Is the Square Root of 3 a Rational Number- Unraveling the Mystery of Irrationality
Is the square root of 3 a rational number? This question has intrigued mathematicians for centuries and remains a fundamental topic in number theory. In this article, we will explore the nature of the square root of 3 and determine whether it is a rational or irrational number.
The square root of 3, denoted as √3, is an irrational number. To understand why, let’s first define what a rational number is. 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.
Now, let’s consider the square root of 3. If √3 were a rational number, it could be expressed as a fraction of two integers, a/b, where a and b are integers and b is not equal to zero. Squaring both sides of this equation, we get:
(√3)^2 = (a/b)^2
3 = a^2/b^2
Multiplying both sides by b^2, we have:
3b^2 = a^2
This implies that a^2 is divisible by 3. Therefore, a must also be divisible by 3, as any integer squared will have the same divisibility properties as the integer itself. Let’s assume a = 3c, where c is an integer. Substituting this into the equation, we get:
3b^2 = (3c)^2
3b^2 = 9c^2
Dividing both sides by 3, we have:
b^2 = 3c^2
This shows that b^2 is also divisible by 3, and hence b must be divisible by 3 as well. However, this contradicts our initial assumption that a and b are coprime (i.e., they have no common factors other than 1). Since we have reached a contradiction, our assumption that √3 is a rational number must be false.
Therefore, the square root of 3 is an irrational number. This means that it cannot be expressed as a fraction of two integers and has an infinite, non-repeating decimal representation. The discovery that √3 is irrational is a significant milestone in the history of mathematics, as it demonstrates the existence of irrational numbers and their unique properties.
