Gelfand: Chunk 13 - Rational Expressions Quickie

February 21, 2017

Section 33 of Gelfand’s Algebra introduces the incredibly simple concept of Rational Expressions. These are in effect a super-set of polynomials, created by slightly relaxing the rules of what makes a polynomial a polynomial and allowing in division.

Did I say “incredibly simple”? How could I possibly be mistaken? Let’s take a step back for a second to bring something subtle to mind.

The Subtleties of Division

In coming back to all this algebra after my last two posts, a few fundamental misconceptions from my early maths history have been challenged. Amongst these was my mistaken belief that multiplication and division were two sides of the same coin in the same way that addition and subtraction actually are. What I’d failed to grasp was that whereas in the latter, a movement in position of terms was always possible in my simple world-view (it’s the Commutative Law, in it’s addition-form)

$a + b = b + a$

$a - b = -b + a$

but the same could not be said for the former (the Commutative Law applies to multiplication but not division remember).

Therefore

$2 \times 3 = 3 \times 2$

but

$2 \div 3 \ne 3 \div 2$

In fact, if you write this in another way, using fractions, then the inequality becomes patently evident.I’ll possibly come back to all this as it seems an area rich in potential discoveries.

$\frac{2}{3} \ne \frac{3}{2}$

It’s nice to note that explicitly - division isn’t a flip side of multiplication. So keep it in a difference place in your head.

But we can keep going. When we do add division, as we have by expanding things to take in rational expressions, there’s also the problem of zero.

$\frac{0}{3} = 0$

$\frac{3}{0} = ...$

A divide by zero basically blows up on us. Boom!Or it does until we get on to the concept of infinity which is another post, perhaps when I get back to Turing and Petzold.

With these in hand, we can now get back to the concept the post should be about: Rational Expressions.

Rational Expressions - Zero-Denominator-Disallowed

In his introduction to these, Gelfand basically points out the latter of the two realisations above - you can’t have a denominator in a rational expression which is zero, or is identically equal to zero ( $x - x$ is his example). Given the above notes, it’s clear why this is the case.

The Actual Work of Rational Expressions

Usually I’d now embark on a long-winded explanation of how I eventually grokked this concept, and the tricks arising, but it was in fact (aside from the above notes) very simple given I’d already got my head round much fo these concepts when confronting Fractions (see Chunk 4 - Fractions Ugh). What you do need to do however is hop over there. You might recall I jumped ahead at that point to discuss the various rules of working with fractionsAdding, multiplying, etc. and I jumped ahead to this section of Gelfand.

Some Form of Concluding Aside

There’s not much else to put in this chunk I don’t think. So to give things some kind of conclusion, I wanted to share a bit of knowledge I extrapolated from Petzold which I’m currently reading in parallel to writing this. Here it comes:

Rational expressions are so called, because they are ratios of something else, i.e. fractions of polynomials.

I’m guessing here, but based on the fact that the “rational numbers” are ones which can be expressed as ratios of two integers (e.g. $\frac{2}{3}$ ). I think this isn’t too much of a leap.