Algebraic Proof

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)

Exam Tip

It is a good idea to write a sentence at the end of your algebraic proof to say word-for-word (copied from the question) what has been proved
- e.g. "this shows that all squares of odd numbers are themselves odd"

Worked example

Prove that the difference of the squares of two consecutive even numbers is divisible by 4.

Write down an algebraic expression for an even number

Write down the algebraic expression for the next consecutive even number after 2n

2n + 2

Write down an expression showing the difference of the squares of two consecutive even numbers
Do the larger value subtract the smaller value

${(2 n + 2)}^{2} - {(2 n)}^{2}$

Expand the brackets and collect like terms

$(2 n + 2) (2 n + 2) - 4 n^{2} = 4 n^{2} + 4 n + 4 n + 4 - 4 n^{2} = 8 n + 4$

Show that the final answer is divisible by 4 (a multiple of 4)
Do this by writing it as 4 × ... and write a conclusion that copies the wording in the question

$4 (2 n + 1)$

$4 \times (2 n + 1)$ is a multiple of 4, so the difference between the squares of two consecutive even numbers is divisible by 4