A6: know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments and proofs

What is the difference between an equation and an identity?

An identity is an equation which is always true, no matter what values are substituted. An equation may not be always true (unless it is an identity)

What is a mathematical proof?

A deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.

What is a proof for an even number?

2n (n stands for ‘number’ as any number, even or odd will be even once doubled)

What is the proof for an odd number?

2n+1 (n standing for number - if 2n is an even number, adding 1 will always make it odd)

2n+3 would also work, or 2n+5 and so on

What is a proof for several consecutive numbers?

N, n+1, n+2

N, n-1, n-2

Prove that the product of two odd numbers is always odd.

Product is the value obtained by multiplying - try examples (3×5=15, 7×9=63)

For these examples, oddxodd=odd. To prove that it is always true, get the final answer to a similar structure as the mathematical proof for all odd numbers (2n+1)

Write two odd numbers as 2n+1 and 2m+1 (n and m are integers)

Multiply these together: (2n+1)(2m+1)=4nm+2n+2m+1

The first three terms have a common factor of 2, so the expression can be written as 2(2nm+n+m)+1

This is a form of 2n+1 as n in this case stands for 2nm+n+m

Prove that the difference between two consecutive square numbers is always an odd number

Choose two consecutive numbers: n, n+1

Find the difference between the square of these numbers: (n+1)²=n²+2n+1, n²+2n+1-n²

Simplify: 2n+1

This is the proof for all off numbers

Disproof by counter example

Providing one example which doesn’t work for the statement always proves that it is wrong

If you square a number and add 1, the result is a prime number. Find a counterexample to prove the statement wrong.

Try several examples until one is incorrect:

1²=1, 1+1=2 (prime)

2²=4, 4+1=5 (prime)

3³=9, 9+1=10 (not prime)

The statement is proved to be false

Algebraic equivalent

They present you with a two sided equation, usually with a triple bar (≡) rather than an equals sign (=). Make both sides the same.

Algebraic proof

Using algebra to simplify an equation or expression

Prove that (n+2)²-(n-2)² is divisible for 8 for any positive whole number n.

We are looking for the result to fit into 8(…)

Expand and simplify the expression: (n+2)²-(n-2)² = (n²+4n+4)-(n²-4n+4) or n²+4n+4-n²+4n-4=8n as the n² and ±4 cancel out.

8n is written as 8(n), proving that it must be divisible by 8.

Proofs by indices: prove that the difference between 24^12 and 15^10 is a multiple of 3.

Write the expression to find the difference: 24^12-15^10 (the formula we will put the result in is 3(…) as it shows that (…) is divisible by three

Take out a factor of three from each term: (24×24^11)-(15×15^9)=(3×8×24^11)-(3×5×15^9)

Factorise by three:3((8×24^11)-(5×15^9))

As this is in the form 3(…) it is a multiple of 3.

Proofs with primes: show that 5^89-401 is not a prime number

Oddxodd=odd, so 5^89 is odd. We do not need to write out this number. By odd-odd=even, so 5^89-401 must be even.

Even numbers are not prime (unless they are 2)