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witch1

Member

Registered: 2006-05-27

Posts: 1

reciprocals

if x is greater than y then the reciprocal of x is less than y, find a pair of values which disprove this.   After wrecking my brain for 2 days I am still no nearer finding an answer, Help!

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: reciprocals

2 and -3

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X'(y-Xβ)=0

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: reciprocals

Now that George has shown the way, can I have a go?

1 and 0

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: reciprocals

MIF is playing with our heads because he knows the reciprocal of 0 is irrational, and not irrational like a never ending decimal of no repeating, but just the normal sense of the word.  By the way, is this number irrational?
5.1121231234123451234561234567123456781234567891243456789012345678901123456789012123456789012312345678901234123456789012345 etc...  I think it is but I haven't checked the definition lately.

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igloo myrtilles fourmis

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: reciprocals

1/0 isn't irrational.  It isn't even real or complex.  It's undefined.  Thus, any talk about being greater or less than anything is nonsense.

And yes John, I'm fairly certain that number is irrational.

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"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..."

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: reciprocals

Ricky, fix your 0/1 to be 1/0
Secondly, if I want to call something irrational in the English sense of the word, then I will.

Last edited by John E. Franklin (2006-05-28 04:23:15)

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igloo myrtilles fourmis

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: reciprocals

Thanks John.  An I see what you mean by irrational now.  Sorry for the confusion.

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"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..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: reciprocals

As for trying to find a counter example, one way to do it is to try to prove it, and see where you run into trouble.

Let x be greater than y:

x > y

So if we divide both sides by xy:

x/xy > y/xy => 1/y > 1/x

But that statement is only true if xy is positive.  Thus, either x is negative or y is negative, but not both, fits as a counter example.

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"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..."

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: reciprocals

That's neat.  I never learned about algebra with a greater-than sign.
What do you call it?  Inequality algebra?  What do I search on to learn more about rules and techniques?

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igloo myrtilles fourmis

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: reciprocals

I love how this conversation has gone - but I didn't intend to be controversial, or even irrational, complex or transcendental!

Witch1's question was "if x is greater than y then the reciprocal of x is less than y"

Which leads one to think about 3>2 therefore 1/3<2, but I gave the counter-example of 1>0 leads to 1/1>0 (then, having typed, my fingers moved on ...)

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: reciprocals

oh!, I thought you had to take the reciprocal of both of the numbers.  Read too fast.

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igloo myrtilles fourmis

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: reciprocals

Oy!  I might need to get a new desk now that I smashed my head against it so much...

At least I wasn't alone

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"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..."