# Binary Search for Rational Numbers without using floating point arithmetic

A rational is represented as p/qb, for example 2/3. Given a sorted array of rational numbers, how to search an element using Binary Search. Use of floating point arithmetic is not allowed.

Example:

Input: arr[] = {1/5, 2/3, 3/2, 13/2} x = 3/2 Output: Found at index 2

**We strongly recommend you to minimize your browser and try this yourself first.**

To compare two rational numbers p/q and r/s, we can compare p*s with q*r.

`// C program for Binary Search for Rationalnal Numbers ` `// without using floating point arithmetic ` `#include <stdio.h> ` ` ` `struct` `Rational ` `{ ` ` ` `int` `p; ` ` ` `int` `q; ` `}; ` ` ` `// Utility function to compare two Rationalnal numbers ` `// 'a' and 'b'. It returns ` `// 0 --> When 'a' and 'b' are same ` `// 1 --> When 'a' is greater ` `//-1 --> When 'b' is greate ` `int` `compare(` `struct` `Rational a, ` `struct` `Rational b) ` `{ ` ` ` `// If a/b == c/d then a*d = b*c: ` ` ` `// method to ignore division ` ` ` `if` `(a.p * b.q == a.q * b.p) ` ` ` `return` `0; ` ` ` `if` `(a.p * b.q > a.q * b.p) ` ` ` `return` `1; ` ` ` `return` `-1; ` `} ` ` ` `// Returns index of x in arr[l..r] if it is present, else ` `// returns -1. It mainly uses Binary Search. ` `int` `binarySearch(` `struct` `Rational arr[], ` `int` `l, ` `int` `r, ` ` ` `struct` `Rational x) ` `{ ` ` ` `if` `(r >= l) ` ` ` `{ ` ` ` `int` `mid = l + (r - l)/2; ` ` ` ` ` `// If the element is present at the middle itself ` ` ` `if` `(compare(arr[mid], x) == 0) ` `return` `mid; ` ` ` ` ` `// If element is smaller than mid, then it can ` ` ` `// only be present in left subarray ` ` ` `if` `(compare(arr[mid], x) > 0) ` ` ` `return` `binarySearch(arr, l, mid-1, x); ` ` ` ` ` `// Else the element can only be present in right ` ` ` `// subarray ` ` ` `return` `binarySearch(arr, mid+1, r, x); ` ` ` `} ` ` ` ` ` `return` `-1; ` `} ` ` ` `// Driver method ` `int` `main() ` `{ ` ` ` `struct` `Rational arr[] = {{1, 5}, {2, 3}, {3, 2}, {13, 2}}; ` ` ` `struct` `Rational x = {3, 2}; ` ` ` `int` `n = ` `sizeof` `(arr)/` `sizeof` `(arr[0]); ` ` ` `printf` `(` `"Element found at index %d"` `, ` ` ` `binarySearch(arr, 0, n-1, x)); ` `} ` |

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Output:

Element found at index 2

Thanks to Utkarsh Trivedi for suggesting above solution.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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