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**witch1****Member**- Registered: 2006-05-27
- Posts: 1

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

2 and -3

**X'(y-Xβ)=0**

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,608

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

1 and 0

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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

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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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

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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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

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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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,608

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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**John E. Franklin****Member**- Registered: 2005-08-29
- Posts: 3,588

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

**igloo** **myrtilles** **fourmis**

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

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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