Suppose q is a rational number such that...?

Oi Kei

2016-09-16 7:24 am

Suppose q is a rational number such that 1 < q ≤ 3-sqrt(2). Write q=m/n, where m,n ∈ N and gcd(m,n)=1.
Prove that sqrt(2) ia a positive distance from q. Specifically, show that |sqrt(2)-m/n| ≥ 1/(3n^2) (**)

Hint: First prove that if gcd(m,n)=1 then 2n^2 and m^2 are distinct integers, then "rationalize the numerator" in the LHS of (**)

how to prove if gcd(m,n)=1 then 2n^2 and m^2 are distinct integers? I have to assume sqrt(2) is a rational number?
how to "rationalize the numerator" |sqrt(2)-m/n|??

Thanks

回答 (1)

Lopez

2016-09-16 10:09 pm

✔ 最佳答案

claim :
If gcd(m,n) = 1 , where m,n ∈ N
Then 2n² and m² are distinct integers
pf :
Suppose m² = 2n²
Then 2︱m
So we may assume that m = 2k , for some integer k.
m² = 2n² implies 4k² = 2n²
n² = 2k²
So 2︱n
2︱m and 2︱n implies gcd(m,n) = 2 , a contradiction.
Hence, m² ≠ 2n²
#

Since 2n² and m² are distinct integers,
︱2n² - m²︱≧ 1 ..... (1)

m/n = q ≦ 3 - √2
√2 + m/n ≦ 3
1/( √2 + m/n ) ≧ 1/3 ..... (2)

︱√2 - m/n︱
=︱n√2 - m︱/ n
=︱( n√2 - m )( n√2 + m )︱/ [ n( n√2 + m ) ] , this step called "rationalize the numerator"
=︱2n² - m²︱/ [ n( n√2 + m ) ]
≧ 1 / [ n ( n√2 + m ) ] , by (1)
= 1 / [ n² ( √2 + m/n ) ]
= ( 1 / n² )[ 1 / ( √2 + m/n ) ]
≧ ( 1 / n² )( 1 / 3 ) , by (2)
= 1 / ( 3n² )

Q.E.D.

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