Proof (Edexcel GCSE Maths): Revision Notes
Proof
Introduction to proof questions
Mathematical proof questions might seem intimidating at first, but they're actually quite manageable once you understand the approach. Most proof questions simply require you to rearrange expressions to demonstrate that one thing equals another. The key is knowing how to represent different types of numbers algebraically and then manipulating these representations systematically.
Don't be put off by the word "proof" – these questions are really just about showing your working in a logical, step-by-step manner. Once you know the basic techniques, they become much more approachable.
Using algebra to show that two things are equal
Basic number representations
Before attempting any proof, you need to understand how different types of numbers can be expressed algebraically. These representations are the foundation of algebraic proof:
Essential Number Representations:
- Even numbers can be written as (where is any integer)
- Odd numbers can be written as
- Consecutive numbers can be written as , , , and so on
These representations work because can be any integer, allowing you to generate all numbers of each type. For example, if , then (even) and (odd).
Proving statements about odd numbers
Let's examine how to prove that the sum of any three odd numbers is always odd. This type of proof demonstrates the systematic algebraic approach.
Worked Example: Proving the sum of three odd numbers is odd
Step 1: Represent three arbitrary odd numbers
- First odd number:
- Second odd number:
- Third odd number:
Step 2: Add them together
Step 3: Simplify and factorise
Step 4: Recognise the form Since is an integer, we can call it , giving us , which is the form of an odd number.
Therefore, the sum of any three odd numbers is always odd.
Identity proofs
Some proofs involve showing that two expressions are identically equal, meaning they're equal for all possible values of the variables involved. The identity symbol () is used to show this relationship. For example, proving that involves expanding the left side and showing it simplifies to the right side for any value of .
When you see the identity symbol , remember that this means the expressions are equal for all possible values of the variable, not just specific values.
Disproving statements using counterexamples
When you're asked to prove that a statement isn't true, you only need to find one example where the statement fails. This is called disproof by counterexample, and it's often much quicker than attempting a full proof.
The counterexample method
The process is straightforward: keep testing examples until you find one that doesn't work. You don't need to check every possible case – just one counterexample is sufficient to disprove the entire statement.
Worked Example: Disproving consecutive square numbers claim
Ross claims: "The difference between any two consecutive square numbers is always a prime number."
Step 1: Test small consecutive squares
- (prime ✓)
- (prime ✓)
- (prime ✓)
Step 2: Continue testing
- (not prime ✗)
Step 3: Conclusion Since 9 is not prime (9 = 3 × 3), we have found a counterexample. Therefore, Ross's statement is false.
Efficient counterexample strategy
You don't need to work through examples systematically. If you can spot a likely counterexample immediately, you can go straight to testing it. The key is to look for cases where the statement might fail, often involving larger numbers or special cases.
Remember: To disprove a statement, you only need ONE counterexample. Don't waste time finding multiple examples – one is sufficient to prove the statement false.
Key proof techniques summary
Understanding these fundamental approaches will help you tackle most GCSE proof questions:
Key Proof Techniques:
- For proving general statements: Use algebraic representations and systematic manipulation
- For disproving statements: Find one counterexample that doesn't work
- For identity proofs: Show that expressions are equal for all values by algebraic manipulation
The most important thing to remember is that proof questions test your ability to think logically and manipulate expressions systematically, not your ability to memorise complex procedures.
Remember!
Essential Points to Remember:
- Even numbers are represented as , odd numbers as
- To prove something is true, use algebraic manipulation to show the statement holds generally
- To disprove something, find just one counterexample where the statement fails
- Identity proofs (using ) must work for all possible values of the variables
- Practice is essential – the more proof questions you attempt, the more confident you'll become with the techniques