why are algebraic manipulations in general derivations valid?

My question somehow embarrasses me because this used to make sense to me before. Sometimes my brain just becomes confused with things I understood previously :(. My question is this:

Derivations of equations are done using variables only, and only rarely do we really specify the numerical values. This is because we like to have a general theorem for something, or whatever. What I don't get is why using only variables which are not really numbers is subject to the same reasoning as numbers. Variables are this sort of "a number capable of being any number" but its not really anything.

I don't know why this doesn't make sense to me right now if it was just sort of a truism before. I'd like to hear an opinion on why this sort of generality is true and valid.

I'm specifically bothered by equations in physics btw.

Thanks and sorry for the question.

• I think it would help to have an example of the kind of manipulation that is bothering you. "Equations in physics" seems to me to refer to something different from "Derivations of equations". – mweiss Jan 13 '15 at 2:46
• Because the equations generalize/abstract the properties that are satisfied by numbers. – user203864 Jan 13 '15 at 2:47
• In most cases, an “algebraic manipulation” is just a way of writing an expression that may be different in appearance from the original, but is equal for all values of the variable-letters that appear. – Lubin Jan 13 '15 at 2:55
• "What I don't get is why using only variables which are not really numbers is subject to the same reasoning as numbers." Because that's how we defined them. We defined the allowed operations based on how numbers behave. – hasnohat Jan 13 '15 at 3:32
• Hi @mweiss . I've made another question which is more specific to my actual confusion. In case you want to see it: math.stackexchange.com/questions/1103461/… – ben ari Jan 14 '15 at 1:23

An equation involving variables such as $$\tag{1} (a+b)(c+d)=ac+ad+bc+bd$$ doesn't in itself mean anything until you choose values for each of the variables. Once all of the variables have values we can do the arithmetic to evaluate each side of the equation to a number and see if the numbers are the same. Then, and only then, does the equation become something that is either true or false.

There are some equations -- such as (1) above -- with the property that they end up being true no matter which values we choose for the variables. That's a useful property; we call those equations "identities" or (in some cases) "rules".

However, we also work with equations that may or may not end up being true when we choose values for the variables -- for example $$\tag{2} 5a+2a=14$$ which becomes true if $a$ is $2$ and false if $a$ is $3$.

We can rewrite such equations using rules, to get for example $$\tag{3} 7a-14=0$$ and the point of the rewriting is that the value choices for $a$ that makes (2) true are exactly the same as the value choices for $a$ that makes (3) true.

This property is what we mean when we say the rewriting is valid, and a rewriting is valid whenever we can prove (in whichever convincing way we can think up) that the value choices that makes one equation true are the same ones that make the other true.

So in the end it all comes down to calculations on actual numbers, and the symbolic algebra on letters is just a slick way to predict how those calculations will come out before we choose which numbers to do it with. It's useful and time-saving but not in itself mysterious.

(The above is true at the algebra-precalculus level. In higher algebra we meet examples where variables really are things in themselves and everything does not straightforwardly reduce to actual values, such as formal polynomials over finite fields. But one shouldn't try to wrap one's head around those cases without being completely comfortable about the less advanced setting).

• Nicely said. The manipulation of symbols captures the pattern of manipulation that one would carry out in any specific instance with actual numbers. +1. – MPW Jan 13 '15 at 3:23