Constant term

[Saumyojit]

Just once check whether the table information is right or wrong .

What I just discovered in the wiki article on "Algebraic expression" is that any constant is also considered as a algebraic expression .eg->Writing only 2 is AE?

Why it is being told every where(tutorial on tube) that a AE has to contain a variable .


@JeffM u said that "A term in an algebraic expression is any part of it that denotes a number"
In "X+2" there are two terms . First term is X and second one is 2 . So in the first term where is the number?

 

[JeffM]

That article gives fair warning that it is a crude first effort (a “start” in wiki’s terminology) and, in the opinion of a group of mathematical volunteers, needs improvement. The table that you are referring to contains informal descriptions of different types of expression in terms of the kinds of objects and operations that the expression may contain.

Here are my informal attempts at definitions. I shall be happy if someone improves on them.

A mathematical expression is a description of a mathematical object using internationally recognized mathematical symbols strung together in a syntactically meaningful way.

An arithmetic expression is an expression that identifies a number by using only numerals and a finite number of the four arithmetic operations.

I’d say that the article itself does a better job of the narrow task of defining an algebraic expression than I can manage. I think its major problem is that it tries to give a fairly technical definition of an algebraic expression mixed in with an attempt to explain informally a large number of technical terms that a student of algebra or calculus may run across.

 

[JeffM]

Just once check whether the table information is right or wrong .

What I just discovered in the wiki article on "Algebraic expression" is that any constant is also considered as a algebraic expression .eg->Writing only 2 is AE?

Why it is being told every where(tutorial on tube) that a AE has to contain a variable .


@JeffM u said that "A term in an algebraic expression is any part of it that denotes a number"
In "X+2" there are two terms . First term is X and second one is 2 . So in the first term where is the number?

What do you think x denotes in x + 2. A tree? A pie? New Dehli? It denotes a number. You are perhaps not aware of the difference between a number and a numeral. The only difference between x and 2 is that the x symbol does not tell you which number is denoted, but you know that it represents A number.

 

[Dr.Peterson]

Another worst thing is that they are allowing anyone to edit in wiki . I just did now in that para of constant term u can see it. i was not asked whether i did a phd in maths or not or the validity of my claim. Its a serious con of wiki.

I will be watching with curiosity to see how long it takes for someone to delete your addition, which is utterly inappropriate in form. An appropriate change would have been just to replace the word "constant" with "constant term".

The good thing about Wikipedia is that errors are usually corrected, when someone notices; I generally do trust Wikipedia for non-controversial topics like mathematics, though I have seen many foolish edits that last for a while. (The one you asked about in the OP appears to have been there since 2011!)

But these basic articles are, on one hand, so trivial that they are probably not examined often; and, on the other hand, notoriously difficult to state clearly so that they are both clear to newcomers and technically accurate. Try writing a dictionary definition for some everyday three-letter word, and you can discover the problem!

 

[Saumyojit]

it denotes a number. You are perhaps not aware of the difference between a number and a numeral.

I just saw and as far as I understood a number is a abstract mathematical idea whereas 5 ,20 these are numerals which are representing each corresponding number .

An arithmetic expression is an expression that identifies a number by using only numerals and a finite number of the four arithmetic operations.

Thats why a variable is not under arithmetic expression in that table because a variable is a symbol but not a numeral. Writing just " forty" - is this a valid arithmetic expression? As forty is a numeral by which we can identify the no "40" or we have to write in number form always like 40.

What I just discovered in the wiki article on "Algebraic expression" is that any constant is also considered as a algebraic expression .eg->Writing only 2 is AE?

Why it is being told every where(tutorial on tube) that a AE has to contain a variable

a constant is under AE or not?

 

[JeffM]

I just saw and as far as I understood a number is a abstract mathematical idea whereas 5 ,20 these are numerals which are representing each corresponding number .

You've got it bang on the nose.

Thats why a variable is not under arithmetic expression in that table because a variable is a symbol but not a numeral. Writing just " forty" - is this a valid arithmetic expression? As forty is a numeral by which we can identify the no "40" or we have to write in number form always like 40.

I think you are very close in your first sentence. I look forward to someone else's take on this, but I think the phrase "arithmetic expression" is merely intended to convey that the expression can be understood and evaluated by anyone who knows basic arithmetic. Different authors probably have slightly different detailed definitions.

Here is a definition I might feel comfortable using. An arithmetic expression is a syntactically valid string of internationally recognized mathematical symbols that identifies a specific number using no operations other than addition, subtraction, multiplication, division, and exponentiation by an integer. Thus, according to that definition, a numeral is the simplest kind of numeric expression. It also means that an arithmetic expression must contain at least one numeral and must not contain any pronumerals (what you called a "variable"). Provided that the symbols are used syntactically, an arithmetic expression may contain a finite number (including zero) of operator symbols for addition, subtraction, multiplication, division, and exponentiation as well as a finite number (including zero) of grouping symbols.

Also according to that definition, an "arithmetic expression" could not contain English words like "forty" because English is not a universal language.

a constant is under AE or not?

Under any definition of "algebraic expression" that I have ever seen, they may contain constants.

As I said, I'd be pleased as punch if someone has comments, criticisms, or a different view.

 

[Dr.Peterson]

a constant is under AE or not?

Please don't use your abbreviation AE in a context where you have been talking about both Arithmetic Expressions and Algebraic Expressions! The goal is to communicate clearly, right? I think you mean algebraic.

The trouble with all of this is that words are used in slightly different ways in different contexts, so it's impossible to give a universal answer.

The word "constant" can refer to a numeral representing a specific number (like 3 or [MATH]\pi[/MATH]), or to a symbol like [MATH]c[/MATH] that would in general be called a variable, but is considered a parameter in a given context, meaning that it is thought of as a fixed value at any given time. So the "[MATH]c[/MATH]" in "[MATH]ax^2+bx+c[/MATH]" is called a constant term. I know that gets complicated! See https://en.wikipedia.org/wiki/Constant_(mathematics) and https://en.wikipedia.org/wiki/Variable_(mathematics). [In the context of this discussion, I am hesitant to refer to Wikipedia, as if I were saying I agree with everything in a given page, even what might be added tomorrow; I do so simply to show some of the ideas involved.]

So not everything that could be called a constant will be found in an arithmetic expression. But just about anything that could be called a constant can be found in an algebraic expression. Algebraic expressions contain more than arithmetic expressions, simply because the former are used in algebra classes, while the latter are used in lower level classes. That's really all that matters. In moving up to algebraic expressions, you don't lose the ability to use mere numbers (constants); you gain the ability to use variables. If someone seems to be telling you that an algebraic expression must contain a variable, they are just forgetting this, because most do, and that is the characteristic feature of algebra.

But you should also be aware that none of this is terribly important. The reason these terms are not defined precisely is that they don't make much difference. If two of us disagree on what should be called a constant or an arithmetic expression (probably because we have different contexts in mind), that won't change any theorems we state.

 

[Saumyojit]

So not everything that could be called a constant will be found in an arithmetic expression. But just about anything that could be called a constant can be found in an algebraic expression

Can you give me some valid and not valid eg of each case.

 

[Dr.Peterson]

Can you give me some valid and not valid eg of each case.

I did.

An arithmetic expression wouldn't contain a constant called "c". It can contain "2".

"Just about anything" is a way of saying, "perhaps everything, but I don't want to commit myself in case I'm missing some detail". It is not a definite statement that I have a counterexample in mind! Maybe someone considers infinity a constant, but that doesn't belong in an expression. Whatever.

Once again, I don't consider these concepts to be precisely defined; they are mostly in the category of "I know it when I see it, and it doesn't really matter anyway". So an attempt at a precise definition is a waste of effort.

 

[mmm4444bot]

… Try writing a dictionary definition for some everyday three-letter word …

Indeed. I'd like to suggest 'run'.

?

 

[Saumyojit]

An arithmetic expression is a syntactically valid string of internationally recognized mathematical symbols that identifies a specific number using no operations other than addition, subtraction, multiplication, division, and exponentiation by an integer.

exponent by a integer doesn't fall under Arithmetic expression acc to wiki table

The word "constant" can refer to a numeral representing a specific number (like 3 or ππ\displaystyle \pi), or to a symbol like cc\displaystyle c that would in general be called a variable, but is considered a parameter in a given context, meaning that it is thought of as a fixed value at any given time. So the "cc\displaystyle c" in "ax2+bx+cax2+bx+c\displaystyle ax^2+bx+c" is called a constant term. I know that gets complicated!

yes it gets complicated and I need to know the context. By the way I saw your article on " parameter vs constant" which showed me different ways of seeing the parameter. Really that article helped a lot!

An arithmetic expression wouldn't contain a constant called "c". It can contain "2

Ok

 

[Dr.Peterson]

exponent by a integer doesn't fall under Arithmetic expression acc to wiki table

Unless Wikipedia gives a source, their claim can't be considered reliable. You know this already. They give no source for this, and admit it is just someone's synthesis.

We've stated that there is no "official" definition of this term, and its use depends on context. I just tried searching for definitions of "arithmetic expression", and almost everything I found is from computer terminology, which is quite different from mathematics. (For example, here it includes variables.) The closest I came to an actual definition outside of Wikipedia is this:

Art of Problem Solving

artofproblemsolving.com

They, too, include variables! I presume this is because "arithmetic" to them refers to working with numbers as opposed to, say, matrices. But it's also possible that this is just a mistake, because they don't consider it any more important than I do.

Anyway, I see no reason not to include squaring a number as an arithmetic operation. Do you? (Reasons are far more important than authority in mathematics.)

By the way I saw your article on " parameter vs constant" which showed me different ways of seeing the parameter. Really that article helped a lot!

I suppose you must mean this, which is by my twin brother, not me!

 

[JeffM]

exponent by a integer doesn't fall under Arithmetic expression acc to wiki table

Do you think about what anyone says or do you want pick nits?

I said “Different authors probably have different detailed definitions.” and then I gave MY definition and the rationale for it. Wiki’s definition is not canonical.

Dr. Peterson said “It is impossible to give a universal definition” and “it doesn’t really matter anyway.”


In any case, the wiki article says “finite product.” Would you care to tell me the significant differences between

[MATH]3^4 \text { and } \prod_{i=1}^4 3[/MATH].

 

[Saumyojit]

Would you care to tell me the significant differences between

[MATH]3^4 \text { and } \prod_{i=1}^4 3[/MATH].

ok its just that i have to experiment a bit . X^2 also means x*x .

 

[JeffM]

ok its just that i have to experiment a bit . X^2 also means x*x .

Yes. I like to define exponentiation by integers as follows

[MATH]x = 0 \implies a^x \equiv 1;\\ x > 0 \implies a^x \equiv a * a^{(x-1)}; \text { and}\\ a \ne 0 \text { and } x < 0 \implies a^x \equiv \dfrac{1}{a} * a^{(x+1)}.[/MATH]Thus, exponentiation by integers is simply a shorthand way of indicating a finite product, and any discrepancy between what wiki says and I say becomes immaterial.

It is this kind of issue that explains why Dr. Peterson says that these categorizations of types of expression need not be rigid.

I want to go back to what I said before. What makes sense to me is that an arithmetic expression is one that can be understood by or easily explained to someone who knows only arithmetic. And an algebraic expression is one that can be understood by or easily explained to someone who knows the rudiments of elementary algebra. Different people may make slight variations around those central ideas for different purposes.

 

[Saumyojit]

And lastly going through capital pie notation I found out about free and bound variable

In Y=X+1
I know for a fact that X is independent variable and Y is dependent variable
but is X a free variable and Y is bound variable ? can we say like that.
As Value of 'Y' is not free it always rely on 'X'

In a video i saw the def of " A bound variable is a variable which stays in the scope of any Quantifier " whereas Free is out of scope .

In wiki it is said : " A bound variable is a variable that was previously free, but has been bound to a specific value or set of values called domain of discourse or universe. For example, the variable x becomes a bound variable when we write:

For all x,
(x + 1)^2 = x^2 + 2x + 1 "

Now what i found out in (https://study.com/academy/lesson/free-variable-vs-bound-variable.html#:~:text=Another criterion is whether or,you have a free variable.)

they said that in f(x)=3x-1 --> 'x' is free variable
But go back to this eg : for all x, (x+1)^2=x^2+2x+1 (In Wiki page)

writing the wiki eg in terms of function : f(x)=x^2+2x+1
so x must be free variable in this case (acc to study.com) as i can put any value in place of x , isn't it ? There are no limitations on x

CONTRADICTION isn't it ?

But why in wiki did they say bound coz they wrote in terms of quantifier not in terms of function?

 

[Saumyojit]

ok its just that i have to experiment a bit . X^2 also means x*x .

This X cannot be a arithmetic expression but it has to be a number.
Srry.

 

[JeffM]

Once again, you ask about differences among three different sources, but give a link to only one. So it is hard to be sure what you are asking.

But I think you may be mixing up two different meanings of "bound."

Your example [MATH]ln(y) = x + 1[/MATH]
does not determine that x is an independent variable and y is a dependent variable. Why do you say so?

 

[Dr.Peterson]

Your example [MATH]ln(y) = x + 1[/MATH]
does not determine that x is an independent variable and y is a dependent variable. Why do you say so?

I think that was "In [the equation] y = x + 1", not a logarithm. I misread it at first, too. I hate fonts in which capital I and lowercase l look the same.

But,

And lastly going through capital pie notation I found out about free and bound variable

Why did you have to concern yourself with those terms at all? You appear to have seen the terms and then searched for whatever you could find about them, without realizing that you were looking in entirely different contexts. Once again, that is not the way to learn anything.

At least, stick with one source to start with, tell us what it is, and focus on that context. I'll guess that maybe it was this: https://en.wikipedia.org/wiki/Multiplication#Capital_pi_notation

(By the way, notice that it is not "capital pie", but the Greek letter "pi". At least learn that!)

All that matters in your context (figuring what JeffM meant by [MATH]\prod_{i=1}^4 3[/MATH], I presume) was the first bit, that it means [MATH]3\cdot 3\cdot 3\cdot 3[/MATH].

Then they say,

The subscript gives the symbol for a bound variable (i in this case), called the "index of multiplication", together with its lower bound (1), whereas the superscript (here 4) gives its upper bound. The lower and upper bound are expressions denoting integers.

The link tells you what they mean by "bound variable" (as usual, taking it far more broadly than what is needed right here); but all it is is the variable used within the summation and not outside it.

The rest that you found is irrelevant, as it is from very different contexts.

 

[Saumyojit]

Free bound see When I saw the terms I went to given link page to see the definition .
After seeing the eg of bound variable especially this
For all x,
(x + 1)^2 = x^2 + 2x + 1 "
Then suddenly independent and dependent as I knew these before hit my mind so are free and bound are same as independent and dependent variable respectively.

Then I checked study.com
Where they are saying
f(x)=3x-1 --> 'x' is free variable
But go back to this eg : for all x, (x+1)^2=x^2+2x+1 (In Wiki page)

writing the wiki eg in terms of function : f(x)=x^2+2x+1
so x must be free variable in this case (acc to study.com) as i can put any value in place of x , isn't it ? There are no limitations on x

CONTRADICTION isn't it ?

What's the problem in my doubt