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I have found a lot of different definitions of coefficient, many of which limit coefficient to a constant multiplier of a variable.

My confusion, though, is then teaching that every unknown has an assumed coefficient of 1...some unknowns aren't variables. I'm thinking of situations in which the unknown we're trying to find might be a constant...we can still view there as being a 1 next to it.

So why do we limit the definition of coefficients to variables alone? Thank you for any light you can shed.

apnorton
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Kate
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  • What is your distinction between "unknown" and "variable"? – Matt Samuel Aug 11 '15 at 22:17
  • I'm thinking of an unknown as any unknown quantity, whether its value is fixed or varies in a situation. My thought would be that the a, b, and c in ax^2 + bx + c = 0 and the b and m in y = mx + b are unknowns in the formulas anyway, but they're not variables, as they stand for constants. Do you define them differently? – Kate Aug 12 '15 at 17:57
  • I would generally just not use the term "unknown" as a noun, but that's my personal preference. In any case, no matter what $a$ represents, it is a symbol, and symbols can have coefficients. In the expression $3a$, $3$ is the coefficient of $a$. The concept is also more general; in the expression $(a+b)c$ you could say that $a+b$ is the coefficient of $c$. – Matt Samuel Aug 12 '15 at 22:41
  • Thanks. In (a + b)5, would you also consider the (a + b) to be a coefficient of 5? Or do you limit the definition of coefficients to symbols. – Kate Aug 13 '15 at 14:40
  • That's a bit of a gray area. A numeral is a symbol, so yes, it could be called a coefficient of 5. – Matt Samuel Aug 13 '15 at 14:42

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Maybe I'm not using the terms correctly myself, but If you declare $C$ an unknown constant and look at the expression $6x^2+2C$, I don't see any problem in saying the coefficient of $C$ is $2$.

For example, in the statement $2x^2=0$ where $x$ is a real number, is $x$ a variable? I would say the coefficient of $x^2$ is $2$ but $x$ is certainly fixed - it cannot vary.

I will continue to use "coefficient" instead of "the number in front of that thing" regardless of the context as it has not produced any ambiguity in my life thus far.

David P
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  • These discussions aren't of mathematical definitions, but of language usage:$\quad$1. Since "unknown" suggests an absence of knowledge, perhaps $C$ is more accurately an "arbitrary constant" rather than an "unknown constant"; the former descriptor sounds like a contradiction in terms but makes clearer that $C$ is a parameter; so, the coefficient $2$ here indeed does belong to a (special) variable. – ryang Mar 27 '22 at 06:05
  • $x$ in $2x^2=0, 2x^2=1$ and $2x^2=-1$ is considered a variable in the sense that these equations are either true or false as we vary its value, even as the first equation is satisfied only by a single particular value; so, the coefficient $2$ here also does belong to a variable.
    • – ryang Mar 27 '22 at 06:05