Question

Anyone available to help with a proof?

Answer 1

hoblos?

Answer 2

prove what ?

Answer 3

b) Use a) and the quotient remainder theorem and case reasoning to prove that the square root of 7 is irrational.

Answer 4

what is a

Answer 5

one moment

Answer 6

A previous proof we had to do here it is:

Answer 7

http://minus.com/mbdyHP1cpI#

Answer 8

You here hoblos?

Answer 9

If √7 is rational, then it can be expressed by some number a/b (in lowest terms). This would mean: (a/b)² = 7. Squaring, a² / b² = 7. Multiplying by b², a² = 7b². If a and b are in lowest terms (as supposed), their squares would each have an even number of prime factors. 7b² has one more prime factor than b², meaning it would have an odd number of prime factors. Every composite has a unique prime factorization and can't have both an even and odd number of prime factors. This contradiction forces the supposition wrong, so √7 cannot be rational. It is therefore irrational.

Answer 10

@joe, yea it isnt being proved by contradiction here though

Answer 11

@Joe, I actually previous;u proved it by contradiction: http://minus.com/mbdI3DAuzr#

Answer 12

Use the quotient remainder theorem and case reasoning to prove that the square root of 7 is irrational.

Answer 13

Thats what I gotta do this time

Answer 14

bonus question:
where in each proof is the method of infinite descent hidden?

Answer 15

Can any of you see how it would be done using the quotient remainder theorem and case reasoning?

Answer 16

what is the quotient remainder theorem? does this involve polynomials? Because there is a short proof using the fact that sqrt 7 is a solution to the equation:\[x^2-7=0\]and by the rational roots theorem, the only possible rational roots are \[\pm 1, \pm 7\]So because none of those are roots, sqrt 7 must be irrational.

Answer 17

http://minus.com/mJX27ETKd#

Answer 18

oh the division algorithm. that leads to the same proof you posted above though.

Answer 19

How so?

Answer 20

every time you say something like \[7\mid a^2\Longrightarrow 7\mid a\]you are using the division algorithm, with r = 0. I dont see how you could create a new direct proof using the division algorithm.

Answer 21

So is the proof I posted that I did previously the only way to do it?

Answer 22

What about the case reasoning