Proof That Root 3 Is Irrational

Proof That √3 is Irrational: A Comprehensive Guide

Meta Description: Learn how to prove the irrationality of the square root of 3 using proof by contradiction, a fundamental technique in mathematics. This guide provides a clear and concise explanation, perfect for students and math enthusiasts alike.

The square root of 3 (√3) is an irrational number. This means it cannot be expressed as a fraction p/q, where p and q are integers and q is not zero. Understanding this proof is crucial for grasping fundamental concepts in number theory. This article provides a step-by-step explanation of the proof, utilizing the popular method of proof by contradiction.

Understanding Proof by Contradiction

Proof by contradiction, also known as reductio ad absurdum, is a powerful indirect proof method. We begin by assuming the opposite of what we want to prove. If this assumption leads to a contradiction – a logically impossible statement – then our initial assumption must be false, thereby proving the original statement true.

Proving the Irrationality of √3

Let's assume, for the sake of contradiction, that √3 is rational. This means it can be expressed as a fraction:

√3 = p/q

where p and q are integers, q ≠ 0, and p and q are coprime (meaning they share no common factors other than 1). This is crucial; we're assuming the fraction is in its simplest form.

Now, let's square both sides of the equation:

3 = p²/q²

Multiplying both sides by q², we get:

3q² = p²

This equation tells us that p² is a multiple of 3. Since 3 is a prime number, this implies that p itself must also be a multiple of 3. We can express this as:

p = 3k (where k is an integer)

Substituting this back into the equation 3q² = p², we get:

3q² = (3k)²

3q² = 9k²

Dividing both sides by 3:

q² = 3k²

This equation shows that q² is also a multiple of 3, and therefore q must be a multiple of 3 as well.

The Contradiction

We've now shown that both p and q are multiples of 3. This contradicts our initial assumption that p and q are coprime (they share no common factors other than 1). We've reached a logical impossibility.

Conclusion

Since our initial assumption that √3 is rational leads to a contradiction, that assumption must be false. Therefore, the square root of 3 (√3) is irrational. This proof demonstrates the elegance and power of proof by contradiction in establishing fundamental mathematical truths about numbers. Understanding this proof provides a solid foundation for exploring more advanced concepts in number theory and related fields.