Proof

Question 1

what is a rational number

Answer 1

• anything that can be expressed as a fraction

Question 2

explain how to prove the irrationality of √2

Answer 2

• assume √2 is rational and is equal to a/b
• a/b is in it’s simplest form
• when solved, it implies that a² is so a is even
• then, let a = 2c
• √2 = 2c/b (when solved) b² = 2c² so b is even
• a and b are not in their simplest form so √2 isn’t rational – is irrational

Question 3

what is the name for the proof of √2

Answer 3

• proof by contradiction

Question 4

explain the proof for the infinity of primes

Answer 4

• assume primes are finite and we have a complete list P1, P2, P3…Pn
• consider the number given by: P1 x P2 x P3 x…Pn + 1 –if this number is prime, then we don’t have all the primes, there’s a contradiction
• the answer will either be prime or be divisible by primes that aren’t in our original list
• there’s a contradiction w/ the initial assumption, so primes are infinite

Question 5

example of the proof for the infinity of primes (contradicting)

Answer 5

2 x 3 x 7 x 43 + 1 = 1807

- 1807 is divisible by 13 and 139 which are two primes