Math Is Fun Forum

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Irrational roots

(a) Give an example of two irrational numbers which, when added, produce a rational number. (sqrt(2) and -sqrt(2) )

Now let's consider just the addition of radicals.

(b) Suppose that a and b are positive integers such that both

are irrational. For what values of a and b is

rational? Prove your answer.

(c) Again assuming a and b positive integers such that both

are irrational, for what values of a and b is

rational? Prove your answer.

Thank you. I'm stumped!

Last edited by thedarktiger (2014-11-28 17:49:34)

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anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Irrational roots

b) When a=b.

c) Always.

You should try to prove this by assuming it's incorrect, i.e. that a and b are different and

is rational.

Last edited by anonimnystefy (2014-11-29 10:42:27)

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“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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Irrational roots

hi Stefy,

Do you have a proof for (b) that is the only solution?

Your answer for (c) ???

Now if, say, a = b = 2, then you've just said 2root2 is rational, I think.

Bob

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Irrational roots

Meant it the other way round. It's always irrational.

Proof for b):

Last edited by anonimnystefy (2014-11-30 12:00:04)

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“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.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Irrational roots

Thanks.  That's very neat. 

Bob

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Irrational roots

Yeah, it's the classic proof of irrationality.

Of course, that proof needs a bit more work to be an actual proof, such as explaining what exactly was done there and the discussion of the a=b case.

The second one is a bit more straightforward.

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“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.

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: Irrational roots

Thanks!

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Good. You can read.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Irrational roots

No problemo. Did you do the other one?

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“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.

thedarktiger

Member

Registered: 2014-01-10

Posts: 91

Re: Irrational roots

yep thanks but please do not post solutions just help

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Good. You can read.