Irrationality of Sums and Products

Master advanced proof techniques! Discover how rational and irrational numbers combine, and learn to prove the irrationality of complex expressions using contradiction.

Advanced Irrationality Proofs!

Explore how rational and irrational numbers interact
through sums, differences, and products

Sum/Difference Rule
Rational ± Irrational = Irrational

When you add or subtract a rational and irrational number, the result is always irrational

Product Rule
Non-zero Rational × Irrational = Irrational

When you multiply a non-zero rational by an irrational, the result is always irrational

Proof: 5 - √3 is Irrational

1
Assumption
Assume 5 - √3 is rational
5 - √3 = a/b (where a, b are integers, b ≠ 0)
2
Rearrange
Solve for √3
√3 = 5 - a/b
√3 = (5b - a)/b
3
Analyze Result
Since 5, a, b are integers:
(5b - a) is an integer
Therefore (5b - a)/b is rational
4
Contradiction!
This makes √3 rational
But we know √3 is irrational!
Our assumption must be false
Therefore: 5 - √3 is IRRATIONAL

Proof: 3√2 is Irrational

1
Assumption
Assume 3√2 is rational
3√2 = a/b (where a, b are integers, b ≠ 0)
2
Rearrange
Solve for √2
√2 = a/(3b)
√2 = a/(3b)
3
Analyze Result
Since 3, a, b are integers:
3b is an integer
Therefore a/(3b) is rational
4
Contradiction!
This makes √2 rational
But we know √2 is irrational!
Our assumption must be false
Therefore: 3√2 is IRRATIONAL

Practice: Prove These Are Irrational

Challenge 1
1/√2

Hint: Use product rule
√2 × (1/√2) = 1

Challenge 2
7√5

Hint: Same method as 3√2
Assume rational, isolate √5

Challenge 3
6 + √2

Hint: Same method as 5 - √3
Assume rational, isolate √2

General Strategy

1. Assume the expression is rational
2. Rearrange to isolate the known irrational part
3. Show this makes the irrational part rational
4. Contradiction! Therefore the original expression is irrational

Proof Technique Summary

🎯 When to Use Contradiction
Proving irrationality of expressions with known irrationals (√2, √3, √5, etc.)
↓
🔄 Standard Process
1. Assume rational (a/b form) → 2. Isolate known irrational
3. Show irrational appears rational → 4. Contradiction!
↓
📚 Key Theorems
• √p irrational for prime p • Rational ± Irrational = Irrational
• Non-zero Rational × Irrational = Irrational
Click any button below to explore advanced irrationality proofs!
Master Advanced Proof Techniques
🧠 Mathematical Insights
These advanced proof techniques show how irrationality "spreads" through arithmetic operations. When you combine rational and irrational numbers, the irrationality dominates, creating new irrational numbers with predictable patterns.