# Set of real numbers

1. Jul 22, 2011

### cragar

If the set of natural numbers is $\aleph$
and when we write a real number we have 10 choices for each position 0-9
so can we say that there are $10^{\aleph}$ real numbers ?

2. Jul 22, 2011

### micromass

Hi cragar!

You probably mean $\aleph_0$, right?

Yes, that is correct, the real numbers have cardinality $10^{\aleph_0}$. Also note that

$$2^{\aleph_0}=10^{\aleph_0}=\aleph_0^{\aleph_0}$$

But, I also must give you a warning. Saying that you have "10 choices for each position 0-9" is not exactly true, there are technical details. For example 1.00000000... and 0.9999999... are the same numbers, so some choice yield the same number. Also, choice like ...9999999.999999.... are not allowed: we must only have a finite number of 1-9 in front of the dot.

These technical matters can be fixed however.

3. Jul 22, 2011

### cragar

how is this true $$2^{\aleph_0}=10^{\aleph_0}=\aleph_0^{\aleph_0}$$

4. Jul 22, 2011

### micromass

Well, to give an intuitive explanation. You showed that the real numbers have cardinality $10^{\aleph_0}$, but you used decimal representation here. We can also use binary representation. In that way, you have numbers of the form 111.0101101 for example. So you have to choose 0 or 1 a countable number of times. So by the same reasoning, the real numbers have cardinality $2^{\aleph_0}$.
When using hexadecimal, you'll obtain $16^{\aleph_0}$ as cardinality of the reals. So

$$2^{\aleph_0}=3^{\aleph_0}=...=10^{\aleph_0}=...$$

5. Jul 22, 2011

### cragar

I seen the proof where the set has 2^n subsets . like for example if i have a sub set {3,2} this would mean I would put a one in the 3rd position and a 2 in the second position and zeros in the rest. but i thought this was a proof where we couldn't repeat numbers. We didn't start with a multiset. So are you saying the reals are all of the subsets of the naturals.

6. Jul 22, 2011

### micromass

Well, the reals aren't the set of all subsets of the naturals, but they certainly have as much elements!!

7. Jul 22, 2011

### cragar

when you use binary for your list count, what do you mean by your 0 or 1 .

8. Jul 22, 2011

### micromass

Just use the binary system: for example 10=2, 11=3, 100=4, 0.1=1/2, etc.