Re: how

Am 25.04.2024 um 02:48 schrieb Chris M. Thomasson:

1, 1.25, 1.5, 1.75, 2
 I this finite section, 1 and 2 are natural, and the other three are not.

Right.

However, they [all] are within the set of natural numbers

Nope, they are not.
Hint: 1.25, 1.5 and 1.75 aren't natural numbers. :-)
This means: 1.25 !e IN, 1.5 !e IN and 1.75 !e IN.

but three of them cannot be directly represented.

Well, the not just "cannot be directly represented", but they simply AREN'T natural numbers (and hence not in IN).

Although, they can be indexed using natural numbers, zero aside for a moment:
 [0] = 1
[1] = 1.25
[2] = 1.5
[3] = 1.75
[4] = 2

Right.
Hint: The rational numbers are countable. I.e the can be indexed using natural number.
Numbers like 1.25, 1.5, 1.75 might be called "finite decimals". They are a proper subset of the rational numbers (and hence are countable).
What I mean her is that, say, the number 1/3 = 0.333... is not a "finite decimal", even though it's a rational number.

There is an infinity [of rational numbers] between 1 and 2... They can be indexed using naturals?

Yes.
Cantor (first) showed that the rational numbers are countable (i.e. can be indexed by the natural numbers.)
Funny, is't it?
On the other hand, the _real numbers_ between 1 and 2 CANNOT be indexed by the natural numbers.

;^) Just having some fun with numbers here.

Sure. We are cool, man.