EDEXCEL GCSE HIGHER MATHS - What is Algebraic Proof

Edexcel GCSE Higher Maths - What is Algebraic Proof?

Introduction

Algebraic proof is a way of proving mathematical statements using algebraic manipulation. It involves using the rules of algebra to demonstrate that a statement is true for all possible values of the variables involved.

Key Concepts

• Algebraic manipulation: This involves using the rules of algebra, such as the commutative, associative, and distributive properties, to simplify expressions and solve equations.
• Assumptions: When proving a statement, you start by assuming that it is true for all values of the variables.
• Logic: Proof relies on logical reasoning to demonstrate that a statement is true based on its assumptions.

Steps Involved in Algebraic Proof

1. State the statement to be proven: This is the statement you want to prove is true.
2. Assume the statement is true: This means assuming the statement is true for all values of the variables.
3. Manipulate the statement algebraically: Use the rules of algebra to simplify the statement or rearrange it to a form that is easier to work with.
4. Reach a conclusion: After manipulating the statement, you should reach a conclusion that either supports or contradicts the original assumption.
5. Draw a conclusion: If the conclusion supports the original assumption, then the statement is proven. If the conclusion contradicts the assumption, then the statement is not true.

Examples

Example 1: Prove that the sum of two consecutive numbers is always odd.

• Statement: The sum of two consecutive numbers is always odd.
• Assumption: Let the first number be n. The next consecutive number is n + 1.
• Manipulation: The sum of the two consecutive numbers is n + (n + 1) = 2n + 1.
• Conclusion: 2n + 1 is always odd, as it is one more than an even number (2n).
• Result: The statement is proven.

Example 2: Prove that (a + b)^2 = a^2 + 2ab + b^2.

• Statement: (a + b)^2 = a^2 + 2ab + b^2
• Assumption: Assume (a + b)^2 = a^2 + 2ab + b^2 is true.
• Manipulation: Expand (a + b)^2 = (a + b)(a + b) using the distributive property:
  • (a + b)(a + b) = a(a + b) + b(a + b)
  • = a^2 + ab + ba + b^2
  • = a^2 + 2ab + b^2
• Conclusion: The expanded form of (a + b)^2 matches the right-hand side of the original statement.
• Result: The statement is proven.

Practice

Practice algebraic proof by trying to prove the following statements:

1. The product of two odd numbers is always odd.
2. The difference of squares of two consecutive numbers is always equal to the sum of those two numbers.
3. The square of an even number is always divisible by 4.

Remember to follow the steps involved in algebraic proof and use logical reasoning to reach your conclusions.