Math is Really Weird: On Strange Sums and Counterintuitive Results
Whenever you add a finite integer to another finite integer, you always get a sum which is, itself, a finite integer. This, by itself, is not very shocking. When you add 1 to 1, you get 2. When you add 5 and -9, you get -4. When you add 0 and 299,792,458, you get 299,792,458. This is all rather unsurprising.
However, math can get weird once you start adding up an infinite collection of numbers. Take Zeno’s Dichotomy Paradox, for example. Numerically, we can represent this problem as an infinite summation: Even though we are adding up an infinite quantity of numbers, we arrive at a finite value– in this case, . Arguably the most famous philosopher in history, Aristotle, would have vehemently objected to this formulation– and, in fact, did object rather loudly in his book Physics, when discussing this particular paradox. However, it has been over three centuries, now, since mathematicians would have found this problem to be controversial; and, in fact, similar cases of infinite summation form the entire basis of integral calculus. High schoolers are introduced to these concepts in their Pre-Calculus classes, nowadays, and you might even remember evaluating some of these limits of convergent sums from your own schoolwork.
But math can get far stranger, still. One of the most peculiar things in all mathematics occurs when you attempt to sum all of the Natural numbers. As absolutely insane as this might sound, today I’m going to demonstrate for you that .
Before we look at that, though, let’s start by looking at a different summation, which is also fairly strange: Grandi’s series. An Italian priest and mathematician named Guido Grandi wrote a book called On Infinite Infinities. In his book, he discussed a number of different peculiar series at length, some of which were convergent (like Zeno’s Dichotomy paradox, above) and others which were divergent. If you remember from those aforementioned Pre-Calculus classes, a divergent function is one which does not have a limit that tends towards a single, finite value. Usually, in pre-Calc, this meant functions which tended towards infinity or negative infinity; but there are other types of divergent functions, as well. The series which is now referred to as “Grandi’s series” was one of the problems discussed at length by the Italian scholar, and it takes the form 1-1+1-1+1-1+…
This is a divergent series because the partial sums do not tend toward a single, finite value. Whenever you take the nth partial sum, you’ll get a value of 1 if n is odd and a value of 0 if n is even. The function bounces back and forth between 0 and 1, never settling any closer to either value. However, mathematicians still wanted to know if this sum could be assigned a definite value. Some argued that it should be assigned a value of 1, others argued that it deserved a value of 0, and still others argued for a stranger result.
Let us assume that there is some number G such that . We still don’t know how to evaluate that summation, yet, but if G is a well-defined number (as we have assumed), then we are allowed to perform mathematical operations on that number. So, let’s ask the question, “What is ?” We can see from our earlier definition that . However, because of the Distributive Property of Multiplication, we know that . Evaluating this operation, a pattern emerges, ; but that is just our original series, G, again! Therefore, we can conclude that , and all it takes is some basic algebra to then show that .
We’ve arrived at our first strange result of the day: .
Oh, but it does not stop there, my friends. In fact, we’re just getting started. Now, let’s look at another series, which is related to Grandi’s series. I’m going to call this one F, and we’re going to define F as Now, we again are assuming that F is a definite number, which allows us to perform numerical operations on F. This time, we’re going to multiply F by 2, giving us . Now, at this point, you may be wondering why I would want to look at something so very obvious as that. The answer is that I’m going to use this opportunity to get a little bit clever.
You see, if we pull the first term out of the summation, we can very easily note that . This means that . Here’s where that cleverness comes in. Thanks to the Commutative Property of Addition, we can combine these two different summation operations into a single one, which we can then simplify. So, we see that . Substituting this back into our equation for 2F yields ; but wait a moment– that looks familiar! We dealt with the right-hand-side of that equation when we were discussing Grandi’s series, above! So, we’ve just demonstrated that and since we know what G is, it’s rather trivial to see that .
We’ve now arrived at our second strange result: .
We now have all the tools necessary to demonstrate something so strange that many people simply refuse to believe it. It’s so very odd that people think it cannot possibly be correct; that there must be some trickery or foul play at hand. However, I assure you, no such trickery is being employed. We are now going to look at the sum of all Natural numbers,
Again, I am operating under the assumption that S is a definite value, which allows me to perform numerical operations on S. This time, I’m going to subtract F from S in order to find a way to evaluate S. So, let’s look at . I’m once again going to use the Commutative Property to combine the two summations into one. This gives us . This can be simplified even further, since it is fairly obvious that whenever n is even, the expression will equal 0; and whenever n is odd, that expression will equal 2. So, we can see that a pattern emerges: . Since those zeroes are extraneous to the equation (thanks to the Identity Property), and the rest of the terms are all even multiples of 4, we can rewrite this as . However, we should recognize that summation! That’s just S! So, now we’ve shown that , and we know the value of F from our previous calculations. Substituting that in, we get . A little bit of simple algebra, and we can solve this revealing that .
So, now we’ve arrived at the strangest result of the day: .
“Preposterous!” you might exclaim. “Incredulous! You’ve fouled something up!” However, I assure you quite sincerely that I have not. I made only three assumptions: that G, F, and S were definite numbers. Beyond that, I only utilized the Commutative Property, the Distributive Property, the Identity Property, and basic arithmetic and algebra. No fancy tricks or crazy axioms needed to be employed. This really was just basic math, despite the strangeness of it all. All three of these fantastic results can also be proved by other methods, and they are wholly consistent with the rest of mathematics. What’s more, these crazy calculations actually have physical ramifications in the real world– the fact that the sum of all natural numbers is is used by particle physicists to calculate Casimir energy, for example.
Sometimes, our intuitions fail us. We see results like these and think that they must be false because they simply do not feel right. However, our intuitions and our feelings do not govern mathematics. Perhaps the most beautiful thing about math is that it manages to surprise us with such oddity. This wonderful strangeness can spur us on into newer and more fascinating discoveries, precisely because the world of mathematics is not always what we would expect it to be.