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krassi_holmz
2006-04-07 18:23:34

agree.

George,Y
2006-04-07 11:05:29

But d ∈ N => exists k ∈ N : k^c=d.

for more illustrative

<=>
⇒ a/b=q∈N ⇒ d=qc

krassi_holmz
2006-04-06 21:40:12

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.

krassi_holmz
2006-04-06 21:33:18

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.

George,Y
2006-04-06 21:17:25

you can simplify

as 2+√5

George,Y
2006-04-06 21:11:17

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.

krassi_holmz
2006-04-06 20:30:03

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

krassi_holmz
2006-04-06 20:28:21

...≈3.80423

ganesh
2006-04-06 18:53:20

a²=10+2√5

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

MathsIsFun
2006-04-06 17:18:53

Software?

George,Y
2006-04-06 11:55:13

it has no solution, my software told me

George,Y
2006-04-05 16:11:33

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