h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

The distinction between transcendental and algebraic irrationals may be appropriate on this page. Drop me a line if you want a hand with definitions.

Could you? It would be helpful. Bear in mind though that this is basic number theory. I will be doing a project on more complex concepts later.

I will try to come up with a brief paragraph giving the differences between the two, and illustrating with examples. Nothing very technical.

Cheers,

SW.

Here you go then:

Real numbers can be classified into two categories; Algebraic and Transcendental. Algebraic are defined as numbers which can be described as being some root of a polynomial equation. The classic example of this is 1.414.... or the square root of 2: x^2 -2=0 is the polynomial, v/2 is its real root. Other examples include 4 (polynomial x^2-16=0); and (1+v/5)/2 (polynomial x^2 -x -1=0).

Transcendental numbers are reals which cannot be described by such a formula.

Note that all transcendental numbers are irrational, whilst algebraic numbers can be either (see proof that v/2 is irrational).

Note also that the set of algebraic numbers is far smaller than the transcendentals. Algebraic numbers can be shown to be countable - that is equal in size to the set of natural numbers, where transcendental numbers have been shown to be uncountable (using Cantor's Diagonal Argument for example).

(BTW Jo, you've probably worked it out, but the ' v/ ' is the surd symbol, ^ the power symbol. The last paragraph may not be worth including at this stage - perhaps not until an article on Cardinal numbers has been written. Having said that, it does give an example of why we might want to use the Diag Arg, something many people find useful to explain why we bother with all this Maths stuff! That's why I suggested (somewhere) adding a few lines in the Complex Numbers article about Capacitance and Inductance in electronic circuitry - it shows why we would want to 'invent' imaginary numbers...)

We didn't invent imaginaries, we discovered them. They had practical applications in early analysis but were (and still are for me) just fun to mess around with.

I am not sure about adding that about Transcendentals. I think it would be better to wait until we have arrived at the next number theory project with set theory under our belt and also some algebra. It may make people feel more comfortable with the concept of cardinals and so on.

Yeah, probably best to leave the cardinality bit 'till later.

Oh, and that's why I put 'invent' in inverted commas

Oh Analysis is going to be fun *rolls eyes*

Going to have to cover algebra first in a lot of detail. That may be a good project for after the Historical Figures one. Algebra will make a lot of the later proofs and concepts easier to follow (try explaining that d/dx (x^2) = 2x without basic algebra and yet rigorously )

Anyway, that is at least 2 months away

Completeness and Symmetry is up for a gander at http://www.h2g2.com/A422010

Let me know what you think.

Nice exposition of Russell's Paradox - short and to the point. Interesting overview of completeness in general, and concise definition of symmetry. A couple of comments though:

Why "Symmetry and Completeness"? I can't really see the connection. Perhaps symmetry should come later in with an article on basic group theory. Something which may fit in better with completeness is consistency. In a logical setting, the following sums the two up:

Every tautology has a proof (Semantic Completeness), every theorem is a tautology (Semantic Consistency)

Or; everything which is always true is provable, and everything which has a proof is always true (respectively), for those who haven't read their intro to Logic...

Oh - and there was a typo somewhere near the end. Can't remember what it was offhand.

HTH.

SW.

Hmm... you are probably right about the symmetry. Perhaps you could write a short article on consistancy and add it to the project.

Bugger! In too deep...

OK then, why not! I'm pretty busy this weekend, but I'll try to get something written over the next week. Will that be OK?

Yep!

I will ask Abi, on Tuesday, to give me an extra week on the project (the due date is 1st September)

Cool.

SW

Hi Joe,

The article on Consistency is now ready, and can be found at: http://www.h2g2.com/A428177 .

Hope it is accessible enough. I can include a proof that all propositions are a consequence of an inconsistent set if you like. It's not too technical, but let me know.

Add the proof as an addendum and it would work. An exercise for the interested reader to follow.

Thanks though!
Brilliant work.

OK. Done.

What happens with these research projects now then?

SW.

Someone is currently doing proofreading. Once that is finished, I email the Towers and they draw subs and scouts attention to it.

Great. Look forward to seeing the results...

We did invent imaginaries. We invented all of mathematics the moment we fixed our axioms. We may often forget this but it is fairly easily shown. It is a self contained system with only tenuous and approximate connections to reality. Indespensible and vastly useful maybe, but only ever an approximation.