Why array index starts from zero ?

Prerequisite : Pointers in C/C++ 

THERE CAN BE MANY REASONS, BUT HERE ARE TWO REASONS:

Reason 1 :
Consider int arr[100]. The answer lies in the fact how the compiler interprets arr[i] ( 0<=i<100). 
arr[i] is interpreted as *(arr + i). Now, arr is the address of the array or address of 0th index element of the array. So, address of next element in the array is arr + 1 (because elements in the array are stored in consecutive memory locations), further address of next location is arr + 2 and so on. Going with the above arguments, arr + i mean the address at i distance away from the starting element of the array. Therefore, going by this definition, i will be zero for the starting element of the array because the starting element is at 0 distance away from the starting element of the array. To fit this definition of arr[i], indexing of array starts from 0. 
 

#include <iostream>
using namespace std;

int main()
{
    int arr[] = { 1, 2, 3, 4 };

    // Below two statements mean same thing
    cout << *(arr + 1) << " ";
    cout << arr[1] << " ";

    return 0;
}

public class Main {
    public static void main(String[] args)
    {
        int[] arr = { 1, 2, 3, 4 };

        System.out.print(arr[1] + " ");
        System.out.print(arr[1] + " ");
    }
}

// This code is contributed by Shivam Gupta

# code
print("GFG")
arr = [1, 2, 3, 4]

# Below two statements mean the same thing
print(arr[1], end=" ")
print(arr[1], end=" ")

// Main function
function main() {
    const arr = [1, 2, 3, 4];

    // Accessing the second element of the array using array subscript notation
    console.log(arr[1] + " ");

    // Accessing the second element of the array using pointer arithmetic (not valid in JavaScript)
    console.log(arr[1] + " ");
}

// Call the main function to execute the example usage
main();

2 2 

The conclusion is, we need random access in the array. To provide random access, compilers use pointer arithmetic to reach i-th element.

Reason 2 :

Modern languages, especially C++ use row-major ordering for storing two-dimensional arrays.

Let us assume a 2D array and write a row-major formula  with two different approaches:

1. array indices starting from 1
2. array indices starting from 0

let the 2D array be arr[m][n] of type int

let &arr be "address"

case 1 ( array indices start from 1 ) :

         &( arr[i][j] ) = address + [ ( i-1 )*n + ( j-1 ) ]*( sizeof(int) ) ] so here we are performing 6 operations

case 2 ( array indices start from 0 ) :

        &( arr[i][j] ) = address + [ ( i )*n + ( j ) ]*( sizeof(int) ) ] and here we are performing only 4 operations

So we see here that we are performing 2 less operations when we store 2D arrays and obtaining an element's address. This looks like it doesn't make sense but it does! While handling with huge size data this may improved performance and speed. case 1 may look user-friendly but case 2 is more efficient. That's why most languages like C++, PYTHON, JAVA use arrays starting with index 0 and rarely languages like Lua arrays starting with index 1.