Methods for Finding Square Root of a Number
Six different methods for finding the square root of a number exist:
- Prime Factorization Method
- Long Division Method
- Average Method (Babylonian Method)
- Newton-Raphson Method
- Binary Search Method
- Using Built-in Functions or Libraries (such as sqrt() in math library of many programming languages)
Check out the square root calculator
Prime Factorization Method
The Prime Factorization Method is a classical method for finding the square root of a number. It involves finding the prime factorization of the number and dividing it into pairs of similar factors. These pairs are then simplified and the square root is extracted from each pair. This method can be time-consuming, especially for larger numbers, but it is considered to be a reliable and accurate method for finding the square root. Prime factorization is done by dividing the number by a prime number until the quotient is no longer divisible. This process is repeated with the quotient until all prime factors are found. The square root of the number is then calculated by taking the square root of each prime factor.
Here’s an example of how to find the square root of a number using the Prime Factorization Method:
Find the square root of 144.
Step 1: Factor the number into its prime components. In this case, 144 = 2^4 * 3^2.
Step 2: Divide the exponent of each prime factor by 2. For example, the exponents of 2 and 3 are 4 and 2, respectively, and when they’re divided by 2, they become 2 and 1.
Step 3: Take the square root of each prime factor. In this case, the square root of 2^2 is 2 and the square root of 3^1 is 3.
Step 4: Multiply the square roots of each prime factor to find the square root of the original number. In this case, the square root of 144 = 2 * 3 = 6.
The square root of 144 is equal to 6.
Long Division Method
The Long Division Method is a step-by-step process for finding the square root of a number. It involves dividing the number into a series of smaller parts and then finding the square root of each part. This method is often used to find the approximate value of a square root. The process starts by choosing a number that is close to the square root of the target number and then dividing the target number by that number. The result is then used as the divisor in the next step of the division, and the process is repeated until the desired level of accuracy is achieved. The Long Division Method is considered to be a more accurate method than the Prime Factorization Method, but it can be more time-consuming and requires more computational effort. However, it can be useful for solving complex square root problems and provides a visual representation of the calculation process.
Here’s an example of how to find the square root of a number using the long division method:
Find the square root of 5776.
Step 1: Determine the pair of digits that form the largest perfect square that’s less than or equal to the first two digits of the number (57). The largest perfect square less than or equal to 57 is 49, which is the square of 7.
Step 2: Divide the number by the pair of digits from Step 1 (57 divided by 7). The quotient is 8, with a remainder of 1.
Step 3: Write down the division result in a pair of parentheses with the quotient (8) on top and the remainder (1) on the bottom.
Step 4: Bring down the next two digits of the number (76). The new number is 176.
Step 5: Double the number in the pair of parentheses from Step 3 (8), and write it outside of the pair of parentheses. This is your new divisor.
Step 6: Divide the new number from Step 4 by the divisor from Step 5 (176 divided by 16), and write the result as a decimal after the pair of parentheses. The result is 10.875.
Step 7: Choose the integer that, when multiplied by itself, is closest to the decimal result from Step 6 (10 or 11). We choose 11.
Step 8: Multiply the number in the pair of parentheses by the integer from Step 7 (8 * 11) and write the result outside of the pair of parentheses. This is your new divisor.
Step 9: Repeat Steps 6-8 until you have all of the decimal places you want. In this case, we’ll stop after just one iteration since we only want one decimal place.
The square root of 5776 is approximately equal to 76.
Average Method (Babylonian Method)
The Average Method, also known as the Babylonian Method, is a method for finding the square root of a number that is based on the principle of averages. It is considered to be one of the most efficient and accurate methods for finding the square root of a number. The method involves finding the average of the number and its divisor and then using the result as the divisor in the next step. This process is repeated until the desired level of precision is attained. The Average Method takes advantage of the fact that the square root of a number is always between the number and its divisor. As the process is repeated, the approximation of the square root gets closer and closer to the actual value. The Average Method is considered to be faster and more accurate than the Long Division Method, and it is commonly used in computer algorithms for finding square roots.
Here’s an example of how to find the square root of a number using the Average Method (also known as the Babylonian Method):
Find the square root of 144.
Step 1: Choose a starting number, x0. This number can be any positive number, but it’s usually a number that’s close to the actual square root. Let’s choose x0 = 12.
Step 2: Calculate the average of the number and the starting number. The formula for this is: x1 = (x0 + n/x0) / 2, where n is the number whose square root you’re trying to find (in this case, n = 144). So, x1 = (12 + 144/12) / 2 = 12 + 12 = 24/2 = 12.
Step 3: Repeat Step 2, using x1 as the new starting number and finding x2 = (x1 + n/x1) / 2 = (12 + 144/12) / 2 = 12 + 12 = 24/2 = 12.
Step 4: Repeat Step 2 and 3 until the desired accuracy is reached. In this case, since we want the square root to 2 decimal places, we’ll repeat until x2 and x1 are close enough. In this case, x2 = 12, which is close enough, so we can stop.
The square root of 144 is approximately equal to 12.
Find out what will be the square root of 6225
Newton-Raphson Method
The Newton-Raphson Method is a repetitious method for finding the square root of a number. It is based on the concept of finding the roots of an equation by linear approximation. The method starts by choosing a value that is close to the square root and then using that value to approximate the actual square root. The approximation is then refined in each iteration by using the formula derived from the linear approximation. The Newton-Raphson Method is considered to be one of the most efficient methods for finding the square root of a number, as it often requires fewer iterations to achieve the desired level of accuracy compared to other methods. It is also well-suited for solving complex square root problems and can be used to find the square roots of real and complex numbers. The method is widely used in engineering, science, and computer programming, and it is often implemented in computer algorithms for finding square roots.
To find the square root of a number using the Newton-Raphson Method, follow these steps:
- Start with an initial guess (e.g. x0 = 1).
- Compute the derivative of the function f(x) = x^2 – a, where a is the number whose square root you want to find.
- Use the formula: x1 = x0 – f(x0)/f'(x0) to find a better estimate of the root.
- Repeat the previous step until the difference between x1 and x0 is sufficiently small.
- The final value of x1 is an approximation of the square root of the number a.
Note: The Newton-Raphson Method is a fast and accurate method for finding the roots of a function, but it requires the derivative of the function to be known.
Binary Search Method
The Binary Search Method is a method for finding the square root of a number by dividing the search space into halves in each iteration. It is based on the idea of a binary search algorithm and uses the concept of bisection. The method starts by choosing two numbers that are close to the square root and then dividing the range between these two numbers into halves. In each iteration, the middle value is compared to the target number, and the search space is reduced based on the result of the comparison. The process is repeated until the desired level of precision is achieved. The Binary Search Method is considered to be a fast and efficient method for finding the square root of a number, as it reduces the search space by half in each iteration. It is well-suited for solving simple square root problems and can be used as a backup method in computer algorithms for finding square roots when other methods fail to provide accurate results.
To find the square root of a number using the Binary Search Method, follow these steps:
- Choose an interval that contains the square root of the number, such as [1, a].
- Find the midpoint of the interval, mid = (left + right)/2.
- If mid^2 is equal to the number, return mid as the square root.
- If mid^2 is less than the number, update the interval to [mid, right].
- If mid^2 is greater than the number, update the interval to [left, mid].
- Repeat steps 2 through 5 until the difference between left and right is small enough or a desired accuracy has been reached.
- The final value of mid is an approximation of the square root of the number.
Note: The Binary Search Method is a simple and efficient method for finding the square root of a number, but it requires the choice of a suitable interval that contains the root.
Using Built-in Functions or Libraries (such as sqrt() in math library of many programming languages)
Using Built-in Functions or Libraries is a convenient and efficient method for finding the square root of a number, especially in computer programming. Many programming languages have a built-in library or function that can be used to find the square root of a number with just a few lines of code. For example, in Python, the sqrt() function in the math library can be used to find the square root of a number. Similarly, in other programming languages such as Java, C++, and R, there are built-in functions or libraries that can be used to find the square root of a number. This method is considered to be one of the fastest and easiest methods for finding the square root of a number, as it eliminates the need for manual calculations and reduces the risk of errors. The results from these functions or libraries are often accurate to a high degree of precision, making them ideal for use in scientific and engineering applications.
Here’s an example of how to find the square root of a number using a built-in function in Python:
This will output:
