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Can anyone solve it explicitly out?
find positive irrantional a, who satisfies a²=10+2√5
X'(y-Xβ)=0
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it has no solution, my software told me
X'(y-Xβ)=0
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Software?
"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman
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a²=10+2√5
It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.
Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.
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...≈3.80423
IPBLE: Increasing Performance By Lowering Expectations.
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But there's another thing:
You have to prove that a is irrational
IPBLE: Increasing Performance By Lowering Expectations.
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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)
X'(y-Xβ)=0
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you can simplify
as 2+√5X'(y-Xβ)=0
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If a = p/q, then:
IPBLE: Increasing Performance By Lowering Expectations.
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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
Last edited by krassi_holmz (2006-04-06 00:31:35)
IPBLE: Increasing Performance By Lowering Expectations.
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for more illustrative
<=> ⇒ a/b=q∈N ⇒ d=q[sup]c[/sup]Last edited by George,Y (2006-04-06 13:09:51)
X'(y-Xβ)=0
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agree.
IPBLE: Increasing Performance By Lowering Expectations.
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