Is equal to ? No, of course not but why do so many students even at university level end up adding numerators to denominators to form these absurdities? I’m not even that annoyed by it. More intrigued because it doesn’t make any sense at all. Well of course it does in a way. You don’t know what you’re supposed to do (or can do) so you make a ‘reasonable guess’ as to what an expression could look like and you just run with it. After all you’ve never been asked to derive or motivate these things (at least not in years) but when these things show up when I’m correcting a test at university level it gets me all depressed. I mean the student in question had applied to an engineering school (in chemistry) but he’s so far behind on basic algebra that he’ll either never make it or will go through hell. Well, well, zero points and onto the pile… Granted I only ever saw this particular error twice while grading tests but other variants of universal linearity such as and were frequent.
But if we focus more on this particular issue of adding numerators and denominators (besides it being kind of strange) is it at least something? Let’s call these alternate rules of algebra the broken system by naively defining
and let’s see what goes wrong. Well even at a brief glance we see that it’s not consistent with standard arithmetic but what is so much worse is that it isn’t even internally consistent! Sometimes absurd errors can be related to real mathemtical concepts. One of my favourites for example is when someone accidentally writes
because this is actually true if you work in an anticommutative algebra such as a Clifford algebra where the property leads to the annihilation of cross-terms. But as to the broken system adding denominators implies that if we are to have a zero (a number which when added does nothing, and we have to have one!) we’re going to have to use and that’s just wrong because once we allow this abomination into our algebra nothing becomes well defined because it clashes with the reducible property of fractions. That is the fact that which is encapsulated in
If zero denominators are allowed then suddenly all numbers end up being equal, and they do absolutely nothing when we combine them. That doesn’t seem right… Well maby the reducibility property (the equivalence relation) needs to go (I never really liked it anyways says the student) but then suddenly you’ve ended up with being distinct from and that just gives us new problems because suddenly you can’t find inverses anymore since
No matter how you twist an turn and redefine you seem to just end up with more problems than you started with. It can’t be made remotely similar to numbers (with the real number axioms). It’s just broken.
I find errors like universal linearity more depressing than annoying because to me it represents the divorce of mathematical symbols from their origin as descriptions of real things. They’re still abstract things, different in nature from whatever we wish to equate them with in the real world but the “axioms” and rules which we use are not arbitrary. An axiom such as is not ‘just because’ but instead always has real intuition behind it. To point to the cutting of a cake or the geometric construction of proportions are not proofs of why adding fractions is the way it is but it’s a damn good indicator of why it ought to be the way it is.
Anyone who looks at three pie charts knows that you can’t add numerators to denominators. It is apparent!
To move to explain the way addition of fraction should be done is of course trickier because of the issue with imperfect metaphores which I think I want to return to in a future post. But the cake metaphore works just fine as a model of why you ought to have to put things on common denominators.
Metaphors are tools for understanding. They’re not the key to understanding. Generalization and abstraction is the key but without metaphors and physical representations of mathematical concepts there is no motivation for what we squiggle on a paper.