# The Mystery of Finding the Square Root of 17

The concept of finding the square root of any number is not something new. It has been around for centuries, and mathematicians have come up with various ways to calculate this value. However, there are some numbers that make the process of finding their square roots challenging, and one such number is 17.

Finding the square root of 17 is a mystery that has puzzled mathematicians for many years. Many have tried and failed to find an exact value for it, and as time goes by, the more complicated the task becomes. So why is it so hard to find the square root of 17?

The answer lies in the fact that 17 is a prime number, meaning that it can only be divided by itself and the number one. This makes it unique from other numbers whose square roots can easily be determined using prime factorization. For instance, if we take the number 36, we know that it is divisible by 2, 3 and 6, making it easy to find its exact square root.

When it comes to 17, however, things get a bit complicated. The most straightforward method of finding the square root of any number is to use long division. In this method, we start by dividing the number whose square root we want to find by a number close to its square root. We then take the average of the two resulting numbers and continue the process until we achieve a satisfactory degree of accuracy.

Suppose we apply this method to find the square root of 17. In that case, we will end up with a repeating decimal with no pattern or sequence, making it impossible to find an exact value for the square root of 17. To illustrate this, let’s consider the first few steps of the process.

To find the square root of 17, we divide it by 4, which is a number that is close to the square root of 17. We get 4.25 as the result, which means that 17 is greater than 16, but less than 17. We then take the average of 4 and 4.25, which is 4.125, and divide 17 by this number. We get 4.12121212… as the result, which goes on indefinitely.

This repeating decimal means that the square root of 17 cannot be represented as a finite decimal or as a fraction. In other words, it is an irrational number. This means that its digits go on infinitely without any pattern or sequence, making it impossible to find an exact value for the square root of 17 using any conventional means.

So what do mathematicians do when faced with such a problem? Well, they have developed various methods to approximate the value of the square root of 17. One such method is the Newton-Raphson method, which involves simplifying the process of long division by using calculus.

The method involves taking an initial guess and applying a formula that moves us closer to the true value of the square root of 17 with each iteration. This formula involves calculating the slope of the tangent line to the function f(x) = x^2 – 17 at the point where x is equal to our initial guess.

If we take an initial guess of 4, we can find the slope of the tangent line to the function f(x) = x^2 – 17, which is equal to 8. We then subtract the value of f(4) from our initial guess divided by the slope, i.e., (4^2 – 17)/8, which gives us a new estimate of 4.125. We can then repeat the process to get closer and closer to the true value of the square root of 17.

Using this method, we can approximate the value of the square root of 17 to any degree of accuracy we want. For instance, if we repeat the process 10 times, we get an estimate of 4.1231056256, which is accurate to 10 decimal places. If we repeat the process 100 times, we get an estimate of 4.1231056256176605498214092591788, which is accurate to 30 decimal places.

In conclusion, finding the square root of 17 is a mystery that has puzzled mathematicians for a long time. Its status as a prime number and the resulting repeating decimal make it impossible to find an exact value using conventional means. However, mathematicians have come up with various methods to approximate its value, such as the Newton-Raphson method. With this method, we can calculate the value of the square root of 17 to any degree of accuracy, making it possible to solve some of the most complex mathematical problems.