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

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

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 15,954

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

The knowledge of some things as a function of age is a delta function.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,376

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

Children are not defined by school ...........The Fonz

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

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 15,954

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

The knowledge of some things as a function of age is a delta function.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,376

why?

Bob

Children are not defined by school ...........The Fonz

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

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 15,954

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

The knowledge of some things as a function of age is a delta function.

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,376

Ok. That'll do me. Thanks.

Bob

Children are not defined by school ...........The Fonz

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

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 15,954

Don't mention it!

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

The knowledge of some things as a function of age is a delta function.

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

Thanks a lot, that is great help.

Apologies about the confusion with posting in two forums.

Jack

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 15,954

You're welome!

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

The knowledge of some things as a function of age is a delta function.

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**mathaholic****Member**- From: Earth
- Registered: 2012-11-29
- Posts: 3,020

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

Mathaholic | 17th most active poster | 681st most recent person to join MIFF | Person | rrr's classmate

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