Chapter 16: Not -1/12

I flicked my cigarette stub over the window and headed for my abandoned students. I found them still, like I left them, and Melinda seated at her desk.

'Okay, so where were we?'

'The sum of the natural numbers'. I looked up to where the sound had come from and the culprit was this boy whose name I can't remember, for the life of me. He looked like a miniature wrestler from the eighties, all bulked-up around his biceps, and slimy, long blonde hair, falling to his shoulders. 'But what Melinda did is not what you taught us'.

I turned to look at the blackboard, all brightened up by the white scribblings of chalk, and even wondered how Melinda could've filled all this in the space of ten minutes, which is the time I spent talking to Ben. I scanned all the way to the end result and there it was, written white on black: negative one-twelfth.

'Melinda, how did you get to that?'

'Adding all natural numbers', replied the girl.

Yeah, I was seeing that, but that's not the result you get. One plus two plus three plus four and so forth to 'n', where n is a natural number is 'n' multiplied with its consecutive ('n'plus one) and divided by two. Basically, it looks like this:

1+2+3+4.......+n = n(n+1)/2

So if you have n=5 then this would be 1+2+3+4+5= 5(5+1)/2= 15.

For n=100 this would translate as 1+2+3+4+.....100= 100(100+1)/2 = 5050 and so forth. What Melinda did is get to the same result an Indian mathematician called Ramanujan arrived at in the 1920s, described as one of the most remarkable results in mathematics at that time. When you learn in school that 1+2+3+...n =n(n+1)/2, it means that the sum of the natural numbers is a divergent series; divergent meaning the series is infinite and doesn't converge to any finite sum. Basically, if you add all natural numbers in succession to infinity, the sum will be infinite. Ramanujan proved, however, that the series could be convergent. It is a manipulated form of mathematics which is now used in quantum theory and other fields of mathematics and physics. Ramanajun managed show that the sum of all natural numbers is minus one-twelfth and not infinite, and Melinda had done the same right in front of my eyes.

'But Melinda; the sum of natural numbers is 'n' multiplied with 'n plus one' and divided by two.

'Yeah, sometimes. But sometimes it's not. Sometimes it's negative minus one-twelfth.'

'How do you know about negatives?'

The girl shrugged. I asked her if she could go again and describe the entire thought process to the rest of the class. She then picked up piece of chalk and went through all the calculations one more time. I was like a referee at a tennis match. I was sharp-focused on both Melinda and the faces of the students. Most of them were fidgeting, clearly unaware of the immensity of the moment, and a few others were trying hard to concentrate but lost focus half-way through.

'Melinda, did your parents help you with this?'

'No.'

'Did you see this on the internet, or read this in a book?"

'We don't have internet at home. My mother says it worries us too much. No, I just know it is minus one twelfth.

'You said that only sometimes it's minus one-twelfth. What do you mean? When is it that you get minus one-twelfth?'

'When you're not looking.'

I felt a cold shiver down my spine, like someone had just fired a gun in school, or one of the chestnut-cabinets in the classroom had fallen to the floor, only that this time there was no bang or thud. What had hit me to the core was just the weirdness of it all coming out from the mouth of a ten year-old. Everything she was saying felt like answers advanced Maths students - heck, no! - more like researchers would say. "When you're not looking", as creepy and weird as it sounds, is, in fact, a more simplistic summarization of the paradox involving Schrodinger's cat, a thought experiment in which a cat, if placed in a box, can be alive and dead simultaneously. It is a principle used to describe how quantum mechanics works at its heart, especially regarding how subatomic particles interact with an outside observer.

I remained kneeling in front of this girl for a while, speechless. I inhaled and exhaled a few times, and, instead of thinking of the right questions to ask Melinda, I was more worried that I was breathing out nicotine smoke into her face.

'Okay. You can go back to your seat now. We'll talk later about this.' And the little girl conformed and I looked at the blackboard one more time, thinking I desperately needed another cigarette.

This is what she did: she started with two other series 'S1' and S2'.

She defined S1 as:

S1= 1-1+1-1+1-1+......

This is what is called "the Grandi's series", and she proved this sum to be one half (1/2). Now, you would be tempted to think S1=0, which is true, if you bracketed it this way:

S1= (1-1)+(1-1)+(1-1)+....=0+0+0 =0

But S1 is not really convergent. In fact, the sum itself has no value, because you can also bracket it this way:

S1= 1+(-1+1)+(-1+1)+....= 1+0+0,

which means S1 is 1.

There is, however, a way to get to 1/2, the value Melinda found for this series, which is like this:

1-S1= 1- (1-1+1-1+1-1+....)

Opening the brackets, you get:

1-S1= 1-1+1-1+1-1+......

1-S1= S1

1=2S1

S1=1/2.

She then went to define another series S2:

S2= 1-2+3-4+5-6+.....

and calculated:

2S2= 1-2+3-4+5-6+....+(1-2+3-4+5-6+....)

2S2= 1-1+1-1+1-1+.....

2S2= S1= 1/2

S2= 1/4

She then subtracted S2 from the original S (the sum of the natural numbers):

S-S2= 1+2+3+4+5+.... -(1-2+3-4+5-6+....)

Eliminating the brackets:

S-S2= 4+8+12+16+.....

S-S2= 4(1+2+3+4+.....)

S-S2= 4S

S-1/4= 4S

-3S= 1/4

S= -1/12