This might be one of those posts that have no purpose because those who will read it already know and those who won’t read it will never need to know. But since One Over Epsilon is principally written for my daughters, I have to let my brain spill out a bit here, since I cannot predict what Kezia and Sylvie will find amusing or crazy about their Dad when they grow older (and they do enjoy crazy).
What I’m about to show you may seem miraculous. If it does, then good! You should be impressed. One thing that distinguishes a normal human from a good mathematician is that the normal person sees such results as `obvious’ or `uninteresting and mundane’, whereas the good mathematician is so idiotic that these seemingly simple results and theorems appear miraculous and they find them irresistible to study and contemplate.
Infinite series are the big game here. Finite sums are kind of boring, although occur all over the place in science and applied mathematics. But it is with infinite sums (infinitely many terms in the sum, not necessarily an infinite valued total sum) that mathematics becomes richer and awesome in power.
Preliminary: you recall what means? It is . Similarly, , and once more, . And so on. These are called powers of . The superscript is called the “exponent”. Exponentiation of a number like “” here is thus just a big word for “repeated multiplication”. I tell you this because in a second I’m going to show you how to sum infinitely many exponents of a number , and I will do this without knowing the value of . I hope you are preliminarily impressed!
There are at least three truly astounding things about summing an infinite bunch of numbers.
(1). You can write the sum as a finite expression even when you do not know the numbers in the sum. (This is not always possible, but it works for geometric series.) Here’s what I mean,
Think of a number . Then contemplate the infinite sum
Even though you do not tell me the value of , I can tell you precisely what the infinite sum is equal to, and my answer is that it equals
Pretty amazing huh? But the cool thing is not simply knowing and appreciating this fact, the cool thing is to try to prove it yourself! So go ahead. You probably want to see an example, so here’s one with
Now that’s ridiculous! On the left we have an infinite sum of increasingly larger positive numbers which should diverge to +∞. On the right we equate this to negative one-half. There’s gotta be something wrong here, right?
Actually, this is not a fallacy, nor pathological, it is a deep (and to you, no doubt mysterious) thing about diverging series. But I’m not going to tell you what it means, since that’d take way too long. I’ll just leave you with it as a tease. (If you must know something of it, then just try thinking of a diverging infinite sum like this as a geometric process in two-dimensions, which the partial series sums of positive numbers is only a one-dimensional shadow of, so you cannot see the whole thing by just looking at the sum as it blows up, and if you could see the entire sum geometrically then you’d see how it can extend out to infinity and than wrap around to a small negative value. How? Because in 2D infinity is not the end of the line, in 2D space ∞ is a circle with infinite radius. Ok, that’s enough mystery for you to feed on for now. Yes, that’s right, mystery is important in mathematics, it’s where everything fresh and new springs from. It’s far more important than all the known textbook results in the world.)
The normal high school teaching tells us the above formula for summing an infinite series is only valid when . So, we really should have used as an example, then I would not have exposed you to the mystery of the diverging infinite series that sums to a negative number. So here’s the simpler example,
So this infinite sum adds up to one-and-a half. And you can actually check this partially on a calculator. Just add up the first six or seven terms, and you’ll get close to 1.499, because the fractions in the sum are getting really small by the time you get to . (There’s no such hope for checking the previous divergent sum when in the same way, since your calculator can never “get all the way out to infinity” which is where the sum finally flips around to the negative 1/2 value.
I will try not to clutter this blog with formulae, even though it is full of mathematics (mathematics is language, not just formulae and Greek-Hindu-Arabic-Latin symbology). But I have to show you a proof of the formula, since that’s where the beauty is found.
(2). You can prove the above result without summing the numbers. All you need is to state a couple of properties that all methods of summation should obey. This really blew my mind when I learned about it! You can sum an infinite series like the one above without using any technique of computation, without using any summation method. You gotta be crazy huh?
Here’s how: you just ask yourself, “what should all proper summation methods (algorithms) have in common?” And you might think of a couple of things:
(Rule A.) If “GS” is short for “General summation method”, and “” and “” are both themselves sums of numbers (possibly infinite sums) then we’d expect
right? That seems reasonable, since the computation of a sum of sums should equal the sum of the computations of the sums. Haha. Really, I’m not trying to confuse you. If I tell you this is all very simple then maybe you’ll relax. The way to think about maths in a relaxing manner is to remind yourself that whatever you think the meaning is, then it is probably what the meaning is, i.e., don’t sweat, you probably have the right idea even if you are not certain. (Kids at school routinely confound this piece of wisdom! But I think that’s only because previously they were told what to learn, and were not told how to learn for themselves.)
BTW, it’s really deep and interesting that the above general property of summation is the only valid way to split sums. If a sum is infinite then it is invalid to rearrange the terms wining (inside) the sum. In fact, astoundingly (again!) it is also invalid to group (associate using brackets) parts of an infinite sum. This is clear when you consider a pathological sum like,
If you rearrange the terms you can make it all=0, like this,
which is equal to zero, clearly, right? But what if you write,
which is equal to one, clearly, right? Both these computations cannot be true, or the universe will vanish in a puff of logic (Doug Adams).
The resolution of this paradox is simple: you cannot using grouping to rearrange an infinite sum. That is, associativity (i.e, bracketing terms together into sub-sums) is not a valid general summation rule, even though it is a perfectly good rule for finite sums.
(Rule B.) Another good rule for any summation algorithm is, if “” is any number, “GS” a general summation procedure as before, and “” a (possibly infinite) sum of numbers as before, then
in other words, if after summing up, you then multiply the result by a number “” then this will be the same value as you get by using the summing procedure on the entire sum multiplied by .
Now, almost insanely beneficent, these two rules (a) and (b) are enough for us to compute exactly the infinite sum,
Here’s how: say to yourself, let the sum of all these powers of be ““. Then suppose we have found a general summation algorithm, which miraculously does the entire infinite sum in a finite span of time! Yep, that’s right, in mathematics you are allowed to dream! Call this unknown procedure “GS”.
(The infinite sum above)
By Rule A. this must be
by Rule B we can then say,
but this is exactly,
your content here we have used the unknown general procedure “GS” to get rid of infinitely many terms! The rest is basic high school algebra,
Or as a mathematician would say, “Q.E.D” (= “quo erat demonstradum” = “which is the thing we wanted to prove”.)
To save you the embarrassment I’ll respond for you. So, “WTF!!!” How is this possible? Was there some trickery, some slight of mind?
Nope! All the above maths is logically impeccable. We truly have summed infinitely many unknown numbers using an unknown summation algorithm!
Please understand: there is no point in writing this post so that you can know go around city streets or wander desert wilderness to find someone and sum their infinite series for them. What’s the point then? It’s simple. Just to blow your mind up a bit. Expand it. Inflate it. But gently. And to show you that mathematicians can see beauty too, just like an artist.
I mean, don’t you think what I’ve just described is wonderful? Maybe I did not describe it very artistically or with pretty prose, but the abstract ideas are beautiful, are they not?
On the other hand, maybe you are like a normal person, and you think nothing above is surprising or miraculous, and it’s all pretty plain and mundane. If so, then that’ll be healthy, since you won’t then go through life thinking all sorts of weird shit is “astounding” and “miraculous” and “abstractly beautiful”, and you may even live your life with better obsessions, like looking after your children and eating heartily, and listening to nature, and enjoying your vacations instead of furiously trying to grok obscure mathematics formulae in your spare time.
So yeah, if you find mathematics hard, then it’s probably because you are not stupid enough. (Half serious ’bout that!)
That was supposed to be the end of this post. But there’s a paradox about teaching and education I thought about when writing it. A satisfactory teacher can impart knowledge and wisdom to students by instruction. A good teacher can infuse students with knowledge and wise sensibilities through mere example and good deeds, which is a graceful art because it also gives students a sense of freedom and independence—they hardly realise they are being “schooled”. I do not agree with traditional schools, but this learning from example is the best kind of schooling.
Then there’s the paradox. A great teacher does not teach at all, but instead just reveals the universe like an open book. This is such a difficult art to perfect. Most teachers (professionals that is), whom I know, simply cannot relinquish the need to instruct and display their knowledge and wisdom. (The whole “teaching is a performance career” model.) I think there is too much ego involved in being a traditional teacher. Also, if a teacher does not instruct and show that they have expert knowledge, there is a prejudice that their students will disrespect them. But this is not true in my experience. If you really have sound expert knowledge and a smattering of wisdom, then students will automatically see it, without you (the teacher) needing to demonstrate your powers. You just have to trust your students, and show them kindnesses.
So you should be wanting to just open the minds of your students, treat them humanely, allow them great freedom, and show, through the invisible force of quiet example, the wonders of the universe. If you cannot do this then undeniably you are teaching the wrong subject (sorry to tell you that, but it’s only my humble opinion).
Be kind and gentle to all people. But avoid the heart-breakers. Treasure the companionship of the wise. I did not invent these pearls, I borrowed them from a mystic seer who made all the laws of spirituality plain and crystal clear for all to see.