For thousands of years, mathematicians have been enthralled by the square root of 2, a mathematical constant. Ancient civilizations like the Greeks and the Egyptians, who utilised it in their geometrical and mathematical computations, are credited with its invention. Since the square root of 2 is an irrational integer, it cannot be written as a straightforward fraction and its decimal representation never ends. Despite this, the square root of 2 has a distinct position in mathematics and is utilised in a wide range of subjects, including calculus, geometry, and trigonometry. We will look at the definition, characteristics, and uses of the square root of 2, as well as its historical importance and effects on mathematics, in this blog.

## Definition and Properties of the Square Root of 2

The **square root of 2** is a mathematical constant that is defined as the positive real number that, when multiplied by itself, equals 2. Symbolically, it can be represented as √2. It is a fundamental constant in mathematics that has many unique properties.

One of its important properties is that it is an irrational number, meaning that it cannot be expressed as a simple fraction. Its decimal representation goes on forever without repeating, making it a non-repeating, non-terminating decimal. This property makes it useful in mathematics and other fields where precision is crucial.

Another **property of the square root of 2** is its non-periodicity. This means that its decimal representation does not repeat in a predictable pattern, making it different from other repeating decimals.

The square root of 2 also has some interesting geometric applications. For example, in Euclidean geometry, it is used to calculate the diagonal length of a square with sides of length 1. In trigonometry, it is used in the calculation of trigonometric functions, such as sine and cosine.

In summary, the square root of 2 is a mathematical constant with unique properties such as being an irrational number and having non-periodic decimal representation. These properties make it useful in a variety of mathematical applications and calculations

## How to Find Square Root of 2 By Long Division Method?

Now, let’s look at how to find the square root of 2 using the method of division. You can begin by utilizing one of the most common methods, which is called division. The following procedures are included in this method: In the first step of this method, you are required to make an estimate, which indicates that you have selected a number that is relatively close by locating at least two roots that are perfect. Your number is between those values because those roots have caused it to be that way.

1st Step: The 2nd step is to divide the number by one of the square roots that you found earlier.

2nd Step: Once you begin and continue with the process of division, you will need to take the average of the result of step 2 and the root.

3rd Step: This will be your final answer. And lastly, you can use the result that you get in step 3 to repeat steps 2 and 3 as many times as necessary until you arrive at a number that satisfies your requirements in terms of how accurate it is.

The illustration of **Calculating the square root of two** can be found down below:

**How to Find the Root of the Square of 2**. You can Try on your Own If you Calculate the value of Square Root of 2, it will promptly give you the result for the given number, which is equal to 1.414.

## Applications in Mathematics

The square root of 2 is a useful mathematical concept that can be put to work in a wide variety of contexts and can be found in many different subfields and fields of mathematics. The following are some of the more prominent applications:

**With the Help of Geometry: **Calculating the length of the diagonal of a square with sides of length one requires using the square root of 2, according to the principles of Euclidean geometry. In numerous geometric calculations and approaches to solving problems, the relationship between the side lengths and diagonal lengths of a square is an important consideration.

**With the Help of Trigonometry:** Calculating trigonometric functions like sine and cosine in trigonometry requires the use of the square root of 2, which is a key concept in this branch of mathematics. These functions are utilized to provide a description of the relationships that exist between the lengths of the sides and the angles that make up a right triangle.

**With the Help of Calculus and Real Analysis:** In calculus and real analysis, the square root of 2 is utilised in the calculation of limits and derivatives. The study of functions and the properties of functions, such as continuity and differentiability, is another application of this technique.

**With the Help of Number Theory: **In number theory, the square root of 2 is a useful tool for researching the characteristics of irrational numbers as well as their connections to other types of numbers, including rational numbers and integers.

**With the Help of Some Miscellaneous Applications: **Other applications of mathematics include the calculation of algorithms and the solution of mathematical problems. The square root of 2 is used in other areas of mathematics, including linear algebra and computer science, where it is used to calculate algorithms and solve mathematical problems.

## Conclusion

To summarize, the square root of 2 has a wide range of applications in mathematics and is utilized in a variety of mathematical domains and subfields. Because of its one-of-a-kind characteristics and significant mathematical role, it is a useful tool for both scientists and mathematicians. The fascinating mathematical constant known as the square root of 2 has been essential to mathematics for thousands of years. Due to its special qualities and historical importance, it is a valuable tool for mathematicians and scientists working in a variety of fields. The square root of 2 is still an important and practical mathematical constant, with applications ranging from geometry and trigonometry to real analysis, calculus, and number theory. Understanding the square root of 2 and its properties is a crucial step in broadening your understanding and appreciation of mathematics, whether you are a student, teacher, or professional mathematician.