Today is Easter Sunday, the most sacred and second most widely observed holy day in the Christian calendar, celebrating the resurrection of Jesus. Friday was Good Friday, the remembrance of the crucifixion. Thus after spending three days buried, Jesus rose from the dead.
Wait, three days? Let’s see… Friday night to Saturday is a couple of hours, Saturday is a full day, and Sunday morning is a few more hours on top of that. So a day and a half, then. But of course all the liturgy and ceremony talks about three days.
This is because the ancients counted time as opposed to measuring it. (The great pop-math book Prime Obsession discusses this in detail) Some modern cultures including our own in some contexts do the same thing. You might work a five-day week, despite only working 40/24 = 1.67 whole days.
This difference between numbers treated as discrete countable entities and numbers as smooth measurements of continuously varying quantities informs large sections of the math used in physics. Let’s grab a function defined as a sum, like this:
In other words, f(3) = log(1) + log(2) + log(3), and so on for other values of N.
This kind of problem is very tricky in general for large N. Maybe you don’t want to add up a few trillion trillion terms, should N happen to be huge. Maybe the thing we’re summing doesn’t really have any tricks that let us sum it exactly using shortcuts – which is generally the case. But if we quit thinking of this as a discrete sum and as more of a continuous measurement we can make progress. Let’s graph it with Mathematica.
The area of each rectangle is equal to one term in the series. The total area of the rectangles is the sum. But we also know that the area under the curve defined by the continuous function f(x) = log(x) is not too far from being the same thing. After all, an integral is just the limit of a sum as the width of the rectangles approaches zero. Nothing prohibits us thinking about that process in reverse. So if N is large enough so that the individual imprecision between each rectangle and its associated section of curve are small, we can say that
Now we can’t be guaranteed that the sum and the integral actually become the same with large N, only that their percent difference goes to zero. And that’s pretty much all we want as physicists.
How good is the approximation? Well, for N = 100 the sum is 363.739. For N = 100 the integral approximation is 361.517. Not half bad, given that the effort involved in evaluating the expression for the integral (it’s 1 – N + N log(N), easy to find if you’ve had calculus) for your specific N. Much more easy than laboriously evaluating 100 or more logarithms in your calculator and adding them up.
You might even call it a salvation of sorts.