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Coefficients beyond Constants, and in Context

Date: 03/24/2014 at 06:18:02
From: rohit
Subject: coefficients in algebra

In a term like 3x^4.yz, would it be correct to say that 3x^2.yz is the
coefficient of x^2?

Are there two coefficients in a term like this -- namely, a numerical one
(in this case, 3) and a literal one for y (3x^4z)?

I found a website that says that an equation like 5x - 9 = 7 x 4 has two
coefficients, viz., 5 and 7. Is that mathematically correct?

Maybe my difficulty is about the very definition of coefficient.



Date: 03/24/2014 at 09:22:05
From: Doctor Ian
Subject: Re: coefficients in algebra

Hi Rohit,

Given a term like 3x^4.yz, saying that 3x^2.yz is the coefficient of x^2
would be an unusual use of the word.

> Are there two coefficients in a term like this -- namely, a numerical 
> one (in this case, 3) and a literal one for y (3x^4z)?

Normally, we would say that 3 is the coefficient of x^4yz. None of the
individual variables would be said to have its own coefficient.

> I found a website that says that an equation like 5x - 9 = 7 x 4 has
> two coefficients, viz., 5 and 7. Is that mathematically correct?

Do you mean this?

  5x - 9 = 7x^4

The coefficient of x would be 5; the coefficient of x^4 would be 7. 

> Maybe my difficulty is about the very definition of coefficient.

Part of the problem here is that normally, we talk about "coefficients" in
the context of polynomials. For example, consider this generic polynomial:

   (a0)x^0 + (a1)x^1 + (a2)x^2 + ... + (an)x^n

Its coefficients are a0, a1, a2, and so on. They are constants rather
than variables.

When you start expanding the usage of the word to other contexts, things
start to get a little blurrier. But that has to do with the nature of
language, not with mathematics.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 03/24/2014 at 11:06:29
From: Doctor Peterson
Subject: Re: coefficients in algebra

Hi, Rohit.

To expand slightly on what Dr. Ian said, I would add that technically
(according to some sources), what we commonly call a "coefficient" is
called the "numerical coefficient." More generally, the word "coefficient"
means simply "the other factor," so that you would be correct in saying
that the coefficient of x^2 in 3x^4.yz is 3x^2.yz, and that 3x^4z is the
coefficient of y.

You would not, however, say that there are two coefficients; merely that
each factor of the term has a coefficient.

As Dr. Ian said, we don't normally use this form of expression; in fact,
most sources don't mention the broader usage, but define "coefficient" as
the numerical (or constant, in the case of parameters) part of a term. I,
myself, wasn't aware of this issue until a few years ago I noticed that a
textbook I was using made the distinction, and I realized that in fact I
have said things like "the coefficient of x in 2xy is 2y" without noticing
that this usage did not fit my stated definition.

For example, Wikipedia and MathWorld both take "coefficient" to mean
"numerical," as most math sources do:

    http://en.wikipedia.org/wiki/Coefficient 

    In mathematics, a coefficient is a multiplicative factor in some
    term of an expression (or of a series); it is usually a number,
    but in any case does not involve any variables of the expression.

    http://mathworld.wolfram.com/Coefficient.html 

    A multiplicative factor (usually indexed), such as one of the
    constants a_i in the polynomial
    
    a_n x^n + a_(n - 1) x^(n - 1) + ... + a_2 x^2 + a_1 x + a_0. 

But some dictionaries give the broader definition, and their examples show
why it is sometimes appropriate:

   The Free Dictionary
    http://www.thefreedictionary.com/coefficient 

    1. A number or symbol multiplied with a variable or an unknown
       quantity in an algebraic term, as 4 in the term 4x, or x in
       the term x(a + b).

    a. a numerical or constant factor in an algebraic term: the
       coefficient of the term 3xyz is 3.
    b. the product of all the factors of a term excluding one or
       more specified variables: the coefficient of x in 3axyz is
       3ayz. 

   Merriam-Webster
    http://www.merriam-webster.com/dictionary/coefficient 

    1. any of the factors of a product considered in relation to a
       specific factor; especially: a constant factor of a term as
       distinguished from a variable 

   Math Glossary
    http://math.about.com/library/blc.htm 

    Coefficient - A factor of the term. x is the coefficient in the
    term x(a + b) or 3 is the coefficient in the term 3y.

The important thing is that the word "coefficient" should always be used
in the form "the coefficient OF" something. Only in linear equations in
one variable can we be so lazy as to say merely "the coefficients in the
equation." When expressed properly, there is no ambiguity.

So, returning to the usual use of "coefficient" to mean "numerical
coefficient" for your third question: in the equation 5x - 9 = 7x + 4,
there are two coefficients OF X, because there are two terms of the form
ax, and each has a coefficient. In fact, even in x - 9 = 4, x has a
coefficient -- namely, 1. I think it is inappropriate to merely say "there
are two coefficients," although perhaps the context makes it clear what
they mean. (The constants -9 and 4 are also coefficients, but not of x!)

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 03/31/2014 at 22:14:16
From: rohit
Subject: Thank you (coefficients in algebra)

Thank you, Dr. Ian and Dr. Peterson. 

I think that the concern is of a lack of global standardization of
definitions of certain math terms. For example, some authors of math
textbooks in India (btw, I am in India) use the word "literals," which is
defined as another word for "variables." For example, in the term -3x^2yz,
-3 is said to be the numerical coefficient, while x^2yz is said to be the
literal coefficient of the term.

Now, to make things more complicated, here is a website that has something
slightly different to say:

   Calculator Edge
    http://www.calculatoredge.com/math/advmath/advans28.htm 

   Question: What is a "Literal Coefficient"?

   Answer: A letter which has fixed value is a Literal Coefficient. 
   Usually the letters at the beginning of the alphabet are used for 
   this purpose.

This is then followed by an example:

   To save time, when you want to write down 56398246 each time in a sum,
   or if its value although fixed is unknown, then this number may be
   called a, b, c, or any other letter of the alphabet.

The author here appears to endorse the use of a letter to represent a
constant!

Anyway, it is a very pleasant experience EVERY TIME I communicate with Dr.
Math!!!! It is a wonderful contribution you are making to Society -- FREE 
OF COST! 

Please keep up the good work.



Date: 04/01/2014 at 17:03:32
From: Doctor Peterson
Subject: Re: Thank you (coefficients in algebra)

Hi, Rohit.

We see differences in terminology all the time, some regional and some
just because textbook authors make their own choices. Sometimes you see a
better way to present a concept, which requires making a slight change in
how you use a word -- so you do that. That can be a good thing; it's just
unfortunate that most texts don't mention that their usage may not be
exactly the same as others'.

The result is that students these days go on the Internet expecting to see
exactly the same information everywhere, and get confused by different
usages.

I've learned not to think someone's spelling or word usage is wrong (even
when I have no idea from their email address where they are), because it
may just be what they were taught. I often ask for the definition they
were given for a word, or the statement of a theorem that they think is
universal, in order to make sure we are talking about the same thing.

Your source here is defining "literal coefficient" as opposed to
"numerical coefficient"; that's not a bad thing to mention, but they
didn't actually give a definition for "coefficient," and what they
describe is really the more general concept of a "literal constant." Such
faulty "definitions" are rather common in elementary texts, I think.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Definitions
High School Polynomials

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