Real numbers

**prometheus** · 11-06-08, 10:08 AM

I was having a chat with one of my colleagues (also a physicist) about the set of real numbers. He is absolutely adamant that there is a way of defining the set of real numbers such that the trancendentals are not in $\mathbb{R}$ . To me this seems crazy because the way reals are taught to school kids is that there are integers, between which are rationals between which are reals. When you define things a bit more rigorously the reals are the limits of all Cauchy sequences of rationals (I don't really understand this definition - I am reciting it from a lecture I attended).

I would be grateful if someone could tell me if my colleague is wrong and why. Personally, I think it's obvious that $\pi$ is a real number, but I could be using the playschool definition of the reals.

**Guest254** · 11-06-08, 10:54 AM

Your colleague is talking rubbish. If you construct some set without the transcendental numbers, then you have not constructed the set of real numbers. The transcendental number pi is "obviously" a real number as it can be written as the limit of the Cauchy sequence (3, 3.1, 3.14, 3.141, ...).

**D H** · 11-06-08, 10:56 AM

You're physicist friend is talking out of an orifice on which the sun does not shine. The reals are defined as

The complete Archimedean ordered field. The reals are the only complete Archimidean ordered field within an isomorphism, and hence the the qualifier.
The set of all convergent Cauchy sequences in the rationals.
The set of all Dedekind cuts in the rationals.

The set of all algebraic numbers is not Dedekind complete. For example, consider the Cauchy sequence {2,9/4,64/27, ...(1+1/n)ⁿ, ...}. This obviously converges to e, which is of course transcendental. Every element in the sequence is algebraic. The sequence has an upper bound (three, for example), and yet the least upper bound is not algebraic.

**AlphaNumeric** · 11-06-08, 11:13 AM

Originally Posted by prometheus

When you define things a bit more rigorously the reals are the limits of all Cauchy sequences of rationals (I don't really understand this definition - I am reciting it from a lecture I attended).

That's the definition I always work with. I think of it as : If the number is expressible as the limit of a sequence of rationals, where the sequence elements get closer and closer, so that $|a_{n}-a_{n+m}| < A_{n}$ for all integer m>0 and $A_{n} \to 0$ as n goes to infinity.

That's a little more stringent than $|a_{n}-a_{n+1}| \to 0$ because the Cauchy convergence means all the elements after $a_{n}$ are within a specific distance of $a_{n}$ , which gets smaller as n gets bigger.

The Wiki page shows why that isn't enough for Cauchy convergence, in the second picture.

To miss out the transcendentals from the reals is to miss out 'most' of the reals! The algebraic numbers are countable, since they are roots of polynomials and the polynomials form a countable set, because there's a bijection between a vector in n+1 dimensional space and n'th order polynomials (ie the polynomial coefficients). Since the reals are not countable, to ignore the transcendentals is to be considering a countable subset of an uncountable set!

This thread is an example of why Hilbert thought physics was too important to leave to physicists