# Cardinality of the set of binary-expressed real numbers

#### Pengkuan

This article gives the cardinal number of the set of all binary numbers by counting its elements, analyses the consequences of the found value and discusses Cantor's diagonal argument, power set and the continuum hypothesis.
1. Counting the fractional binary numbers
2. Fractional binary numbers on the real line
3. Countability of BF
4. Set of all binary numbers, B
5. On Cantor's diagonal argument
6. On Cantor's theorem
7. On infinite digital expansion of irrational number
8. On the continuum hypothesis

Cardinality of the set of binary-expressed real numbers
PengKuan on Maths: Cardinality of the set of binary-expressed real numbers
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#### CRGreathouse

Forum Staff
Your proof in section 1 is incorrect -- you can't assume that set cardinalities will behave like exponents in the limit, despite the suggestive notation. (The result is correct, though.)

Your claim in section 2, explored in section 3, is incorrect. The set you actually work with is the dyadic rationals, but you conflate them with the set of fractional binary numbers (your $\mathbf{B_F}$, i.e., the set of real numbers between 0 and 1, which you choose to represent in binary).

Section 4 is incorrect by virtue of relying on the incorrect result from sections 2 and 3. Section 5 consists of a proof by assertion. Section 6, like section 4, relies on incorrect results and hence is also wrong. I did not bother checking further.

Review: Is this a mathematical breakthrough?

#### Country Boy

Math Team
Surely you understand that the "number of real numbers" in a given interval does NOT depend upon how the numbers are expressed? So I don't understand your reference to "binary expressed numbers" and "binary numbers".

#### Pengkuan

Your proof in section 1 is incorrect -- you can't assume that set cardinalities will behave like exponents in the limit, despite the suggestive notation. (The result is correct, though.)

Your claim in section 2, explored in section 3, is incorrect. The set you actually work with is the dyadic rationals, but you conflate them with the set of fractional binary numbers (your $\mathbf{B_F}$, i.e., the set of real numbers between 0 and 1, which you choose to represent in binary).

Section 4 is incorrect by virtue of relying on the incorrect result from sections 2 and 3. Section 5 consists of a proof by assertion. Section 6, like section 4, relies on incorrect results and hence is also wrong. I did not bother checking further.

Review: Is this a mathematical breakthrough?
Thank you for checking my article. I'm sorry to have replied late. I'm not alerted by email of your message.

#### Pengkuan

Surely you understand that the "number of real numbers" in a given interval does NOT depend upon how the numbers are expressed?
Yes.
So I don't understand your reference to "binary expressed numbers" and "binary numbers".
I thought the set of binary numbers in the unit interval contains irrationals. It turned out no. Because when the number of digits goes to infinity, the digits are not actually infinite many.

#### v8archie

Math Team
Because when the number of digits goes to infinity, the digits are not actually infinite many.
Marvellous! I do applaud your willingness to suspend preconceptions when dealing with the infinite, but not all counterintuitive results are correct.
I thought the set of binary numbers in the unit interval contains irrationals.
It does.

There is no such thing as "a binary number". Any real number is expressible in decimal, binary or any other base you care to name (as long as you allow me to have infinite strings of numbers). Rather neatly, if you choose any rational base, the set of numbers that you can't express without recourse to infinite strings of numbers is the same (here I am making use of notations like $13 \div 22 ={13 \over 22}= 0.5\dot0\dot9=0.5090909\ldots$ all of which are finite representations as is the equivalent binary fraction $0.\dot1000001001010011110\dot0$).

So your use of binary is completely redundant and flawed.

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#### skipjack

Forum Staff
13/22 = 0.590909090..., whereas 0.509090909... is the expansion of 28/55.

#### Pengkuan

13/22 = 0.590909090..., whereas 0.509090909... is the expansion of 28/55.
Yes. But they do not have the same binary expressions.

#### skipjack

Forum Staff
Your analysis of Cantor's theorem doesn't seem to quote any part of Cantor's proof of it.