Edexcel A-Level Further Mathematics Core Pure with Q & A
Edexcel A-Level Further Mathematics Core Pure with Q & A Conjecture ANSWER Statement that has not yet been proven What is mathematical proof? ANSWER Proceeding from given assumptions through a series of logical steps to a conclusion Proof sqrt(2) is irrational using proof by contradiction ANSWER - assume sqrt(2) is rational so it can be written as a/b where a and b are integers and a/b is in it's simplest form - sqrt(2)=a/b ⇒ a²=2b² ⇒ a² is even ⇒ a is even - if a is even it can be written as 2n where n is an integer so subbing this in (2n)²=2b² ⇒ b²=2n² ⇒ b² is even ⇒ b is even - a and b can't both be even since that means the fraction a/b wasn't in it's simplest form meaning contradiction so sqrt(2) isn't rational Prove there are infinite primes by contradiction ANSWER - suppose there are finite number of primes 2, 3, 5, ..., p - let q = 2x3x5x...xp + 1 - if q is prime then it is a new prime number not in list - if q isn't prime then it has a prime factor but 2 can't be a factor because it's one more than a multiple of 2 3 can't be a factor because it's one more than a multiple of 3 similarly, none of the primes in the list can be a factor of q so q must have a prime factor that isn't in list
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