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July 10th, 2017, 05:48 AM   #21
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Originally Posted by Maschke View Post
That is really a very philosophical question that's deeper than it seems. For example, take a simple quadratic expression like $ax^2 + bx + c$. We are told that $a$, $b$, and $c$ are "constants" and $x$ is a "variable." What does that mean? They are all letters of the English alphabet that stand for numbers. But $a$, $b$, and $c$ stand for particular fixed numbers during the evaluation of the expression, while $x$ "ranges over" some domain like the real numbers.

And what can all that possibly mean? How can a man from Mars sort out the meaning? After all, $a$, $b$, and $c$ do in fact freely range over the real numbers. I think we need a philosopher to sort this out. I don't think this is as clear as we all think it is. Maybe it has something to do with free and bound variables but I'm not actually sure.

As an example, we might ask the question: If we equate the quadratic expression to $0$, what condition on $a$, $b$, and $c$ results in $x$ being a real number? The answer is that the discriminant $b^2 - 4ac \geq 0$. But in this question $a$, $b$, and $c$ are the "unknowns," they're the things we're trying to find out about.

The more I think about the question of what is a variable, the less I understand it.
a, b, and c can be any numbers, but their values are given in the equation, which is why they are called constants. x is a variable because it is what is being solved for. The difference between the constants and the variable(s) is not because of what numbers the domain ranges over.

Originally Posted by awholenumber View Post
No, I am not sure why.
If x = 2, the fraction = 0/0, which is undefined.

Last edited by skipjack; July 10th, 2017 at 02:41 PM.
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