Is the Square Root of 98 a Rational Number- A Deep Dive into the World of Irrational Numbers

Is the square root of 98 a rational number? This question might seem simple at first glance, but it delves into the fascinating world of mathematics and the properties of numbers. In this article, we will explore the nature of the square root of 98 and determine whether it is a rational or irrational number.

The square root of 98, denoted as √98, is an intriguing number that has sparked curiosity among mathematicians for centuries. To understand whether it is a rational number, we need to delve into the definitions of rational and irrational numbers.

A rational number is a number that can be expressed as a fraction of two integers, where the denominator is not zero. In other words, a rational number can be written in the form a/b, where a and b are integers, and b is not equal to zero. On the other hand, an irrational number is a number that cannot be expressed as a fraction of two integers. These numbers have decimal expansions that neither terminate nor repeat.

To determine whether the square root of 98 is rational, we can try to express it as a fraction of two integers. If we can find integers a and b such that √98 = a/b, then it is a rational number. However, if we cannot find such integers, then it is an irrational number.

Let’s assume that √98 is a rational number. In that case, we can write it as √98 = a/b, where a and b are integers. Squaring both sides of the equation, we get 98 = a^2/b^2. Multiplying both sides by b^2, we obtain 98b^2 = a^2. This implies that a^2 is divisible by 98, which means that a is also divisible by 7 (since 98 = 7 14).

Now, let’s express a as a multiple of 7, i.e., a = 7k, where k is an integer. Substituting this into the equation 98b^2 = a^2, we get 98b^2 = (7k)^2, which simplifies to 98b^2 = 49k^2. Dividing both sides by 7, we obtain 14b^2 = 7k^2. This implies that b^2 is divisible by 7, which means that b is also divisible by 7.

Since both a and b are divisible by 7, we can express them as a = 7m and b = 7n, where m and n are integers. Substituting these values into the equation √98 = a/b, we get √98 = 7m/7n, which simplifies to √98 = m/n. This shows that we can express √98 as a fraction of two integers, making it a rational number.

However, this conclusion is incorrect. The square root of 98 cannot be expressed as a fraction of two integers, as we have shown. Therefore, the square root of 98 is an irrational number. This means that its decimal expansion neither terminates nor repeats, and it cannot be represented as a fraction of two integers.

In conclusion, the square root of 98 is not a rational number. This discovery highlights the beauty and complexity of mathematics, as it showcases the existence of numbers that defy our intuition and require deeper understanding to comprehend.