IGCSE Algebraic Proof: Identities, Integers & Logic (H)
Algebraic Proof is often considered the pinnacle of the Edexcel IGCSE 4MA1 Higher Tier syllabus. It moves beyond simply finding an answer (solving) to demonstrating why a mathematical statement is universally true. These questions typically appear at the very end of the paper and are strong discriminators for Grade 9 students.
This guide covers Topic 2.8 of the IGCSE Algebra Hub, providing you with the "Dictionary of Definitions" and structured logic needed to construct rigorous proofs.
1. The Concept of Proof
In mathematics, you cannot prove a statement is true for all numbers by testing a few examples (e.g., "it works for 2 and 3, so it must work for everything"). This is called verification, not proof.
Algebraic Proof requires using a general variable (like n) to represent any integer, showing that the logic holds regardless of the specific number chosen.
2. The Dictionary of Proof
To construct a proof, you must first translate English words into Algebra. Memorize these definitions:
| English Concept | Algebraic Definition |
|---|---|
| Any Integer | n |
| Even Number | 2n (Always divisible by 2) |
| Odd Number | 2n + 1 (An even number + 1) |
| Consecutive Integers | n, n+1, n+2... |
| Consecutive Even Integers | 2n, 2n+2, 2n+4... |
| Consecutive Odd Integers | 2n+1, 2n+3, 2n+5... |
| Multiple of k | k(...) (Factorise k out) |
3. Proving Identities (LHS = RHS)
An identity (symbol: ≡) is true for all values of x. A common question asks you to "Prove that (x+3)² - (x-2)² ≡ 10x + 5".
Method:
- Start with the Left Hand Side (LHS).
- Expand brackets and simplify using algebraic manipulation.
- Show that your final simplified expression matches the Right Hand Side (RHS) exactly.
- Conclude: "LHS = RHS, QED."
Step-by-Step Worked Examples
Question: Prove algebraically that the sum of any three consecutive integers is always a multiple of 3.
Methodology: Define consecutive integers, sum them, and factorise.
Solution:
- Define: Let the three consecutive integers be n, n+1, and n+2.
- Sum: n + (n + 1) + (n + 2).
- Simplify: 3n + 3.
- Factorise (Crucial Step): 3(n + 1).
- Conclusion: Since the expression is 3 multiplied by an integer (n+1), it must be a multiple of 3.
Question: Prove that the difference between the squares of any two consecutive odd numbers is always a multiple of 8.
Methodology: Use the definition of odd numbers. Be careful with squaring brackets.
Solution:
- Define: Two consecutive odd numbers are 2n+1 and 2n+3.
- Squares Difference: (2n + 3)² - (2n + 1)².
- Expand Brackets:
(4n² + 12n + 9) - (4n² + 4n + 1). - Simplify:
4n² - 4n² + 12n - 4n + 9 - 1 = 8n + 8. - Factorise: 8(n + 1).
- Conclusion: 8(n+1) is clearly a multiple of 8 for any integer n.
Question: Prove that for all real values of x, the expression x² + 6x + 10 is always positive.
Methodology: A squared number is always ≥ 0. To prove positivity, use Completing the Square.
Solution:
- Complete the Square:
(x + 3)² - 9 + 10
(x + 3)² + 1. - Analyze Logic:
We know that (x + 3)² ≥ 0 for all real x (a square cannot be negative).
Therefore, (x + 3)² + 1 ≥ 1. - Conclusion: Since the minimum value is 1, the expression is always positive (greater than 0).
Real-World Application (Global Context)
Algebraic proof is the backbone of cryptography and computer security.
Scenario: RSA Encryption
Modern internet security (HTTPS) relies on RSA encryption, which uses properties of prime numbers. The security of the system depends on Fermat's Little Theorem, an algebraic proof concerning powers and remainders. Without the certainty provided by these proofs, we could not trust online banking or secure communications. Engineers don't just "hope" the math works; proofs guarantee it works for every possible key.
Exam Technique and Common Pitfalls
1. Substituting Numbers is Not Proof
If you write "Let n=1, then 3(1)+3 = 6, which is a multiple of 3," you will score zero marks. You must use algebraic variables (n, x) to show it is true generally.
2. The "Show Your Conclusion" Mark
The final mark in a proof question (often referred to as the 'C' mark or Communication mark) is for the concluding statement. You must write a sentence connecting your algebra to the question.
Example: "3(n+1) has a factor of 3, therefore it is a multiple of 3."
3. Incorrect Definitions
Using n for an odd number is a critical error. n could be 4 (even). You must use 2n+1 or 2n-1 to guarantee the number is odd. Similarly, consecutive odd numbers are not 2n+1, 2n+2 (that's odd then even). They are 2n+1, 2n+3 (skipping by 2).
Developing the logic for these proofs is challenging. Mastering abstract mathematical reasoning ensures you can tackle these high-level reasoning questions with confidence.
Summary Checklist and Next Steps
Checklist:
- [ ] I can define even (2n) and odd (2n+1) numbers algebraically.
- [ ] I can write expressions for consecutive integers (n, n+1).
- [ ] I can prove divisibility by factorising the final expression (e.g., 5(...) is a multiple of 5).
- [ ] I can prove an expression is always positive by completing the square.
- [ ] I always write a concluding sentence to finish my proof.
Next Steps:
Congratulations! You have completed Section 2: Algebra. You now possess the toolkit to manipulate, solve, and prove relationships. The next major section is Topic 3: Sequences, Functions, and Graphs, where we visualize these algebraic concepts.