[pcmonk]

That's not what larger means, that's what (maps into a) strict subset means. In finite numbers, that's the same as larger (greater cardinality), but it's very much not the case for infinite numbers.

There are more than two sizes of infinite sets, but there are just as many real numbers as imaginary numbers for the same reason there's just as many integers as rational numbers.

Try reading this: https://en.wikipedia.org/wiki/Cardinality#Infinite_sets

[Retric]

We agree that {A,B,C} has a lower cardinality than {A,B}.

Now, feel free to try and map the set of Real numbers to the set of irrational numbers. ex: e + ei.

[pcmonk]

{A,B,C} is of greater cardinality than {A,B}. However, the argument that "a rule works for finite numbers, so it must work for infinite numbers" is clearly false.

For a your mapping, see: http://math.stackexchange.com/questions/512397/is-there-a-si...

[Retric]

The distance between 0 and 1 is smaller than the distance between 0 and 2. The number of points between 0 and 1 is larger than the number of rational numbers.

[pcmonk]

> The distance between 0 and 1 is smaller than the distance between 0 and 2.

This is correct. However, the number of real-valued points between 0 and 1 is the same as the number of real-valued points between 0 and 2.

> The number of points between 0 and 1 is larger than the number of rational numbers.

This is also true because there are uncountably many real-valued points between 0 and 1 and countably many rational numbers.

[yongjik]

1. e + ei is not real.

2. f(x) = x + sqrt(2) if there exists an integer k>=0 such that x - k * sqrt(2) is rational; f(x) = x otherwise.

This function maps all real numbers to irrational numbers, 1-to-1.

[Retric]

e is real, e + ei is irrational.

[pcmonk]

I believe you mean "complex". You can form a bijection between the reals and the complex numbers by interleaving the digits, as any Google search can tell you.

[Retric]

For every point on your mapimg I define two points X + 1 and x + 1i. You can assign infinity to one of them but not both.

[yongjik]

I'm afraid you have some basic confusion with mathematical concepts, including "mapping", "irrational", "complex", and "infinity".

So, after you define x+1 and x+i, now what? Also please bear in mind that "infinity" is neither real nor complex: if you have a well-defined mapping into complex numbers, then by definition, it never maps to infinity.

(Yes, there are some "functions" like y = 1/x that "maps to infinity", but it's simply mathematicians being lazy and abusing notations because everybody around them understands what's going on.)

[jakub_h]

e is irrational, e+ei can't be irrational because it isn't real (all irrationals are reals).

[Retric]

Sorry, imaginary number composed of two irrational numbers.