Exploring the Rationality of Square Roots: The Cases of √98 and √2

Understanding whether the square roots of numbers are rational or irrational can be a fascinating journey into the realms of mathematics. In this article, we will delve into the specific examples of whether √98 and √2 are rational or irrational numbers. We will explore the proofs behind these facts and provide a comprehensive explanation for each.

√98 - An Irrational Number

The square root of 98, √98, is an irrational number. This is because 98 is not a perfect square, meaning it cannot be expressed as the product of two equal integers.

To prove that √98 is irrational, we can follow a proof similar to the one for the square root of 2. Let's start by squaring both sides of the equation √98:

Proof:

Consider the product √98 × √2. When squaring this product, we get: (√98 × √2)2 98 × 2 196. So, √98 × √2 √196. Since 196 is a perfect square, √196 14. Therefore, √98 × √2 14, which implies √98 14/√2 8√2.

Since √2 is known to be irrational, any multiple of √2, such as 8√2, is also irrational. Thus, √98 is an irrational number.

√2 - A Classic Case of an Irrational Number

The square root of 2, √2, is a well-known example of an irrational number. There are several proofs available to demonstrate that √2 cannot be expressed as a ratio of two integers. One of the most famous proofs is as follows:

Proof that √2 is irrational:

Assume, for the sake of contradiction, that √2 is a rational number. This means we can express √2 as a fraction a/b, where a and b are integers and b ≠ 0. This implies that (a/b)2 2, or a2 2b2. Since a2 is even, a must also be even. Let a 2k for some integer k. Substitute a 2k into the equation a2 2b2. This gives (2k)2 2b2, or 4k2 2b2, which simplifies to 2k2 b2. Since b2 is even, b must also be even. This contradicts our assumption that a and b have no common factors other than 1. Thus, our initial assumption that √2 is rational must be false, and √2 is irrational.

Comparing √98 and √2

The key difference between √98 and √2 lies in their forms. While √2 is in its simplest form and is inherently irrational, 98 can be factored as 49 times; 2. Therefore, √98 can also be written as √(49 times; 2) 7√2. Since √2 is irrational, 7√2 is also irrational.

However, the problem states that √982 is equal to √100, which is 10, a rational number. This is because √982 is not the square root of 98, but rather, the square of the square root of 98:

(√98)2 98, and the square root of 98 is not rational, but when squared, it becomes a rational number.

On the other hand, the individual square roots, √98 and √2, are both irrational. This confirms that neither square root of 98 is rational by itself.

Understanding these concepts important for grasping the broader principles of number theory and the classification of numbers into rational and irrational categories. Such knowledge is useful in various fields, including computer science, physics, and engineering.