variable is being applied.POST-NOTE: I did not yet read the posts after #1 which included the use of variables in computer programming when I made my responses #13 and #14.

 

[symbolipoint]

So the unknown variable name X can either be fixed or it varies depending on the equation ?
i think that is why i was trying to narrow it down like this ,
"x" as an unknown object that is fixed or varies ?

Is that more correct ?

YES.

 

[laymanhobbist]

@ symbolipoint

Thanks a lot for your reply
I am not that confused now

:)

 

[laymanhobbist]

I have this one more question ,

If x is an unknown quantity that makes up the equation , find x

Isn't that also correct ?

 

[fresh_42]

I have this one more question ,

If x is an unknown quantity that makes up the equation , find x

Isn't that also correct ?

Yes. The adjective unknown is actually used like a noun in this case: $x$ is an unknown. You do not need to say unknown quantity, but of course it's not wrong. The short mnemonic is indeed:

function $\to$ variable
equation $\to$ unknown

Ok, a function is also written as an equation, but the $x's$ are not fixed in the context of a function, whereas they are fixed in the context of an equation, meant to calculate the $x's$.

 

[laymanhobbist]

Thank you for answering my strange questions .
I found this book called , homework helpers basic math and pre algebra and i feel like refreshing and improving my basic maths . This book is a really good book to refresh maths . One of the best books i have read .

I have to start with numbers again ,
Then expressions ,
Then equations ,
After that functions ,
Trigonometry
Calculus
Differential equation

Which is why i keep looking for the best definitions so that i can easily remember it .

So for the purpose of easy understanding , this is what i wrote ...

If x is an unknown quantity in an equation ,find the value of x

I am also preparing for a government exam soon ... so i really want to refresh my mathematics

 

[fresh_42]

This is correct.

However, I wouldn't bother too long with that issue. To be honest, I have never had thought about it until you asked. It is in mathematics as in real life: the role is important, not the name. Names come into play when concepts, a set of properties, or even a certain single property should be abbreviated or emphasized. Sure, unknown or variable distinguishes between the two different roles, but they occur so often, often simultaneously that they don't deserve to be named.

Is "int k = 0;" a declaration, a initialization, or both? And who cares? Everybody sees what is going on, so nobody bothers the names.

The same is it with $x$ in mathematics. It makes a quantity manageable, be it a fixed one or a varying one. The only purpose is to calculate with it either way.

If you like it sophisticated, then it is the difference between an existence quantor and an all quantor:
unknown: there exists a number $n$ such that $x=n$
variable: for all numbers $n$ of the domain we can substitute $x=n$

 

[laymanhobbist]

OK , i understand at least half of it

Thanks a lot for the detailed explanations

 

[mpresic3]

laymanhobbist has asked a very good question, that has never occurred to me. Perhaps, it has been asked of algebra teachers, in the past. In the example given x + 2= 7, it is clear that the word "variable" is seriously misleading. The word unknown is a better word in this case as x does not vary.

I should add a further related caution that may come up in the math refresher that layman may be using.
The refresher seems to cover math up to differential equations and calculus. Many math refreshers, and some textbooks do not illustrate the following point.

Often, in mathematics, equations are written with unknowns, x, y, z, and sometimes w to be solved for. These letters come from the end of the alphabet. However, in more advanced mathematics, like differential equations or calculus, we have letters a, b, c which represent CONSTANTS (forgive the shouting but this is important). These constants are assumed to be fixed and are not variables and not to be solved for. I say this because one of my former students came across a system of equations with a's and b's , and x's and y's on a test. The student was completely baffled what needed to be solved for. We wanted the student to solve for x and y in terms of a and b.

In light of this, it would be better for textbooks to ask the question, solve for the unknown, a + 2 = 7, or b + 5 = 7. The constants (at the beginning of the alphabet) are to be solved for.
I only add this complication because the math refresher may get to some pretty complicated math, and may use a, b, c, for constants, and x, y, z, for unknowns, (called variables)

 

[symbolipoint]

mpresic3, what you discuss and describe in post #22 is generally what is taught in Introductory Algebra in high schools (and in college below the University level).

 

[mpresic3]

mpresic3, what you discuss and describe in post #22 is generally what is taught in Introductory Algebra in high schools (and in college below the University level).

Good. I'm impressed. In1st year algebra, I remember the a, b, c's were used in the solution of the quadratic formula. However, I have seen freshman undergrads confused on this point. Perhaps under pressure of the test, they do not generalize the notation.

 

[Evo]

@ fresh_42

I was also trying to prepare for a government test and it has lot of mathematics and quantitative aptitude tests .
So all these things came to my mind , so i thought i would ask here
I will ask here if i have more doubts

Thanks for the all the important points .

Just a heads up, in English, we don't say "doubts" if we have a question, we say "question", "doubt" means you do not believe something to be true. I know that in your country it is being taught wrong, they even know it is wrong, but the printers and people in power have refused to correct the mistake, they say it would be too much trouble and too expensive, the material that incorrectly interpreted "question" to mean "doubt" is too widespread. But if you wish to be correct and teach correctly, you should say "I have a question about..."

Here we have become used to people from your area saying "doubt" instead of question, so we know you actually mean you have "questions".

Just thought you'd like to know what is correct.

 

[laymanhobbist]

@Evo

The problems is my english is not that good , that is why the errors in the usage .

Yes i actually meant "questions" itself .

Thanks for correcting it

 

[Evo]

@Evo

The problems is my english is not that good , that is why the errors in the usage .

Yes i actually meant "questions" itself .

Thanks for correcting it

Your English is actually quite good!