As humans, we are always in pursuit of discovering the secrets of the world we live in. One such mystery that has puzzled mathematicians for centuries is the square root of 45. It is a number that holds an infinite amount of decimals and can never be expressed precisely as a whole number. In this article, we will delve into the roots (pun intended) of this enigma and uncover what makes the square root of 45 so intriguing.
Let’s begin by defining what a square root is. The square root of a number is simply the value that we must multiply with itself to get the given number. For example, the square root of 9 is 3 because 3 multiplied by 3 gives you 9. Now, when it comes to the square root of 45, things become a little more complicated.
The square root of 45 is an irrational number, meaning it cannot be expressed as a simple fraction. It is a non-repeating, non-terminating decimal, which makes it an infinite decimal. The decimal representation of the square root of 45 goes on forever and doesn’t terminate or repeat.
So, why is the square root of 45 irrational? This can be proven using the “proof by contradiction” method. Suppose the square root of 45 is a rational number, which can be expressed as a ratio of two integers a/b. We can then simplify this fraction by dividing both a and b by their greatest common factor, which would give us the reduced fraction p/q. If we square p/q, we get 45. This can be written as follows:
(p/q)^2 = 45
Multiplying both sides by q^2, we get:
p^2 = 45q^2
This means that p^2 is an odd multiple of 5, which implies that p must be odd as well. Furthermore, since p^2 is odd, p must be an odd integer. Now, let’s consider the prime factorization of 45:
45 = 3 x 3 x 5
This implies that q^2 is an odd multiple of 3, which means that q must also be odd. However, this contradicts our assumption that a/b is a reduced fraction. We initially assumed that a/b is the reduced form of p/q, which means that both a and b are even. This contradiction proves that the square root of 45 cannot be expressed as a rational number.
Now, we know that the square root of 45 is irrational, but how can we approximate its value? One method is to use a calculator or a computer program, but that doesn’t give us an exact value. Instead, we can use a technique called long division to get as many decimal places as we want.
When we perform long division, we start by dividing the divisor (in this case, 45) by the square of the first digit of the square root. We then multiply this result by 2 and subtract it from the dividend. The result is the remainder, which we bring down to the next line and continue with the long division process. This cycle repeats until we get the desired number of decimal places.
Using this method, we can get the square root of 45 to any number of decimal places we desire. For example, the square root of 45 to 10 decimal places is approximately 6.7082039325. However, it’s important to note that this approximation is not exact, and there will always be some degree of error involved.
In conclusion, the square root of 45 is a fascinating mathematical mystery that has puzzled mathematicians for centuries. Its irrational nature means that it can never be expressed precisely as a whole number, and its infinite decimal representation goes on forever without repeating. However, with the help of techniques like long division, we can approximate its value to any number of decimal places we desire. Despite its elusive nature, the square root of 45 reminds us of the endless possibilities and wonders that exist within the world of mathematics.