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Old 01-24-2009, 01:03 PM #1
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Default [ch8730]2 is irrational

My calc 3 professor showed us this the other day and I thought I would try to explain it to you. When a number is irrational it means that it cannot be represented as a fraction of two integers nor can it be represented as a terminating or repeating decimal number. Pi is irrational. It is a decimal number that will go on forever and is non repeating. So to prove that [ch8730]2 is irrational you can justify an incorrect condition and then contradict it.

So say [ch8730]2 is rational.

Then [ch8730]2=p/q * The fraction p/q is irreducible. (This is important you will see why at the end).

p=[ch8730]2q

p[sup]2[/sup]=2q[sup]2[/sup] * 2q[sup]2[/sup] must be an even number. Therefor p[sup]2[/sup] must be even and p must also be even. Squaring an odd number yields a odd number, it's the same with even numbers too.

So since p is even we can say p=2k * k must be an integer.

By substituting we have (2k)[sup]2[/sup]=2q[sup]2[/sup]

4k[sup]2[/sup]=2q[sup]2[/sup]

2k[sup]2[/sup]=q[sup]2[/sup]
* By the logic explained above we can conclude that q must also be even.

so [ch8730]2=p/q but if p and q are even then the fraction p/q is reducible by at least 2.

By contradicting the initial condition that p and q are not coprime

[ch8730]2 must not be rational because the math showing it could be represented as a irreducible fraction contradicted itself.

Isn't math fun! ;D

-Tony



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Old 01-24-2009, 01:56 PM #2
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Default Re: [ch8730]2 is irrational

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Old 01-24-2009, 04:33 PM #3
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Default Re: [ch8730]2 is irrational

Quote:
so [ch8730]2=p/q but if p and q are even then the fraction p/q is reducible by at least 2.
That is because if p and q are even, then p/q = 1, am I right?

I've seen this explanation lots of times but I had never dedicated time to really read it until now.
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Old 01-24-2009, 05:53 PM #4
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Default Re: [ch8730]2 is irrational

This is called reductio ad absurdum if I remember correctly. It's a clever way to prove things.

What's with all the math threads lately?
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Old 01-25-2009, 02:20 AM #5
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Default Re: [ch8730]2 is irrational

I think there should be an official math thread. But I'm not going to make it, because my threads always die in a day or two.

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Old 01-25-2009, 02:22 AM #6
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Default Re: [ch8730]2 is irrational

NO, WE NEED A MATH SECTION! as well as a sections for flaming
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Old 01-25-2009, 02:40 AM #7
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Default Re: [ch8730]2 is irrational

um.... :-? u lost me after the 1st number... lol
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Old 01-25-2009, 02:42 AM #8
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Default Re: [ch8730]2 is irrational

Quote:
Originally Posted by [ch12290
]NO, WE NEED A MATH SECTION! as well as a sections for flaming

No flame section, yes to the math thread. Will [s]spam[/s] create it right away!
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Old 01-25-2009, 07:02 AM #9
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Default Re: [ch8730]2 is irrational

Quote:
Originally Posted by Niko
That is because if p and q are even, then p/q = 1, am I right?
Well no. 2368378/789217892 is not 1 and those numbers are even. :P

A fraction of an even number over an even number is reducible. 2/4 is 1/2. A fraction that is irreducible must have an odd numerator or denominator.

So setting by [ch8730]2=p/q and saying that p/q is irreducible then through mathematical operations we prove that p and q are even p/q is reducible. Since the logic contradicts itself we can conclude that [ch8730]2 cannot be set equal to a ratio of two integers.

I love math because it is derived from logic and reasoning. You can write a paper or story that some may not like. I hate English and history! Who the hell cares what happened yesterday! This country does not learn from its mistakes. So don't pull that card. But math is either right or wrong (period!!!).

Does anyone get the proof? I cant be the only math person on this forum.

-Tony
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Old 01-25-2009, 04:13 PM #10
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Default Re: [ch8730]2 is irrational

I got lost at
Quote:
2k2=q2 By the logic explained above we can conclude that q must also be even.

so [ch8730]2=p/q but if p and q are even then the fraction p/q is reducible by at least 2.
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Old 01-25-2009, 05:31 PM #11
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Default Re: [ch8730]2 is irrational

Quote:
Originally Posted by Xplorer877
[quote author=Niko link=1232805818/0#2 date=1232818418]That is because if p and q are even, then p/q = 1, am I right?
Well no. 2368378/789217892 is not 1 and those numbers are even. :P

A fraction of an even number over an even number is reducible. 2/4 is 1/2. A fraction that is irreducible must have an odd numerator or denominator.

So setting by [ch8730]2=p/q and saying that p/q is irreducible then through mathematical operations we prove that p and q are even p/q is reducible. Since the logic contradicts itself we can conclude that [ch8730]2 cannot be set equal to a ratio of two integers.

I love math because it is derived from logic and reasoning. You can write a paper or story that some may not like. I hate English and history! Who the hell cares what happened yesterday! This country does not learn from its mistakes. So don't pull that card. But math is either right or wrong (period!!!).

Does anyone get the proof? I cant be the only math person on this forum.

-Tony[/quote]


OH MY CHUCK

I've just realised what I wrote, LOL.


I completely get it now, thanks! :P
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