If you are a budding Python programmer and looking to learn how to calculate square roots in Python without using math functions, you have come to the right place. In this article, we will explore various methods to calculate the square roots of a number without the use of built-in functions such as math.sqrt().

Before we dive into the different methods, it is important to understand what square roots are and why they are important. A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, as 5 multiplied by 5 equals 25.

Calculating square roots is a common requirement in many programming tasks, and Python provides several built-in functions for this. However, it can be beneficial to know alternative methods for calculating square roots without relying on these built-in functions.

Table of Contents

## Method 1: Using Exponents

One of the simplest methods to calculate a square root in Python is by using exponents. The square root of a number can be expressed as that number raised to the power of 0.5. Here’s how you can implement this method:

```
number = int(input("Enter a number: "))
sqrt = number ** 0.5
print("The square root of", number, "is", sqrt)
```

In this method, we take the input number from the user and raise it to the power of 0.5 using the exponent operator (**). The result is stored in a variable called sqrt, which is then displayed using the print() function.

## Method 2: Using Newton’s Method

Another popular method to calculate square roots is by using Newton’s method. This method involves repeatedly calculating the average of a number and its reciprocal until the difference between the two becomes sufficiently small. Here’s how you can implement this method:

```
number = int(input("Enter a number: "))
guess = number / 2
while abs(guess * guess - number) > 0.0001:
guess = (guess + number / guess) / 2
print("The square root of", number, "is", guess)
```

In this method, we first take the input number from the user and initialize a variable called guess to half of the number. We then use a while loop to repeatedly calculate the average of guess and number divided by guess until the difference between the square of guess and the input number becomes less than 0.0001. The final value of guess is then displayed using the print() function.

## Method 3: Using Binary Search

Binary search is another method that can be used to calculate square roots. This method involves repeatedly dividing the search interval in half until the desired accuracy is achieved. Here’s how you can implement this method:

```
number = int(input("Enter a number: "))
epsilon = 0.0001
low = 0
high = number
guess = (low + high) / 2
while abs(guess * guess - number) >= epsilon:
if guess * guess < number:
low = guess
else:
high = guess
guess = (low + high) / 2
print("The square root of", number, "is", guess)
```

In this method, we first take the input number from the user and initialize variables called epsilon, low, and high. We then initialize a variable called guess to the average of low and high. We use a while loop to repeatedly divide the search interval in half until the difference between the square of guess and the input number becomes less than epsilon. Finally, the value of guess is displayed using the print() function.

## Method 4: Using Logarithms

One more method to calculate square roots is by using logarithms. This method involves taking the logarithm of the input number, dividing it by 2, and then taking the antilogarithm of the result. Here’s how you can implement this method:

```
import math
number = int(input("Enter a number: "))
sqrt = math.exp(math.log(number) / 2)
print("The square root of", number, "is", sqrt)
```

In this method, we first import the math module to use the log() and exp() functions. We then take the input number from the user and calculate its logarithm using the log() function. We then divide the result by 2 and take the antilogarithm of the result using the exp() function. Finally, the value of sqrt is displayed using the print() function.

## Final Thoughts

Calculating square roots is a fundamental mathematical operation that is frequently used in programming tasks. Python provides several built-in functions for this, but it can be beneficial to know alternative methods for calculating square roots without relying on these functions.

In this article, we explored four different methods for calculating square roots without using built-in functions: using exponents, Newton’s method, binary search, and logarithms. Each of these methods has its advantages and disadvantages, and the best method to use depends on the specific requirements of the task at hand.

By understanding these methods, you can not only become a better Python programmer but also gain a deeper understanding of the underlying mathematical concepts.