In high school you probably learned about the properties of infinite series. Things like the sum of the reciprocals of the powers of 2: 1/1 + 1/2 + 1/4 + 1/8 + 1/16 + ... converges on 2, but the sum of the reciprocals of the positive integers diverges: 1/1 + 1/2 + 1/3 + 1/4 + 1/5 + ... , growing to infinity. There are any number of interesting convergent series related to really interesting numbers like e and Pi, not to mention it is useful to know when something blows up to infinity. Math just works, eh?
A friend sent a link to a video that seemed wrong. It made a completely nonsense claim: that the sum of all positive integers was not infinity. 1 + 2 + 3 + 4 + 5 + ... + infinity. Stranger yet was the claim it was not an integer and also negative. The video came from real mathematicians who sounded confident. He looked for an April first posting date, but didn't find anything suspicious. This had to be some kind of joke. What was the trick?
It turns the video is correct. In fact you need to be able to do sums like this if you want to use the Standard Model in physics as well as string theory and a few other areas. The same folks put out a second video with a bit more math, but still at a high school level. If you're really curious take a look.
Something extremely interesting is going on.
Imagine math in the days when the only numbers people used were real and rational. Some clever person came along and started using square roots. The square root of 1, 4, 9 and 16 made sense, but 2 was a problem. You could approximate the square root of 2, but it wasn't a rational number. A new type of math had been discovered that was not only was in another context, but expanded the notion of what math was.
On the surface the square root of a negative number seems pure nonsense - so much that the operation was defined as invalid on that side of the number line. But its just math, so why not? The square root of a -1 was defined to be a new type of number - an imaginary number. A new number system had been invented. It was shown to be rigorous in the proper context with certain definitions and has enormous value in physics and all of engineering.
This is the story of math. It is not a pristine temple where people work with pure logic using only existing truth to find new truths. Rather it is a bubbling sea of conjecture and relation. Ultimately a rigorous path needs to be found, but the frontiers are messy. Change comes with rebellion.
Leonhard Euler was just such a rebel. He wondered if sums that diverged to infinity had deeper meanings. Some, like the sum of all positive integers, had a regular structure. How might they differ from other infinite sums? Was there a story different from the final tabulation at the end?
He came up with an ingenious technique and showed the sum of positive integers was -1/12 in a fashion not too far from that of the first video. The first time I saw it in college was electrifying. It was like sorting through an infinite pile of dirt and rock to find a tiny diamond. When a similar series appears in quantum electrodynamics - an area of physics that gives us the highest precision of all measurements to date - you substitute in the little diamond and, low and beware, the calculation produces a meaningful result. The most accurate measurements humans have made agree with a theory that makes use of this type of sum.
Euler worked out several other series. The sum of the squares of all positive integers turns out to be 0 and that of the cubes is -1/120. In fact the all of the sums of even powers of the the integers is 0 and the odd powers is not. (Impress your friends by noting the sum of xi10 over i = 1 to infinity is 0.) He was unable to prove this rigorously. That had to wait about a century for Bernhard Riemann who was able to show a deep connection with something that fascinates to this day - the Riemann zeta function.
So what seems both wrong and crazy turns out to make perfect sense in a certain context - a context that is fundamentally related to how we know the physical world at the deepest level. Why does Nature seem to speak math? Now there is a deep mystery...
I should stress that the equivalence here is something of an issue. This gets deep into the analytic continuation of the Reimann Zeta function at s = -1 ... which happens to correspond to the sum of the positive integers. The videos don't stress it enough, but analytic continuation is probably too abstract to include.
I don't know where the okra is coming from these days, but some is very high quality. I threw together a sort of crispy bhindi. Not exactly traditional, but delicious. As usual the quantities are recollections of what I think I did
° a pound of okra with the tops trimmed and sliced lengthwise into thin strips
° 1-1/2 tsp red chili powder
° 1-1/2 tsp garam masala
° 1/2 tsp turmeric
° 2 tbl chickpea flour
° 1 tsp cornstarch
° juice of one lemon
° vegetable oil
° mix the chili powerder, garm masals, turmeric, chickpea flour and cornstarch and toss the okra in it moistening with the lemon juice
° heat some vegetable oil in a wok to something under its flash point and fry a handful of okra at a time until golden. Drain on paper towels and let cool.
° I tried a second frying on some which makes them even crispier -- experiment!