# Square root of an integer

Given an integer x, find it’s square root. If x is not a perfect square, then return floor(√x).

Examples :

```Input: x = 4
Output: 2
Explanation:  The square root of 4 is 2.

Input: x = 11
Output: 3
Explanation:  The square root of 11 lies in between
3 and 4 so floor of the square root is 3.
```

## Recommended: Please solve it on “PRACTICE ” first, before moving on to the solution.

There can be many ways to solve this problem. For example Babylonian Method is one way.

Simple Approach: To find the floor of the square root, try with all-natural numbers starting from 1. Continue incrementing the number until the square of that number is greater than the given number.

• Algorithm:
1. Create a variable (counter) i and take care of some base cases, i.e when the given number is 0 or 1.
2. Run a loop until i*i <= n , where n is the given number. Increment i by 1.
3. The floor of the square root of the number is i – 1
• Implementation:

## C++

 `// A C++ program to find floor(sqrt(x) ` `#include ` `using` `namespace` `std; ` ` `  `// Returns floor of square root of x ` `int` `floorSqrt(``int` `x) ` `{ ` `    ``// Base cases ` `    ``if` `(x == 0 || x == 1) ` `    ``return` `x; ` ` `  `    ``// Staring from 1, try all numbers until ` `    ``// i*i is greater than or equal to x. ` `    ``int` `i = 1, result = 1; ` `    ``while` `(result <= x) ` `    ``{ ` `      ``i++; ` `      ``result = i * i; ` `    ``} ` `    ``return` `i - 1; ` `} ` ` `  `// Driver program ` `int` `main() ` `{ ` `    ``int` `x = 11; ` `    ``cout << floorSqrt(x) << endl; ` `    ``return` `0; ` `} `

## Java

 `// A Java program to find floor(sqrt(x)) ` ` `  `class` `GFG { ` `     `  `    ``// Returns floor of square root of x ` `    ``static` `int` `floorSqrt(``int` `x) ` `    ``{ ` `        ``// Base cases ` `        ``if` `(x == ``0` `|| x == ``1``) ` `            ``return` `x; ` ` `  `        ``// Staring from 1, try all numbers until ` `        ``// i*i is greater than or equal to x. ` `        ``int` `i = ``1``, result = ``1``; ` `         `  `        ``while` `(result <= x) { ` `            ``i++; ` `            ``result = i * i; ` `        ``} ` `        ``return` `i - ``1``; ` `    ``} ` ` `  `    ``// Driver program ` `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``int` `x = ``11``; ` `        ``System.out.print(floorSqrt(x)); ` `    ``} ` `} ` ` `  `// This code is contributed by Smitha Dinesh Semwal. `

## Python3

 `# Python3 program to find floor(sqrt(x) ` ` `  `# Returns floor of square root of x ` `def` `floorSqrt(x): ` ` `  `    ``# Base cases ` `    ``if` `(x ``=``=` `0` `or` `x ``=``=` `1``): ` `        ``return` `x ` ` `  `    ``# Staring from 1, try all numbers until ` `    ``# i*i is greater than or equal to x. ` `    ``i ``=` `1``; result ``=` `1` `    ``while` `(result <``=` `x): ` `     `  `        ``i ``+``=` `1` `        ``result ``=` `i ``*` `i ` `     `  `    ``return` `i ``-` `1` ` `  `# Driver Code ` `x ``=` `11` `print``(floorSqrt(x)) ` ` `  `# This code is contributed by Smitha Dinesh Semwal. `

## C#

 `// A C# program to  ` `// find floor(sqrt(x)) ` `using` `System; ` ` `  `class` `GFG ` `{ ` `    ``// Returns floor of ` `    ``// square root of x ` `    ``static` `int` `floorSqrt(``int` `x) ` `    ``{ ` `        ``// Base cases ` `        ``if` `(x == 0 || x == 1) ` `            ``return` `x; ` ` `  `        ``// Staring from 1, try all ` `        ``// numbers until i*i is  ` `        ``// greater than or equal to x. ` `        ``int` `i = 1, result = 1; ` `         `  `        ``while` `(result <= x)  ` `        ``{ ` `            ``i++; ` `            ``result = i * i; ` `        ``} ` `        ``return` `i - 1; ` `    ``} ` ` `  `    ``// Driver Code ` `    ``static` `public` `void` `Main () ` `    ``{ ` `        ``int` `x = 11; ` `        ``Console.WriteLine(floorSqrt(x)); ` `    ``} ` `} ` ` `  `// This code is contributed by ajit `

## PHP

 ` `

Output :

`3`
• Complexity Analysis:

• Time Complexity: O(√ n).
Only one traversal of the solution is needed, so the time complexity is O(√ n).
• Space Complexity: O(1).
Constant extra space is needed.

Thanks Fattepur Mahesh for suggesting this solution.

Better Approach: The idea is to find the largest integer i whose square is less than or equal to the given number. The idea is to use Binary Search to solve the problem. The values of i * i is monotonically increasing, so the problem can be solved using binary search.

• Algorithm:
1. Take care of some base cases, i.e when the given number is 0 or 1.
2. Create some variables, lowerbound l = 0, upperbound r = n, where n is the given number, mid and ans to store the answer.
3. Run a loop until l <= r , the search space vanishes
4. Check if the square of mid (mid = (l + r)/2 ) is less than or equal to n, If yes then search for a larger value in second half oF search space, i.e l = mid + 1, update ans = mid
5. Else if the square of mid is less than n then search for a smaller value in first half oF search space, i.e r = mid – 1
6. Print the value of answer ( ans)
• Implementation:

## C/C++

 `// A C++ program to find floor(sqrt(x) ` `#include ` `using` `namespace` `std; ` ` `  `// Returns floor of square root of x          ` `int` `floorSqrt(``int` `x)  ` `{     ` `    ``// Base cases ` `    ``if` `(x == 0 || x == 1)  ` `       ``return` `x; ` ` `  `    ``// Do Binary Search for floor(sqrt(x)) ` `    ``int` `start = 1, end = x, ans;    ` `    ``while` `(start <= end)  ` `    ``{         ` `        ``int` `mid = (start + end) / 2; ` ` `  `        ``// If x is a perfect square ` `        ``if` `(mid*mid == x) ` `            ``return` `mid; ` ` `  `        ``// Since we need floor, we update answer when mid*mid is  ` `        ``// smaller than x, and move closer to sqrt(x) ` `        ``if` `(mid*mid < x)  ` `        ``{ ` `            ``start = mid + 1; ` `            ``ans = mid; ` `        ``}  ` `        ``else` `// If mid*mid is greater than x ` `            ``end = mid-1;         ` `    ``} ` `    ``return` `ans; ` `} ` `  `  `// Driver program ` `int` `main()  ` `{      ` `    ``int` `x = 11; ` `    ``cout << floorSqrt(x) << endl; ` `    ``return` `0;    ` `} `

## Java

 `// A Java program to find floor(sqrt(x) ` `public` `class` `Test ` `{ ` `    ``public` `static` `int` `floorSqrt(``int` `x) ` `    ``{ ` `        ``// Base Cases ` `        ``if` `(x == ``0` `|| x == ``1``) ` `            ``return` `x; ` ` `  `        ``// Do Binary Search for floor(sqrt(x)) ` `        ``int` `start = ``1``, end = x, ans=``0``; ` `        ``while` `(start <= end) ` `        ``{ ` `            ``int` `mid = (start + end) / ``2``; ` ` `  `            ``// If x is a perfect square ` `            ``if` `(mid*mid == x) ` `                ``return` `mid; ` ` `  `            ``// Since we need floor, we update answer when mid*mid is ` `            ``// smaller than x, and move closer to sqrt(x) ` `            ``if` `(mid*mid < x) ` `            ``{ ` `                ``start = mid + ``1``; ` `                ``ans = mid; ` `            ``} ` `            ``else`   `// If mid*mid is greater than x ` `                ``end = mid-``1``; ` `        ``} ` `        ``return` `ans; ` `    ``} ` ` `  `    ``// Driver Method ` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``int` `x = ``11``; ` `        ``System.out.println(floorSqrt(x)); ` `    ``} ` `} ` `// Contributed by InnerPeace `

## Python3

 `# Python 3 program to find floor(sqrt(x) ` ` `  `# Returns floor of square root of x          ` `def` `floorSqrt(x) : ` ` `  `    ``# Base cases ` `    ``if` `(x ``=``=` `0` `or` `x ``=``=` `1``) : ` `        ``return` `x ` `  `  `    ``# Do Binary Search for floor(sqrt(x)) ` `    ``start ``=` `1` `    ``end ``=` `x    ` `    ``while` `(start <``=` `end) : ` `        ``mid ``=` `(start ``+` `end) ``/``/` `2` `         `  `        ``# If x is a perfect square ` `        ``if` `(mid``*``mid ``=``=` `x) : ` `            ``return` `mid ` `             `  `        ``# Since we need floor, we update  ` `        ``# answer when mid*mid is smaller ` `        ``# than x, and move closer to sqrt(x) ` `        ``if` `(mid ``*` `mid < x) : ` `            ``start ``=` `mid ``+` `1` `            ``ans ``=` `mid ` `             `  `        ``else` `: ` `             `  `            ``# If mid*mid is greater than x ` `            ``end ``=` `mid``-``1` `             `  `    ``return` `ans ` ` `  `# driver code     ` `x ``=` `11` `print``(floorSqrt(x)) ` `     `  `# This code is contributed by Nikita Tiwari. `

## C#

 `// A C# program to  ` `// find floor(sqrt(x) ` `using` `System; ` ` `  `class` `GFG ` `{ ` `    ``public` `static` `int` `floorSqrt(``int` `x) ` `    ``{ ` `        ``// Base Cases ` `        ``if` `(x == 0 || x == 1) ` `            ``return` `x; ` ` `  `        ``// Do Binary Search  ` `        ``// for floor(sqrt(x)) ` `        ``int` `start = 1, end = x, ans = 0; ` `        ``while` `(start <= end) ` `        ``{ ` `            ``int` `mid = (start + end) / 2; ` ` `  `            ``// If x is a  ` `            ``// perfect square ` `            ``if` `(mid * mid == x) ` `                ``return` `mid; ` ` `  `            ``// Since we need floor, we  ` `            ``// update answer when mid *  ` `            ``// mid is smaller than x,  ` `            ``// and move closer to sqrt(x) ` `            ``if` `(mid * mid < x) ` `            ``{ ` `                ``start = mid + 1; ` `                ``ans = mid; ` `            ``} ` `             `  `            ``// If mid*mid is  ` `            ``// greater than x ` `            ``else`  `                ``end = mid-1; ` `        ``} ` `        ``return` `ans; ` `    ``} ` ` `  `    ``// Driver Code ` `    ``static` `public` `void` `Main () ` `    ``{ ` `        ``int` `x = 11; ` `        ``Console.WriteLine(floorSqrt(x)); ` `    ``} ` `} ` ` `  `// This code is Contributed by m_kit `

## PHP

 ` `

Output :

`3`
• Complexity Analysis:

• Time complexity: O(log n).
The time complexity of binary search is O(log n).
• Space Complexity: O(1).
Constant extra space is needed.

Thanks to Gaurav Ahirwar for suggesting above method.

Note: The Binary Search can be further optimized to start with ‘start’ = 0 and ‘end’ = x/2. Floor of square root of x cannot be more than x/2 when x > 1.

Thanks to vinit for suggesting above optimization.

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