variable: A symbol used to represent one or more numbers.
Or alternatively: A symbol used to represent any member of a given set.
High school students are justifiably confused by the two distinct concepts:
- a variable as something that “varies” in an
expression, such as the h in the expression $4.50\cdot h$;
and - an unknown quantity that is a "specific unknown” in an
equation
The definition of a variable as being a symbol used to
represent both of these cases explicitly states this
as: “a symbol used to represent one or more numbers.”
Where the “one number” case is the “specific unknown” in a simple
equation, such as $18.00=4.50\cdot h\;$ where the $h$ can only
represent the one number $4$ to turn the open sentence into
a true statement.
The more than one number case being the letter h standing
for values in a table such as: $1, 2, 3,\>$ or $4$ being
substituted into an equation to form a pattern such as
$C=4.50\cdot h$, the definition of a variable now being
interpreted as a symbol, h, used to represent one number
when that number is substituted in for it and a symbol
used to represent more than one number when other numbers
are substituted in for it. Thus generating a table of
values of $C$ such as: $4.50, 9.00, 13.50,\>$ and $18.00$.
Another example, the variable, say x, in the quadratic equation represents a parabola when it "varies" over a given domain. But the variable in the equation $0=x^2+2x+1$ is "an unknown quantity. It does not vary." Thus, in this case the vari-able has lost its ability to "vary." Yet in both situations they are referred to as a variable, and this duality is embodied in the definition of a variable as "a symbol used to represent one or more numbers." Could the motivation behind this definition be such that we don't have to make the distinction between an "unknown specific quantity" and a "varying" quantity?
Does anyone agree that the above argument has been logically developed or
is there some flaw in my reasoning? Thank you.
Best Answer
I don't like the word "variable" in math. When we're solving an equation, $x$ is nothing more than the name of a number whose value we do not yet know, $x$ is not in any sense "variable". It's not as if the value of $x$ can change.
And if we are defining a function $f$, we might say something like, if $x$ is a number, then $f(x) = x^2 + 7$ . Even here $x$ is not "variable". We are just saying that if $x$ is a (specific) number, then $f(x)$ is the number $x^2 + 7$.
Now, in computer programming, you have variables whose value can actually change. That's different.