A LEVEL MATHS: Proof

A LEVEL MATHS: Proof What does the following notation mean? {x:_______} The set of values for x such that ... What does the symbol → mean? x __ implies ____ or if...then.... What does the symbol ← mean? x____is implies by ____ What does the symbol ↔ mean? p implies q and q implies p or p ____ if and only if (iff) Even number notation 2n odd number notation 2n+1 consecutive even numbers notation 2n, 2n + 2 consecutive odd numbers notation 2n+1, 2n+3 rational number notation a= p/q What is the proof behind the irrationality of surds? e.g. prove that √3 is a surd Let us assume to the contrary that √3 is a rational number. It can be expressed in the form of p/q where p and q are co-primes and q≠ 0. ⇒ √3 = p/q ⇒ 3 = p²/q² (Squaring on both the sides) ⇒ 3q² = p²......................................(1) It means that 3 divides p² and also 3 divides p because each factor should appear two times for the square to exist. So we have p = 3r where r is some integer. ⇒ p² = 9r²......................................(2) from equation (1) and (2) ⇒ 3q² = 9r² ⇒ q² = 3r² Where q² is multiply of 3 and also q is multiple of 3. Then p, q have a common factor of 3. This runs contrary to their being co-primes. p/q is not a rational number. This demonstrates that √3 is an irrational number. How do you prove that there are infinitely many even numbers? Proof by contradicition: First you make the assumption that there are finite number of even numbers. The biggest one has a value of N, where N = 2n and n is an integer however if you do N+2=2n+2 =2(n+1) which is larger than N I have contradicted my initial assumption How do you prove that there are infinitely many prime numbers? Proof by contradiction Assume that there are infinitely many prime numbers e.g. n p₁=2, p₂=3, p₄= 5 p₁, p₂, p₃ .....pₙ₋₁, pₙ multiply all of these prime numbers and call this number P (multiple of every prime number) (P+1) / p₁ = p₂p₃....pₙ₋₁pₙ with a remainder of 1 The same applies if you divide by p₂, p₃..... P+1 is not divisivle by any prime numbers therefore either it is also a prime number or it is the product of other prime numbers which we do not know about When you are carrying out proof by contradiction, what do you first do? You assume that the statement made is not true and opperate on this assumption to prove the statement m²-n² is factorised to what? (m+n)(m-n) Also it is important to remember that m+n is larger than m-n Define real numbers all rational and irrational numbers (positive or negative) Define natural numbers positive integers Use proof by contradiction to prove that there are no positive integer solutions to x²-y²=1 1) recognise that x²-y² can be factorise to (x-y)(x+y) 2) if this is the case then we know that the only two positive integers which multiply to make 1 are 1 and 1 3) make up simultaneous equations where x-y=1 and x+y=1 4) when you solve it you find that x=1 and y=0 which contradicts statement 'If m and n are consecutive even numbers then mn is divisible by 8' Either Prove or Disprove this statement This statement is true because with two consecutive even number, one will have two as a factor and the consecutive one must have four as a factor. Therefore the product of these two will form a number which has 8 as a factor Prove that sin²x + cos²x =1 using a triangle labelled: A (next to the θ) then in a clockwise direction, the next point is labelled C and the next point is labelled B How would you factorise 3²ⁿ-1? (3ⁿ+1)(3ⁿ-1) How would you fully factorise n³-n? n(n-1)(n+1)