What are Irrational Numbers?

Before we understand irrational numbers, let's recall what rational numbers are.

Rational Number = Any number that can be written as a fraction $\frac{p}{q}$ where p and q are whole numbers (integers) and q ≠ 0

Examples of Rational Numbers:

• 5 = 5/1 ✓
• 0.5 = 1/2 ✓
• -3 = -3/1 ✓
• 0.333... = 1/3 ✓

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So What is an Irrational Number?

Irrational Number = A number that CANNOT be written as a simple fraction $\frac{p}{q}$

No matter how hard you try, you cannot express these numbers as a fraction of two whole numbers!

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How to Spot an Irrational Number?

Look at the decimal expansion:

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Famous Irrational Numbers

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🎯 The Big Proof: Why is √2 Irrational?

This is one of the most beautiful proofs in mathematics! Don't worry - we'll go step by step.

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First, What is "Proof by Contradiction"?

Imagine you want to prove: "There is no elephant in this room"

How would you prove it?

1. First, ASSUME an elephant IS in the room
2. If an elephant is here, we should see it, hear it, smell it
3. But we don't see, hear, or smell any elephant!
4. This is a CONTRADICTION - something impossible happened
5. So our assumption must be WRONG
6. Therefore, there is NO elephant in the room ✓

This is called "Proof by Contradiction" - we assume the opposite of what we want to prove, show it leads to something impossible, and conclude our assumption was wrong!

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🎬 Watch the Animated Proof First!

Before reading the detailed steps, watch this interactive animation to get the big picture:

Use ◀ ▶ buttons or click the dots to navigate through the 6 steps!

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Now Let's Prove √2 is Irrational!

We'll use the same method. We want to prove √2 is irrational (cannot be written as p/q).

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🔵 Step 1: Make the Assumption

Let's ASSUME √2 IS rational (the opposite of what we want to prove).

If √2 is rational, we can write: $\sqrt{2} = \frac{p}{q}$

Important: We choose p and q such that they have no common factors.

What does "no common factors" mean?

For example, 6/8 can be simplified to 3/4 (divide both by 2)

But 3/4 cannot be simplified further - 3 and 4 share no common factors!

We say 3/4 is in lowest terms or p and q are co-prime.

So we assume: $\sqrt{2} = \frac{p}{q}$ where p and q have NO common factors.

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🟣 Step 2: Square Both Sides

Let's square both sides of our equation:

If $\sqrt{2} = \frac{p}{q}$

Then $(\sqrt{2})^2 = \left(\frac{p}{q}\right)^2$

$2 = \frac{p^2}{q^2}$

Now multiply both sides by $q^2$:

$2q^2 = p^2$

Key Finding: $p^2 = 2q^2$

This means p² is equal to 2 times something, so p² is EVEN!

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🟢 Step 3: If p² is Even, then p is Also Even

Wait, how do we know p is even if p² is even?

Let's think about this:

• If p = 3 (odd), then p² = 9 (odd)
• If p = 5 (odd), then p² = 25 (odd)
• If p = 7 (odd), then p² = 49 (odd)

See the pattern? Odd × Odd = Odd

So if p² turns out to be EVEN, p cannot be odd. p must be even!

Conclusion: Since $p^2 = 2q^2$ is even, p is EVEN.

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🟡 Step 4: Write p as 2k

Since p is even, we can write p = 2k for some whole number k.

(Every even number can be written as 2 × something: 4 = 2×2, 6 = 2×3, 8 = 2×4, etc.)

Now substitute p = 2k in our equation $p^2 = 2q^2$:

$(2k)^2 = 2q^2$ $4k^2 = 2q^2$

Divide both sides by 2: $2k^2 = q^2$

or $q^2 = 2k^2$

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🟠 Step 5: q² is Also Even!

Look at what we just found: $q^2 = 2k^2$

This means q² is also 2 times something, so q² is EVEN!

Using the same logic as Step 3:

If q² is even, then q must also be EVEN!

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🔴 Step 6: THE CONTRADICTION! 💥

Let's look at what we've discovered:

• p is even (from Step 3)
• q is even (from Step 5)

But wait! In Step 1, we said p and q have NO common factors!

If both p and q are even, they both have 2 as a factor!

For example:

• If p = 6 and q = 4, both are even
• 6 = 2 × 3 and 4 = 2 × 2
• They share the factor 2!
• So 6/4 is NOT in lowest terms!

This is a CONTRADICTION!

We said p/q has no common factors, but we just proved both have factor 2!

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✅ The Conclusion

Since assuming "√2 is rational" led to a contradiction (an impossible situation), our assumption must be WRONG.

Therefore, √2 is NOT rational.

√2 is IRRATIONAL! ✓

🎉 Congratulations! You just understood one of the oldest and most important proofs in mathematics!

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📋 QUICK REVISION (For Exams!)

Proof that √2 is Irrational (6 Steps):

Quick Answer: "Both p and q have 2 as common factor, but we assumed no common factors. Contradiction! So √2 is irrational." ✓

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Quick Reference: Is √n Rational or Irrational?

Rule: √n is irrational if n is NOT a perfect square

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📝 Practice Question

Can you prove √3 is irrational using the same method?

Hint: Follow the same steps, but instead of "p² = 2q²", you'll get "p² = 3q²". Then show p is divisible by 3, and so is q!