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 MESSAGE BOARDS >> GENERAL DISCUSSIONS POLL: Math time!

Which set has more numbers?
by Neo7
 Rational Numbers Irrational Numbers They have the same magnitude (infinity)

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 Neo7 send PM view profile add to friends User #: 475 Posts: 7776 Joined: Jan 17, 2004 Last visit: Dec 05, 2013
 Posted: Oct 10, 2011 6:51:56 PM - Report post (0) (0) post reply
 Just a quick review:Rational numbers are real numbers that can be expressed in the form a/b where b is not zero.Irrational numbers are real numbers that cannot be simplified into a/b and exhibit a non-terminating nor repeating decimal.Let us call the "both are the same" option (3rd one) as the default that does not require proof. If you do pick the other two, let's see if you can come up with a reason why you picked your answer (because mathematics is all about trying to prove why a statement is true).[Edited by Neo7, 10/10/2011 6:54:40 PM]
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 benduhova send PM view profile add to friends User #: 961777 Posts: 2744 Joined: Apr 26, 2010 Last visit: Nov 27, 2013
 Posted: Oct 10, 2011 7:03:15 PM - Report post (0) (0) post reply
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 Neo7 send PM view profile add to friends User #: 475 Posts: 7776 Joined: Jan 17, 2004 Last visit: Dec 05, 2013
 Posted: Oct 10, 2011 7:14:02 PM - Report post (0) (0) post reply
 A "non-real" number (formally known as a Complex Number) is a number that has the square root of -1 in it's expression.Actually a "non-real number" is known as an imaginary number and revolves around the square root of -1 and it being raised to various powers.A Complex Number is a number containing a real component and an imaginary component. It is expressed as the form a+bi where a is any real number and bi is a scalar (b) times the square root of -1 (i).[Edited by Neo7, 10/10/2011 7:18:25 PM]
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 lamile send PM view profile add to friends User #: 198571 Posts: 4935 Joined: Jan 22, 2007 Last visit: Dec 06, 2013
 Posted: Oct 10, 2011 8:24:30 PM - Report post (0) (0) post reply
 Cheated a little for explanation.There is a proof, called Cantor's diagonalization argument, which shows that the set of irrational numbers (non-terminating, non-repeating decimals) is uncountably infinite (while rational numbers are countably infinite).
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 Neo7 send PM view profile add to friends User #: 475 Posts: 7776 Joined: Jan 17, 2004 Last visit: Dec 05, 2013
 Posted: Oct 10, 2011 8:47:02 PM - Report post (0) (0) post reply
 And there goes the entire topic. Was kind of hoping it would last longer.
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 benduhova send PM view profile add to friends User #: 961777 Posts: 2744 Joined: Apr 26, 2010 Last visit: Nov 27, 2013
 Posted: Oct 11, 2011 9:18:55 AM - Report post (0) (0) post reply
 if you follow the view of constructive mathematics then a collection of numbers is subcountable if there is a partial surjection from the natural numbers onto it(or that a collection is no bigger than the counting numbers) so the answer could be c depending on your stance
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 Neo7 send PM view profile add to friends User #: 475 Posts: 7776 Joined: Jan 17, 2004 Last visit: Dec 05, 2013
 Posted: Oct 11, 2011 11:11:59 AM - Report post (0) (0) post reply

 quote: originally posted by benduhova:if you follow the view of constructive mathematics then a collection of numbers is subcountable if there is a partial surjection from the natural numbers onto it(or that a collection is no bigger than the counting numbers) so the answer could be c depending on your stance

I think in this case, you need the bijection function (not the surjection function). You can represent an infinite amount of irrational numbers between 0 and 1 by using infinite strings of just 1s and 0s while you can represent the natural numbers as 1,2,3,~

Using Cantor's diagonalization, you will generate a particular number that will not be able to pair up with another rational number so long as they go out infinitely long (which makes creating a bijection impossible)

Day[9] probably explained it much better here: Link

[Edited by Neo7, 10/11/2011 11:14:13 AM]

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 benduhova send PM view profile add to friends User #: 961777 Posts: 2744 Joined: Apr 26, 2010 Last visit: Nov 27, 2013
 Posted: Oct 11, 2011 11:31:30 AM - Report post (0) (0) post reply
 The interpretation of Cantor's result will depend upon your view of mathematics. cantors argument shows no more than that there is no bijection between the natural numbers and T( be a set consisting of all infinite sequences of 0s and 1s). It does not rule out the possibility that the latter are subcountableCantor's argument contradicts epsilon-delta proof.According to the epsilon-delta proof, width/length of the list converges to zero.When we increase digits of the list, width/length monotonically decreases.So, we can easily accept the result.However, Cantor's argument is as follows.When the number of digits reaches the actual infinity, suddenly width/length becomes 1.
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