To determine if √5 is a rational number, we first need to understand what a rational number is. A rational number is any number that can be expressed as the fraction of two integers, where the denominator is not zero.
Now, let’s analyze √5. The square root of 5 is approximately 2.236. We can see that it does not terminate or repeat when expressed as a decimal. This suggests that √5 cannot be expressed in the form of a fraction with integers.
Furthermore, we can prove that √5 is irrational by contradiction. Assume that √5 is rational; then it can be expressed as a fraction in its simplest form: √5 = a/b, where a and b are integers with no common factors, and b ≠ 0. Squaring both sides gives us 5 = a²/b², which leads to a² = 5b². This implies that a² is a multiple of 5, and hence a must also be a multiple of 5 (since the square of a non-multiple of 5 cannot produce a multiple of 5). Let’s say a = 5k for some integer k. Plugging this back in gives us (5k)² = 5b², or 25k² = 5b². Dividing by 5 leads to 5k² = b², which means b² is also a multiple of 5, and thus b must also be a multiple of 5.
This contradicts our initial assumption that a and b have no common factors, as both are divisible by 5. Therefore, our assumption that √5 is rational must be false.
In conclusion, √5 is not a rational number; it is an irrational number.