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#1 2006-04-04 18:11:33

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

√ (10+2√5)=??

Can anyone solve it explicitly out?
find positive irrantional a, who satisfies  a²=10+2√5


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#2 2006-04-05 13:55:13

George,Y
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Registered: 2006-03-12
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Re: √ (10+2√5)=??

it has no solution, my software told me


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#3 2006-04-05 19:18:53

MathsIsFun
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Registered: 2005-01-21
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Re: √ (10+2√5)=??

Software?


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#4 2006-04-05 20:53:20

ganesh
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Registered: 2005-06-28
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Re: √ (10+2√5)=??

a²=10+2√5


There are two possible roots, a and -a. Therefore, a is the positive irrational root.


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#5 2006-04-05 22:28:21

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: √ (10+2√5)=??

...≈3.80423


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#6 2006-04-05 22:30:03

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: √ (10+2√5)=??

But there's another thing:
You have to prove that a is irrational


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#7 2006-04-05 23:11:17

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

Re: √ (10+2√5)=??

eazy, if a is a rational, it can be presented as N/M , where N and M are both integers.
N²/M²=a², thus a is a rational. but a is not.

to be strict 10+2√5 is rational <=>√5 is rational (proved by simple fraction algebra)

The last part is very difficult, it  usually lies on a Math Analysis book's page.
Proporsition: √5 cannot be expressed as N/M, where N and M are both integers.
Proof:
suppose √5 can be expressed as N/M, thus its simpified form would be p or p/q, where p and q are both integers. it cannot be p alone, since no integer p satisfy pp=25
p²/q²=5, p²= p p =5, thus p|5 , then p²|25 and q|5 is invalid(don't know the english words)
∴when p²/q² is an integer L, L|25 but L cannot.
Hence the assumption is false.

Last edited by George,Y (2006-04-05 23:14:54)


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#8 2006-04-05 23:17:25

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

Re: √ (10+2√5)=??

you can simplify

as 2+√5


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#9 2006-04-05 23:33:18

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: √ (10+2√5)=??

If a = p/q, then:

,
where p,q elem N.
But √5 is irrational, and the right side of the equation is rational, so there don't exist (p,q) elem N: p/q=a, so a is irrational.


IPBLE:  Increasing Performance By Lowering Expectations.

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#10 2006-04-05 23:40:12

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: √ (10+2√5)=??

Sorry for my late post, George.

For the other thing:
There' s more generalized formula:
Let Q be the set of all retional numbers:
Q={p/q|p,q ∈ N}, where N = {1,2,3...}
Let Ir is the set of all numbers of the kind x^(1/y):
Ir={x^(1/y)|x,y ∈ N}.
Then:
Q || Ir = N.
Proof:
Let a,b,c,d ∈ N and




But d ∈ N => exists k ∈ N : k^c=d.
But then
.
Particularry, this means that if an integer n is not a square, then √n is irrational.

Last edited by krassi_holmz (2006-04-06 00:31:35)


IPBLE:  Increasing Performance By Lowering Expectations.

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#11 2006-04-06 13:05:29

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

Re: √ (10+2√5)=??


But d ∈ N => exists k ∈ N : k^c=d.

for more illustrative

  <=>
⇒ a/b=q∈N ⇒ d=q[sup]c[/sup]

Last edited by George,Y (2006-04-06 13:09:51)


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#12 2006-04-06 20:23:34

krassi_holmz
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Registered: 2005-12-02
Posts: 1,908

Re: √ (10+2√5)=??

agree.


IPBLE:  Increasing Performance By Lowering Expectations.

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