hw8	Kishan Patel	10/10/21

1)
sqrt(2)=p/q
p and q cannot have the same root
2=p^2/q^2
p^2=2q^2
This makes p^2 even, but then q must be divisible by 4, so both p & q are even numbers, however they cannot have any common factors making the statement false. thus cannot be rational

2)
we have 2 right triangles, one within another. The big hypotneus is M+N, the leg is M. The smaller triangle has 2 legs, that are both equal as well as the other leg subtracted by the hypotneus of the small triangle are all the same length. The hypotenuse length is M-(N-M) making a whole number.
sqrt(2)=sqrt(2)(sqrt(2)-1)/(sqrt(2)-1)
=(2-sqrt(2))/(sqrt(2)-1)
=(2-n/m)/(n/m-1)
=(2m-n)/(n-m)

3)
a)
29/16=1+1/(1+1/(4+1/3))

b)
32/19=1+1/(1+1/(2+1/6))

4)
1+sqrt(2)=1+1+1/(2+1/(2+1/(2+...)))
[2;2]
This is an irrational number, as the continued fraction developed, repeats the 2+1/(2+1/(...)) thus creating an infinite looping of the fraction.
5)
(1+sqrt(3))/2=1+1/(2+1/(1+1/(2+1/(1+...))))
[1;2,1]
With the repeating number of 2 and 1 alternating forever, this number becomes an irrational number