What Is a Coefficient?

Date: 01/29/2004 at 09:17:25
From: victoria
Subject: Numerical coefficient

My book says

  4xy contains a numerical coefficient which is 4.

What's a coefficient?

Date: 01/29/2004 at 11:55:01
From: Doctor Ian
Subject: Re: (no subject)

Hi Victoria,

When we have a term with a bunch of variables multiplied together,
such as 

    2 3
  xy z

we sometimes multiply it by a constant factor, like 

     2 3
  7xy z

In this case, we say the the constant (7, in this case) is the
"numerical coefficient" of the term.  

Sometimes we have parameters--symbols that look like variables, but
don't really change their values.  For example, suppose we know that
the density of something has some value, but we don't know what it is.
 We might call it "D".  The weight of a cube with edge length x, made
of this substance, would be 

  Dx^3

Here, the D looks like a variable, but it's really a coefficient,
since whatever it is, it is, and we're not going to change it the way
we would substitute different values of x.  (Does that make sense? 
It's a little complicated, so write back if you're not quite getting
it.)  In this case, we would say that D is the "coefficient" of the
term, but it's not a "numerical" coefficient, since we're not using
numerals to represent it. 

Usually we talk about coefficients in conjunction with polynomials, e.g., 

  __x^4 + __x^3 + __x^2 + __x^1 + __x^0       (Note that x^0 = 1)

This is a general description of a certain kind of curve, which always
looks sort of the same--but by putting different values in the slots,
we can change the exact shape, to make parts of it taller or shorter,
or sqeezed in or stretched out.  The exponents on the variables tell
us, in a general sense, what _type_ of curve we have; but the
coefficients (the actual numbers we choose to put in the slots) tell
us how each one of those curves differs from the others.  

Does this make sense?

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