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#1 2007-08-22 11:50:48
- MathsIsFun
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- Registered: 2005-01-21
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Rational and Irrational Numbers
I have just updated these two pages:
Rational Numbers
Irrational Numbers
Do they stand up under your watchful gazes?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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#2 2007-08-22 11:53:51
- mikau
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- Registered: 2005-08-22
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Re: Rational and Irrational Numbers
the rational one was really good!
Pythagoras drowned Hippasus for his cleverness?!! WAHAHA!
the irrational one is also cool.
It might be worth it to make a page explaining the method by which you can convert any repeating decimal value to a ratio of two ints. (oh no! 0.99.. = 1 strikes again!) and post a link to it in the rational numbers page.
Just a thought. Anyway, really cool!
Last edited by mikau (2007-08-22 11:58:40)
A logarithm is just a misspelled algorithm.
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#3 2007-08-22 15:00:03
- John E. Franklin
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- Registered: 2005-08-29
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Re: Rational and Irrational Numbers
Looks neat.
I didn't know the square root of 99 was irrational.
How did you know that?
And I love your calculator; 32 digits is a lot! Where do I get one?
I also wrote a computer program to do square roots and cosines to a thousand digits as you know from that old post with the cosine answers. I didn't check the answers with other software because I don't have maple or mathematica.
Is the square root of every whole number between 65 and 80 irrational??
igloo myrtilles fourmis
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#4 2007-08-23 03:21:42
- Ricky
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- Registered: 2005-12-04
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Re: Rational and Irrational Numbers
I didn't know the square root of 99 was irrational.
How did you know that?
The square root of any non-perfect square is irrational.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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#5 2007-09-15 01:10:29
- landof+
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- Registered: 2007-03-24
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Re: Rational and Irrational Numbers
hey, why don't you include why he was drowned? because he was irrational!
I shall be on leave until I say so...
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#6 2007-09-15 02:16:07
- Identity
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- Registered: 2007-04-18
- Posts: 934
Re: Rational and Irrational Numbers
hey, why don't you include why he was drowned? because he was irrational!
that's just a point of view
i personally dislike the pythagoreans because they were like the early church in that they suppressed knowledge from the masses.
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#7 2008-01-19 17:35:19
- JohnnyReinB
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- Registered: 2007-10-08
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Re: Rational and Irrational Numbers
Just one observation on the list of famous irrationals, there are two pi
"There is not a difference between an in-law and an outlaw, except maybe that an outlaw is wanted" 
Nisi Quam Primum, Nequequam
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#8 2008-01-19 18:02:50
- Jai Ganesh
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Re: Rational and Irrational Numbers
But for the 2 pi's as JohnnyReinB pointed out, the pages are lovely.
A separate page for transcendental numbers would be much useful.
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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#9 2008-01-19 18:08:17
- JohnnyReinB
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- Registered: 2007-10-08
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Re: Rational and Irrational Numbers
yes, they were great!
"There is not a difference between an in-law and an outlaw, except maybe that an outlaw is wanted"
Nisi Quam Primum, Nequequam
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#10 2008-02-27 05:48:53
- LuisRodg
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- Registered: 2007-10-23
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Re: Rational and Irrational Numbers
Does anyone know what the student's proof was as to why sqrt(2) could not be represented as fraction?
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#11 2008-02-27 05:56:16
- mathsyperson
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Re: Rational and Irrational Numbers
I'm guessing it was something like this, since it's the only one I know:
Assume that √2 is rational. Then there is a fraction p/q in its lowest form (so, p and q are coprime) such that (p/q)² = 2.
∴ p²/q² = 2
∴ p² = 2q².
Because p² is equal to 2 multiplied by an integer, p² must be even. But that means that p must be even, so we can write it as 2k, for some integer k.
We now have (2k)² = 2q².
∴ 4k² = 2q²
2k² = q².
By similar reasoning to above, q must be even and therefore can be written as 2m, for some integer m.
Then we have p/q = 2k/2m. This contradicts with p/q being in its lowest form, and so √2 cannot be rational.
Why did the vector cross the road?
It wanted to be normal.
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#12 2008-02-27 05:57:23
- LuisRodg
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- Registered: 2007-10-23
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Re: Rational and Irrational Numbers
It says it was a geometric proof?
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#13 2008-02-27 05:59:00
- mathsyperson
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- Registered: 2005-06-22
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Re: Rational and Irrational Numbers
Ah, so it does. In that case I have no idea. 
Why did the vector cross the road?
It wanted to be normal.
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#14 2008-02-27 06:02:49
- LuisRodg
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- Registered: 2007-10-23
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Re: Rational and Irrational Numbers
Ya. I seen your proof since its the most common one, and as you said, the only one I seen which is why I was interested in seeing the geometric one.
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#15 2008-05-08 22:54:10
- JohnnyReinB
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Re: Rational and Irrational Numbers
Geometric proof
Another reductio ad absurdum showing that √2 is irrational is less well-known. It is also an example of proof by infinite descent. It makes use of classic compass and straightedge construction, proving the theorem by a method similar to that employed by ancient Greek geometers.
Let ABC be a right isosceles triangle with hypotenuse length m and legs n. By the Pythagorean theorem, m/n = √2. Suppose m and n are integers. Let m:n be a ratio given in its lowest terms.
Draw the arcs BD and CE with centre A. Join DE. It follows that AB = AD, AC = AE and the ∠BAC and ∠DAE coincide. Therefore the triangles ABC and ADE are congruent by SAS.
Since ∠EBF is a right angle and ∠BEF is half a right angle, BEF is also a right isosceles triangle. Hence BE = m − n implies BF = m − n. By symmetry, DF = m − n, and FDC is also a right isosceles triangle. It also follows that FC = n − (m − n) = 2n − m.
Hence we have an even smaller right isosceles triangle, with hypotenuse length 2n − m and legs m − n. These values are integers even smaller than m and n and in the same ratio, contradicting the hypothesis that m:n is in lowest terms. Therefore m and n cannot be both integers, hence √2 is irrational.
Last edited by JohnnyReinB (2008-05-08 22:55:53)
"There is not a difference between an in-law and an outlaw, except maybe that an outlaw is wanted"
Nisi Quam Primum, Nequequam
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