As mathematicians, a fundamental concept that we all learn in our early education is the notion of square roots. And if you are reading this article, then it is likely that you have reached a level where you understand that the square root of a number is equivalent to finding the value that, when multiplied by itself, gives that number. However, what about when we are given an irrational number, such as 58? The question arises: What is the square root of 58? How do we evaluate it? The answer to this mystery is what we will tackle in this article.

The first thing that we need to establish is that 58 is, indeed, an irrational number. This means that it can’t be expressed in the form of a fraction between two integers. The decimal representation of the square root of 58 is infinite and non-repeating, making it an irrational number. It is clear that the task of finding the exact value of the square root of 58 is not an easy one, so we must rely on mathematical methods to do so.

One way to approximate the square root of 58 is by using the Babylonian method, which is a method that dates back to ancient times. This method involves repeatedly taking the average of two numbers until we reach a desired degree of accuracy. In this case, we can start with an initial guess, take the average of that guess with 58 divided by the guess, continue taking the average of the result with 58 divided by that new guess, and so on. This process can be continued until we reach a degree of accuracy that we are satisfied with.

Using this method, we can find that the square root of 58 is approximately equal to 7.62. However, this is only an approximation, and it is not the exact value of the square root of 58. To find the exact value, we must turn to a more sophisticated method.

One such method is to use the Taylor series expansion of the square root function. The Taylor series is a representation of a function as an infinite sum of terms that are computed from the function’s derivatives at a specific point. In this case, we can use the Taylor series expansion of the square root function around the point x=1 (since 58 is greater than 1).

The first few terms of the Taylor series expansion of the square root function are:

√x = 1 + (x – 1)/2 – (x – 1)²/8 + (x – 1)³/16 – …

By plugging in x=58, we get:

√58 = 1 + (58 – 1)/2 – (58 – 1)²/8 + (58 – 1)³/16 – …

Simplifying this expression, we arrive at:

√58 = 7 + 11/23 + 5/253 – 3/1105 + …

This series is infinite, and the terms converge towards the exact value of the square root of 58. However, it is impractical to evaluate an infinite series, so we must truncate it after a sufficient number of terms. In this case, if we take the first four terms, we get:

√58 ≈ 7.63324958

This is the exact value of the square root of 58, up to the given degree of accuracy. It is worth noting that this value is not rational, meaning that it cannot be expressed as a fraction between two integers. Rather, it is an irrational number, and its decimal representation goes on infinitely without repeating.

In conclusion, while the task of finding the square root of 58 may have appeared daunting at first, there are mathematical methods that we can use to approximate and even determine the exact value with a degree of accuracy. The Babylonian method is a simple approach that can be used for approximation, while the Taylor series expansion of the square root function provides an exact value. These methods are just the tip of the iceberg in the vast field of mathematics, and there are always more tools to discover and utilize.