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#1 2006-05-27 23:30:46

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!

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#2 2006-05-28 00:42:52

George,Y
Member
Registered: 2006-03-12
Posts: 1,306

Re: reciprocals

2 and -3


X'(y-Xβ)=0

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#3 2006-05-28 01:14:32

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,552

Re: reciprocals

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

1 and 0


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

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#4 2006-05-28 01:23:54

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,570

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.


igloo myrtilles fourmis

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#5 2006-05-28 02:42:55

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.


"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..."

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#6 2006-05-28 04:21:57

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,570

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)


igloo myrtilles fourmis

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#7 2006-05-28 04:22:47

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.


"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..."

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#8 2006-05-28 04:25:20

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.


"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..."

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#9 2006-05-28 05:58:10

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,570

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?


igloo myrtilles fourmis

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#10 2006-05-28 09:46:31

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,552

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 ...)


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

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#11 2006-05-28 12:16:03

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,570

Re: reciprocals

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


igloo myrtilles fourmis

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#12 2006-05-28 12:23:47

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 tongue


"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..."

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