Proof That The Square Root Of 5 Is Irrational

Proof That the Square Root of 5 is Irrational

Meta Description: This article provides a clear and concise proof, using proof by contradiction, demonstrating that the square root of 5 is an irrational number, meaning it cannot be expressed as a fraction of two integers. Learn about the fundamental concepts of rational and irrational numbers and understand the elegant logic behind this mathematical demonstration.

The square root of 5, denoted as √5, is a fascinating number in mathematics. It's an irrational number, meaning it cannot be expressed as a simple fraction (a ratio) of two integers. Understanding why this is true requires a grasp of fundamental mathematical concepts and a classic proof technique: proof by contradiction.

Understanding Rational and Irrational Numbers

Before diving into the proof, let's clarify the definitions:

• Rational numbers: These are numbers that can be expressed as a fraction p/q, where p and q are integers, and q is not zero. Examples include 1/2, 3/4, -2/5, and even integers like 5 (which can be expressed as 5/1).

• Irrational numbers: These are numbers that cannot be expressed as a fraction of two integers. Famous examples include π (pi), e (Euler's number), and, as we'll prove, √5. Irrational numbers have decimal representations that are non-terminating and non-repeating.

Proof by Contradiction: Showing √5 is Irrational

This method starts by assuming the opposite of what we want to prove and then showing that this assumption leads to a contradiction. Let's assume, for the sake of contradiction, that √5 is rational. This means we can express it as a fraction:

√5 = p/q

where p and q are integers, q ≠ 0, and the fraction p/q is in its simplest form (meaning p and q have no common factors other than 1). We can also assume that p and q are positive integers for simplicity.

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

5 = p²/q²

Multiplying both sides by q², we get:

5q² = p²

This equation tells us that p² is a multiple of 5. Since 5 is a prime number, this implies that p itself must also be a multiple of 5. We can write p as:

p = 5k (where k is an integer)

Substitute this back into the equation 5q² = p²:

5q² = (5k)²

5q² = 25k²

Dividing both sides by 5:

q² = 5k²

This equation shows that q² is also a multiple of 5, and therefore, q must be a multiple of 5 as well.

Here's the contradiction: We initially assumed that p/q was in its simplest form, meaning p and q have no common factors. However, we've just shown that both p and q are multiples of 5, meaning they do have a common factor of 5. This is a contradiction!

Conclusion

Because our initial assumption (that √5 is rational) leads to a contradiction, the assumption must be false. Therefore, we conclude that the square root of 5 is irrational. This elegant proof highlights the power of proof by contradiction in establishing fundamental mathematical truths. The concept extends beyond just proving the irrationality of √5 and is a cornerstone of many mathematical proofs.