Irrational numbers Watch

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manps
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Can someone please explain to me, if x=2 + √3 and y=2 - √3, why is (x - y) always irrational? I know that in this case x - y = 2√3 but why does it always happen? I have this proof, (x - y) = (x + y) -2y but dont know what it means :confused:

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beauford
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(Original post by manps)
Can someone please explain to me, if x=2 + √3 and y=2 - √3, why is (x - y) always irrational? I know that in this case x - y = 2√3 but why does it always happen? I have this proof, (x - y) = (x + y) -2y but dont know what it means :confused:

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Well x+y is rational and 2y is irrational so (x+y) - 2y is irrational. Simpler yet, root 3 is irrational, hence 2root 3 is (derive a simple contradiction if you assume it's rational).
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Fermat
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(Original post by manps)
Can someone please explain to me, if x=2 + √3 and y=2 - √3, why is (x - y) always irrational? I know that in this case x - y = 2√3 but why does it always happen? I have this proof, (x - y) = (x + y) -2y but dont know what it means :confused:

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(x-y) is always irrational because (x-y) is always 2√3.
√3 is irrational, so its multiple is irrational also.
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Bigcnee
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You can prove √3 is irrational simply. But simply stating that is is irrational is sufficient.
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hatton02
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You say "why is x-y always irrational?" ------- you are right, in this case it DOES equal 2root3. HOWEVER...it will not always equal two root 3....and sometimes x-y may be rational...depending on the values of x and y. If x is 8, and y is 5, then x-y=3 which is rational! However, if any irrational numbers are involved and then subtracted, and they don't cancel out (i.e. root 3 - root 3 = 0)then you will be left with an irrational number.

The "proof" you have suggested has been thought up because someone spotted they needed to get rid of that root 3 for it to become rational. To do this, if x=2+root3 and y=2-root 3, then adding them cancels the root 3s (described above). We now have x+y = 4. BUT we want x-y. To get x-y (from x+y) we now have to subtract 2y...so the answer is 4-2y, but y is 2-root 3...so we have 4-2(2-root3).

This is 4-4+2root3
Which is 2 root 3, as before.

However hard we try to get rid of those root 3's by adding/subtracting, we won't. You CAN get rid of them though... multiply x and y together:

xy = (2+root3)(2-root3)
xy = 4 - 2root3 + 2root3 - root9
xy = 4-3
xy=1

A rational number!

Infact i have two questions to mathematicians/you.

I have said xy = 4-root 9. Surely root 9 is not only 3, but also -3. Surely then, xy = 4-(-3) and so xy can also equal 7!?

One last question, how do you get those "root" signs up on your computer please!

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"Countdown" Kirk
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Fermat
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(Original post by hatton02)
I have said xy = 4-root 9. Surely root 9 is not only 3, but also -3. Surely then, xy = 4-(-3) and so xy can also equal 7!?
It all depends what you mean when you say root(9).
The (square) root of 9 is both '+3' and '-3', since both (-3)² = 9 and (+3)² = 9.
But when you write down root(9) or √9, then you are using a function, and the function you are using is defined to give the positive root only.
When/If you want to write down both roots of 9 then you would write +√9 (or simply √9) or -√9, with +√9 = 3, and -√9 = -3.
If you see an expression involving surds (roots) then you always take the positive value. That's why you use the ± sign in the quadratric formula, x = -b ± √(b² - 4ac) (all over) 2a, to show that both the positive and negative roots of the discriminant (b² - 4ac) are taken.

(Original post by hatton02)
One last question, how do you get those "root" signs up on your computer please!

Cheers
"Countdown" Kirk
Some of the maths symbols are gotten by using the Alt key and the numeric keypad. Others are HTML constructs.
± is Alt+241
√ is &#8730 ; <- but no space between the last digit and the ';'.
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Fermat
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Special Characters
Using the Alt key and the Numeric Keypad

x² : 253
x³ : 252
± : 241
÷ : 246
3° : 248
¤ : 207
ø : 155
Ø : 157
½ : 171
¼ : 172
« : 174
» : 175
µ : 230

Using HTML constructs
These constructs take the form of &#dddd; where dddd is a 3- or 4-digit number given as follows.

Δ : 916
≡ : 8801
η : 951
¾ : 190
∫ : 8747
λ : 955
≠ : 8800
ω : 969
∂ : 8706
π : 960
√ : 8730
ρ : 961
∑ : 8721
σ : 963
ε : 949

You can go to this web-site for more HTML constructs for other symbols.
http://www.chami.com/tips/internet/050798I.html
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hatton02
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test test:

≠ and ∑ oh and λ for fun

Why don't these work on MSN messenger?

Cheers "Fermat"!
Kirk
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