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#1 2012-12-17 05:43:43

jackme12
Member
Registered: 2012-12-17
Posts: 6

Prove √5 is irrational...

As the subject says, can anybody prove that route 5 is irrational?

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#2 2012-12-17 06:51:05

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Prove √5 is irrational...

Let's assume that the opposite is true, that is  that square root 5 is rational. This would mean it can be written in the form p/q, where gcd(p,q)=1.

So:
sqrt(5)=p/q
Then we square both sides:
5=p^2/q^2
5*q^2=p^2
So this means that p must be divisible by 5, and we can write p=5*r and our equation becomes:
5*q^2=25*r^2
q^2=5*r^2,
But this would mean that q is also divisible by 5, and we said that p and q have a gcd of 1! This is a contradiction, so our initial assumption that sqrt(5) is rational is false!


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#3 2012-12-17 07:03:50

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: Prove √5 is irrational...

hi Stefy,

He posted again in help and I replied there.

If p^2 is divisible by 5, how do you know that p is also divisible by 5 ?

(I know it is, but I want a justification please.  Otherwise every root is irrational.)

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#4 2012-12-17 07:05:17

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Prove √5 is irrational...

If p weren't divisible by 5 then neither would be p^2.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#5 2012-12-17 07:07:12

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: Prove √5 is irrational...

why?

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#6 2012-12-17 07:12:28

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Prove √5 is irrational...

This can be shown by assuming that p isn't divisible by 5 and then going through 4 cases p=5k+1,5k+2,5k+3,5k+4 and squaring p in each case to see that the square of p is respectively 1, 4, 4, 1 in each case.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#7 2012-12-17 07:22:00

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,426

Re: Prove √5 is irrational...

Ok.  That'll do me.  Thanks.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#8 2012-12-17 07:29:19

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Prove √5 is irrational...

Don't mention it!


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#9 2012-12-17 09:42:37

jackme12
Member
Registered: 2012-12-17
Posts: 6

Re: Prove √5 is irrational...

Thanks a lot, that is great help. 

Apologies about the confusion with posting in two forums.

Jack

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#10 2012-12-17 09:50:53

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,544

Re: Prove √5 is irrational...

You're welome! smile


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#11 2012-12-17 23:38:41

mathaholic
Member
From: Juliania
Registered: 2012-11-29
Posts: 2,845
Website

Re: Prove √5 is irrational...

Jackme12, welcome to the forum. Sorry if I was a bit late in greeting ya. Anyways, I cannot prove yet that route 5 is irrational 'cause I am just too young to learn irrational numbers like those.
Have fun in the forum, Jackme12!

Julian


"Double the fun, double the thrill, double the coolness" - Julianthewiki

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