Is the Square Root of 15 a Rational Number- Exploring the Intricacies of Irrationality

Is the square root of 15 a rational number? This question may seem simple at first glance, but it delves into the fascinating world of mathematics and the properties of numbers. To answer this question, we need to understand the definitions of rational and irrational numbers and apply them to the square root of 15.

Rational numbers are those 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 p/q, where p and q are integers and q is not equal to zero. On the other hand, irrational numbers cannot be expressed as a fraction of two integers and have non-terminating, non-repeating decimal expansions.

To determine whether the square root of 15 is rational or irrational, we can use the method of contradiction. Assume that the square root of 15 is a rational number. Then, we can express it as a fraction of two integers, a/b, where a and b are integers with no common factors other than 1 (this is to ensure that the fraction is in its simplest form).

Now, let’s square both sides of the equation to eliminate the square root:

(a/b)^2 = (sqrt(15))^2
a^2/b^2 = 15

Multiplying both sides by b^2, we get:

a^2 = 15b^2

This implies that a^2 is divisible by 15. Since 15 is a product of the prime numbers 3 and 5, a^2 must also be divisible by 3 and 5. By the fundamental theorem of arithmetic, this means that a must be divisible by 3 and 5 as well.

Let’s denote a as 3c and b as 5d, where c and d are integers. Substituting these values into the equation, we get:

(3c)^2 = 15(5d)^2
9c^2 = 75d^2

Dividing both sides by 3, we have:

3c^2 = 25d^2

Now, we can see that c^2 is divisible by 25. This means that c must be divisible by 5. Let’s denote c as 5e, where e is an integer. Substituting this value into the equation, we get:

3(5e)^2 = 25d^2
75e^2 = 25d^2

Dividing both sides by 25, we have:

3e^2 = d^2

This implies that d^2 is divisible by 3. Therefore, d must be divisible by 3 as well. However, this contradicts our initial assumption that a and b have no common factors other than 1. Since a and b are both divisible by 3, they must have a common factor, which is not 1.

This contradiction arises from our initial assumption that the square root of 15 is a rational number. Therefore, we can conclude that the square root of 15 is an irrational number. This finding is significant because it highlights the unique properties of square roots and the infinite nature of irrational numbers.