Three dots, minus one by twelve4 min read . Updated: 13 Feb 2014, 06:17 PM IST
Whatever inferences you might draw from a finitude of numbers, you should be prepared to toss them when you have an infinity of numbers
This column is about adding numbers. Something you’re all good at, no doubt. But this is about adding not just one or two numbers, not even five or six. I’m thinking of adding up many numbers. Specifically, this addition:
1 + 2 + 3 + 4 + 5...
Those three dots at the end mean this adding process goes on forever.
What happens when you keep adding those numbers, and never stop?
Well, let’s start by stopping. If you stopped after 2 terms, you’d have 1 + 2 = 3. After 3 terms, 6. After 4, 10. And so on. What about if you stopped after, let’s say, 238 terms?
There’s a lovely story about the German mathematical genius, Carl Friedrich Gauss. As a child, he had been naughty in class. To punish him, his teacher told him to add the numbers from 1 to 100. (the above sum, stopping at 100 terms). To the teacher’s amazement, Gauss produced the result in seconds. Not because he was fabulously swift at addition, but because he found an elegant way to consider the problem.
Here’s what Gauss’s teacher set him:
S = 1 + 2 + 3 + 4 + 5... + 99 + 100. Find S.
You will note that you can also write the sum in descending order:
S = 100 + 99 + 98... + 2 + 1.
Right, so now add these together, all 100 of the terms, pair by pair:
S + S = (1 + 100) + (2 + 99) + ... + (99 + 2) + (100 + 1)
Or: 2 x S = 101 + 101 + 101 + ... + 101 + 101 = 100 x 101.
And thus S = 50 x 101 = 5,050.
In effect, young Gauss found the formula to calculate this sum to any number of terms. That formula is why I can tell you that if you stopped after 238 terms, you’d have a total of 28,441. After 2,388 terms, 2,852,466. And so on. Getting larger all the time, of course.
The operative word in that paragraph, though, is stopped. It’s when you stop adding that you produce a sum and can announce it to the world.
But those little dots...they mean you don’t stop. What happens when you don’t stop adding? In other words, what happens when you go on till infinity? What would you have, if anything at all? Some unimaginably vast number, you think? Infinity, in fact?
A recent New York Times (NYT) article about just this problem has set off a degree of kerfluffle among mathematically-inclined laypersons of my acquaintance. It’s called In the End, It All Adds Up to -1/12. And as you might guess from the title, it makes the rather astounding claim that when you keep adding those numbers and never stop...what you have is the negative fraction -1/12.
That is: 1 + 2 + 3 + 4 + 5... = -1/12
Wow. Where did that come from? The NYT article references a video clip in which two British mathematicians explain how to derive this result.
There’s no magic, no hand-waving; it really is true. They prove it with some clever but really elementary number manipulations. And for now, the best answer to “where did that come from?" might just be...infinity. When you fiddle with infinity, odd things happen. Whatever inferences you might draw from a finitude of numbers—in this case, that the first so many positive integers add up to a certain figure—you should be prepared to toss them when you have an infinity of numbers.
Toss them, fine, but this -1/12 result is still astonishing and remarkable. More than that, it’s completely valid. It’s also one example of certain mathematical beasts called the Riemann zeta functions. These pop up in probability theory, in the study of prime numbers and in physics. They have also given mathematics one of its most enduring unsolved problems, the Riemann hypothesis. (Of that, perhaps another time, but know that if you prove the hypothesis, you’ll win a million dollars).
But there’s something interesting about that -1/12 that the British pair doesn’t even mention in the clip. The first-time ever the formula was derived and written was in a firm, slanting hand in a certain famous notebook. The owner of that slanting hand and the notebook then wrote a letter, briefly explaining the result, to a famous mathematician of his time. Knowing how remarkable his findings were, and how improbable they would seem to the famous man, he went on: “You will at once point out to me the lunatic asylum as my goal."
But of course the famous man, that earlier British mathematician G.H. Hardy, did not point his correspondent to any lunatic asylum. Instead, he recognized the mathematical brilliance in the notebook and...well, the rest is well-known history.
Hardy’s correspondent was, of course, the young genius from Madras, Srinivasa Ramanujan.
Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences. To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza-