My Math Forum Real Numbers are a Subset of the Rationals

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 May 18th, 2018, 08:27 AM #1 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 Real Numbers are a Subset of the Rationals The following table, which is countable (google), includes all pos rationals and reals. The reals are a subset of the rationals if you define: Real: m/100000........., all m Note: Left column and Top row are endless, \begin{matrix} &1 & 2 & 3 & 4 & . & . &. \\ 1 &\frac{1}{1} & \frac{1}{2} & \frac{1}{3} &\frac{1}{4} & . & . &. \\ 2 & \frac{2}{1} & \frac{2}{2} & \frac{2}{3} & \frac{2}{4} & .&. &. \\ 3 & \frac{3}{1}& \frac{3}{2} & \frac{3}{3} &\frac{3}{4} &. & . &. \\ 4 & \frac{4}{1} &\frac{4}{2} &\frac{4}{3} & \frac{4}{4} &. &. &. \\ . &. &. & . & . & . & . & .\\ .&. &. & . & . & . & . &. \\ . &. & . & . & .& . & . & . \end{matrix}\\ 0 assumed included.
 May 18th, 2018, 09:40 AM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,207 Thanks: 1160 $\pi \in \mathbb{R}$ $\pi \not \in \mathbb{Q}$ $\therefore \mathbb{R} \not \subset \mathbb{Q}$ Thanks from zylo
May 18th, 2018, 10:11 AM   #3
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Quote:
 Originally Posted by romsek $\pi \in \mathbb{R}$ $\pi \not \in \mathbb{Q}$ $\therefore \mathbb{R} \not \subset \mathbb{Q}$
True

$\displaystyle \pi$ $\displaystyle \equiv$ 3 "+ " 1415926....../1000000.......

That's awkward and requires refinement: my original version was:

The real numbers in [0,1] are a subset of the rational numbers in [0,1]. From table.

Real numbers in [0,1] = m/1000......... , all m

In that case, a real number is N.r which is still countable: N is a subset of the rationals and so is r.

May 18th, 2018, 10:29 AM   #4
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Quote:
 Originally Posted by romsek $\pi \in \mathbb{R}$ $\pi \not \in \mathbb{Q}$ $\therefore \mathbb{R} \not \subset \mathbb{Q}$
Romsek

I doubt zylo will accept your proof. He simply assumes again that an infinite set has the same cardinality as any other infinite set.

There is no doubt that he has described an infinite set that includes all rationals. In fact, it includes every possible decimal representation of every rational number. He still has not shown how he matches each real to one of these representations. Obviously, he will have no difficulty matching a unique rational to one of those representations. But he neglects to show that the irrationals match up with the remaining representations. Instead he implicitly relies on his assumption.

Last edited by JeffM1; May 18th, 2018 at 10:32 AM.

May 18th, 2018, 10:33 AM   #5
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Quote:
 Originally Posted by zylo True $\displaystyle \pi$ $\displaystyle \equiv$ 3 "+ " 1415926....../1000000....... That's awkward and requires refinement: my original version was: The real numbers in [0,1] are a subset of the rational numbers in [0,1]. From table. Real numbers in [0,1] = m/1000......... , all m In that case, a real number is N.r which is still countable: N is a subset of the rationals and so is r.
You don't seem to be able to discern between an actual and a limiting value. Archie has tried to pound this difference into your head apparently to no avail.

It's my opinion you just want attention. Have fun.

 May 18th, 2018, 10:41 AM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,515 Thanks: 2515 Math Focus: Mainly analysis and algebra I recommend closing this thread and stopping Zylo from creating more like it. I can't see any value at all in rehashing this nonsense again.

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