Number Plane: Filling Holes with Irrationals

[Mentallic]

First came the natural counting numbers.
Second came all integers, positive, negative and 0.
Third came the rationals.

At this point I would've thought that would be all. All the holes in the number plane would've been filled by using infinitesimally closer rationals.

Fourth came the irrationals and now the number plane has been completely filled.

How was it known that there were holes in the plane? I guess my common sense is defying logic...

 

[tiny-tim]

… Fourth came the irrationals and now the number plane has been completely filled.

How was it known that there were holes in the plane? I guess my common sense is defying logic...

Hi Mentallic!

(plane? how many fingers am i holding up? )

The Pythagoreans knew there was a gap because they could construct a length of √2 which they proved isn't rational.

In other words, they knew there was a gap because they could actually see what filled it!

(But there's still gaps … you can define in-between numbers which represent sequences which converge to the same limit but at different rates )

 

[Mentallic]

Sorry I have mistaken the complex number plane and number line

Ahh that makes sense. The idea of irrationals filling in all the spots only occurred after understanding them in detail.

(But there's still gaps … you can define in-between numbers which represent sequences which converge to the same limit but at different rates )

um.. please elaborate? If they converge to the same limit then they must be the same, no matter what rate they converge at, right?

 

[maze]

Legend has it that Hippasus discovered that √2 is irrational on a boat. Pythagoras was so pissed off about this that he threw him overboard and drowned him!

 

[matt grime]

I'm not sure what tim meant exactly, not least because there are a lot of impersonal pronouns flying around that seem to refer to different things.

However, what one can say is:

It has been known for millennia that the rationals are not enough - the square root of 2 is not rational. But one can construct more numbers from rationals with algebraic operations such as taking roots. Let us call a number algebraic if it is the root of a polynomial with rational (or integer by clearing denominators) coefficients, like x^2-2. Are the real algebraic numbers all we need? No, there are real numbers that are not the roots of such polynomials, such as e and pi. I recall that the first number that was shown to be transcendental has the property that its rational continued fraction approximations converge too slowly - there are results about the rates of convergence of continued fraction approximations.

 

[tiny-tim]

um.. please elaborate? If they converge to the same limit then they must be the same, no matter what rate they converge at, right?

The sequences {1/n} and {e_-n} both converge to 0, but at different speeds, and they can be defined as different numbers.

(But {1/2n} and {1/(n+1)} are defined as the same number as {1/n}.)

With that definition, obviously there are infinitely many numbers whose distance from each other is zero.

But there is a perfectly good ordering (<), and addition also works.

(I thought they were called "constructive numbers", but I've tried to look them up, and not found anything yet. )

 

[matt grime]

I've not come across tiny-tim's notion of things converging at different rates, but it strikes me that the OP might need some more info about real numbers to see where the idea comes from.

So, what are the real numbers? That's a surprisingly difficult question to give a proper answer to - we all know what they ought to be, but that's not the same thing.

One way to define the real numbers is via sequences of rational numbers.

Consider the sequence of rational numbers

3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...

this converges to pi (if carried on correctly). This is not rational. Thus taking limits of rational numbers leads to numbers that are not rational.

We can define the reals as the set of all limits of all rational sequences. To do this properly we take the space of ALL Cauchy* sequences of rational numbers. We declare that two sequences {x_n}, {y_n} are equivalent if {x_n-y_n} converges to zero. This is an equivalence relation, and divides the space of all sequences up into disjoint sets - sequences that are equivalent to each other.

We can freely add these equivalence classes by adding sequences and so on. But to make things nice and to see that these are the real numbers we need to choose a representative of each equivalence class. Normally we choose the sequence of increasing decimal approximations, i.e. the decimal expansion of the number, to be the canonical number that represents the class. There are other choices - it is better to think of the sequence x_n=1/3 for all n as representing one third than 0.333... and we would prefer to write 1/3 for this equivalence class. I mentioned continued fractions before - they often are nicer than decimal expansions for numbers that are roots of polynomials: sqrt(2) and phi (golden ratio) have nice continued fraction representations.Tiny-tim's notion says that you can refine the idea of when two sequences converge to 'the same thing', but this is definitely leading out of the realm of ordinary calculus.* Don't worry about the word Cauchy: it is a way of saying 'sequences that converge but where we don't know what they converge to necessarily'. Normally we say something like x_n converges to x if ... but here we're attempting to define the set of x's that are limits so I cannot use x in the definition like that.

 

[Mentallic]

Ahh I like the notion of representing an irrational as a continued fraction; it allows me to understand these "holes" in the number line more clearly.

Sorry Matt grime, I didn't understand most of the terms you were using but in your conclusion I was able to see where you were getting at.

tiny-tim do you think you could give an example of two such converging values? I'm having trouble believing that two numbers converging at different rates but to the same point have different values.

 

[tiny-tim]

tiny-tim do you think you could give an example of two such converging values? I'm having trouble believing that two numbers converging at different rates but to the same point have different values.

Hi Mentallic!

The two sequences don't have different values, they have different names (and one is defined to be bigger than the other).