Find Square Root of a Number in Java Without Using Math.sqrt() – Explanation, Logic, Examples, and Program

Find Square Root of a Number in Java Without Using Math.sqrt()

The square root of a number is a value which, when multiplied by itself, gives the original number. In Java, we normally use the built-in Math.sqrt() function — but interviewers often ask you to implement your own logic without using it.

A powerful and efficient way to do this is by using the Binary Search Method, which works for both perfect squares and non-perfect squares.

Binary Search Approach

Binary search helps us approximate the square root by repeatedly narrowing the range of possible answers.

Steps:

• Start with a range: low = 0, high = number
• Find the mid: (low + high) / 2
• If mid × mid is close to the number → mid is the square root
• If mid × mid is greater than the number → reduce the range
• Else → increase the range

We continue until we reach the required precision, such as 3, 4, or 5 decimal places.

Example

Find √40 (up to 3 decimal places)

40 lies between 36 (6²) and 49 (7²), so √40 is between 6 and 7.

• Mid = 6.5 → 6.5² = 42.25 (too high)
• Mid = 6.25 → 6.25² = 39.06 (too low)
• Mid = 6.30 → 6.30² = 39.69 (too low)
• Mid = 6.32 → 6.32² = 39.93 (too low)
• Mid = 6.324 → 6.324² = 39.98 (almost there)

So, the approximate square root of 40 is around 6.324.

Java Program (Binary Search Method)

public class SquareRootWithoutSqrt {

    // Function to find square root with binary search
    static double squareRoot(double num, double precision) {
        double low = 0;
        double high = num;
        double mid = 0;

        // Binary search for integer part
        while (high - low > precision) {
            mid = (low + high) / 2;

            if (mid * mid > num) {
                high = mid;
            } else {
                low = mid;
            }
        }

        return (low + high) / 2;
    }

    public static void main(String[] args) {
        double num = 40;
        double precision = 0.0001; // 4 decimal places

        double result = squareRoot(num, precision);
        System.out.println("Square root of " + num + " is: " + result);
    }
}

How Accurate Is This?

The precision value determines how close the result is to the actual square root. For example:

• 0.01 → accuracy up to 2 decimals
• 0.0001 → accuracy up to 4 decimals
• 0.000001 → accuracy up to 6 decimals

Binary search guarantees fast and accurate results.

Practice Challenges

1. Modify the program to print the steps of binary search as it narrows down the value.
2. Write a function that calculates both floor and ceil square roots without using Math.sqrt().
3. Find square roots for all numbers from 1 to 100 and print them with 3 decimal precision.