Welcome to the Power of Logical Deduction!
In your H3 Mathematics journey, you’ve already seen how to build a proof step-by-step using a "Direct Proof." But what happens when the direct path is blocked? What if proving something is true seems impossible, but proving that its opposite is "crazy" is much easier?
That is the heart of Proof by Contradiction (sometimes called reductio ad absurdum). It is one of the most elegant and powerful tools in a mathematician's toolkit. By the end of these notes, you’ll be able to dismantle complex statements by showing that their "alternative reality" simply cannot exist.
Section 1: The Big Idea
Imagine you are a detective. You want to prove that "Suspect A" was at the scene of the crime. Instead of looking for a photo of him there, you assume he was actually at home. If you then find out he was also seen at a grocery store at that exact same time, you have a contradiction. He can't be in two places at once! Therefore, your starting assumption (that he was at home) must be false, which proves he was at the scene.
How it works in Math:
1. We want to prove statement \( P \) is true.
2. We begin by assuming the opposite: Suppose \( P \) is false (this is called the negation, \( \neg P \)).
3. Use logical steps to reach a conclusion that is clearly impossible or contradicts a known fact (e.g., concluding that \( 1 = 0 \) or that a number is both even and odd).
4. Since our logic was sound, the only "mistake" must have been our starting assumption.
5. Therefore, the original statement \( P \) must be true.
Quick Takeaway:
If assuming "A" is false leads to a total "logic meltdown," then "A" must be true!
Section 2: The Step-by-Step "Recipe"
Don't worry if this seems tricky at first! Just follow these four steps every time:
Step 1: State your assumption clearly.
Start with: "Suppose, for the sake of contradiction, that [the opposite of what you want to prove] is true."
Step 2: Use your "Math Toolbox."
Work through the problem using definitions, algebra, and known theorems (like those from H2 Math).
Step 3: Find the "Ouch!" moment.
Look for the point where two facts clash. This is your contradiction.
Step 4: The Mic Drop.
Conclude by saying: "This is a contradiction. Therefore, our assumption was false, and [the original statement] must be true."
Section 3: A Classic Example — The Irrationality of \( \sqrt{2} \)
This is a favorite in the H3 syllabus and a perfect way to see the "recipe" in action.
The Goal: Prove that \( \sqrt{2} \) is irrational.
Step 1 (Assumption): Suppose \( \sqrt{2} \) is rational. This means it can be written as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers with no common factors (the fraction is in its simplest form).
Step 2 (The Math):
\( \sqrt{2} = \frac{p}{q} \)
Square both sides: \( 2 = \frac{p^2}{q^2} \)
Rearrange: \( p^2 = 2q^2 \)
This means \( p^2 \) is an even number, which implies \( p \) itself must be even. Let's write \( p = 2k \).
Substitute back in: \( (2k)^2 = 2q^2 \) → \( 4k^2 = 2q^2 \) → \( 2k^2 = q^2 \).
This means \( q^2 \) is also even, so \( q \) must be even.
Step 3 (The Contradiction):
Wait! We just found that both \( p \) and \( q \) are even. This means they share a common factor of 2. But in Step 1, we said they have no common factors. Contradiction!
Step 4 (Conclusion):
Our assumption that \( \sqrt{2} \) is rational must be false. Thus, \( \sqrt{2} \) is irrational.
Section 4: Common Pitfalls to Avoid
Even top students can trip up here. Keep these in mind:
• Weak Negation: Make sure you negate the statement correctly. If you want to prove "All \( x \) are \( y \)", the negation is "There exists at least one \( x \) that is not \( y \)." (Don't assume "No \( x \) are \( y \)"!)
• Circular Reasoning: Don't accidentally use the fact you are trying to prove in your middle steps.
• Stopping too soon: Always explicitly state what the contradiction is. Don't leave the examiner guessing!
Did you know?
The Greek mathematician Hippasus is said to have discovered the proof for the irrationality of \( \sqrt{2} \) while at sea. Legend says his fellow Pythagoreans were so upset (because they believed everything was a clean ratio of whole numbers) that they threw him overboard! Math can be high-stakes!
Section 5: When should I use Contradiction?
You might be wondering: "How do I know when to use this instead of a direct proof?"
Try Proof by Contradiction if:
1. The statement contains the word "not" or "no" (e.g., "There are no integers such that...").
2. You are trying to prove a statement is unique or irrational.
3. A direct proof feels like you are trying to "catch smoke" — it's easier to show that the opposite leads to a mess than to build the truth from scratch.
Section 6: Quick Review Box
Prerequisite Check: To do this well, you need to be comfortable with negation (turning "If P then Q" into "P and not Q").
Key Term: Contradiction — A situation where a statement and its opposite are both claimed to be true (which is impossible).
Memory Trick: Think of it as the "Mirror Proof." You look in the mirror (the assumption), see a monster (the contradiction), and realize you shouldn't have looked in the mirror in the first place!
Final Summary:
Proof by contradiction is the "indirect" path. We assume the world is the opposite of what we think, follow the logic until it breaks, and then conclude that our original thought was right all along. It’s perfect for proving that things don't exist or that numbers are not rational.