In mathematics, big-O notation is a symbolism used to describe and compare the limiting behavior of a function. A function's limiting behavior is how the function acts as it trends towards a particular value and in big-O notation it is usually as it trends towards infinity. In short, big-O notation is used to describe the growth or decline of a function, usually with respect to another function.

[NOTE: x^2 is equivalent to x * x or 'x-squared']

For example we say that x = O(x^2) for all x > 1 or in other words, x^2 is an upper bound on x and therefore it grows faster. The symbol of a claim like x = O(x^2) for all x > n can be substituted with x <= x^2 for all x > n where n is the minimum number that satisfies the claim, in this case 1. Effectively, we say that a function f(x) that is O(g(x)) grows slower than g(x) does.

Comparitively, in computer science and software development we can use big-O notation in order to describe the time complexity or efficiency of algorithms Specifically when using big-O notation we are describing the efficiency of the algorithm with respect to an input: n, usually as n approaches infinity. When examining algorithms, we generally want a lower time complexity, and ideally a time complexity of O(1) which is constant time. Through the comparison and analysis of algorithms we are able to create more efficient applications.

As an example, we can examine the time complexity of the bubble sort algorithm and express it using big-O notation.

// Function to implement bubble sort
void bubble_sort(int array[], int n)
{
    // Here n is the number of elements in array
    int temp;
    for(int i = 0; i < n-1; i++)
    {
        // Last i elements are already in place
        for(int j = 0; j < n-i-1; j++)
        {
            if (array[j] > array[j+1])
            {
                // swap elements at index j and j+1
                temp = array[j];
                array[j] = array[j+1];
                array[j+1] = temp;
            }
        }
    }
}

Looking at this code, we can see that in the best case scenario where the array is already sorted, the program will only make n comparisons as no swaps will occur. Therefore we can say that the best case time complexity of bubble sort is O(n).

Examining the worst case scenario where the array is in reverse order, the first iteration will make n comparisons while the next will have to make n - 1 comparisons and so on until only 1 comparison must be made. The big-O notation for this case is therefore n * [(n - 1) / 2] which = 0.5n^2 - 0.5n = O(n^2) as the n^2 term dominates the function which allows us to ignore the other term in the function.

We can confirm this analysis using this handy big-O cheat sheet that features the big-O time complexity of many commonly used data structures and algorithms

It is very apparent that while for small use cases this time complexity might be alright, at a large scale bubble sort is simply not a good solution for sorting. This is the power of big-O notation: it allows developers to easily see the potential bottlenecks of their application, and take steps to make these more scalable.

For more information on why big-O notation and algorithm analysis is important visit this video challenge!