Algebraic Proof (OCR GCSE Maths)

What is algebraic proof?

• Algebraic Proof is the process of showing something is true in every case, using algebra
• Typical algebra skills include expand brackets and collect like terms
  • At the harder end, knowing the "difference of two squares" factorisation is useful

How do I prove results about odd and even numbers?

• Give things letters (use as few letters as possible):

• • n is “any integer” (or m or k or…)
    • Integer means whole number
  • n + 1 is the consecutive integer after n (the one immediately after n)
  • 2n is an even integer (2n + 2 is the next one)
  • 2m is a different even integer (not necessarily consecutive, but any other even integer)
  • 2n + 1 is an odd integer (and 2n + 3 is the next one, or 2n - 1 is the one before, etc)

• A "multiple of k” means it can be written as k(……), ie. k × …
• To prove something is even, show that the algebraic result can be written as 2 × (...)
  • Make sure whatever is inside the brackets is an integer
• To prove something is odd, show that the algebraic result can be written as 2 × (...) + 1
  • Make sure whatever is inside the brackets is an integer
• When dealing with prime numbers, remember that primes only have factors of 1 and themselves
  • If p is prime then 1 × p or p × 1 are the only ways to write it as a product of two integers

What is the difference between an equation and an identity?

• An equation is true for certain values only
  • For example, 3x − 1 = 5 is an equation and is only true when x = 2
  • Or another example, x² = 9 is an equation and is true only when x = 3 or when x = −3
• An identity is true for all values
  • For example, 2(3x) ≡ 6x is an identity because it is true for all values of x
    • Note that the symbol for an identity is 3 horizontal lines (like an equals sign but with an extra line)