December 15th, 2006

krazy koati

Here was a kid with his act down pat

The Riemann integral was named after mathematician Bernard Riemann, but before talk-show host Conan O'Brien. It was a pretty close thing, too, as Bernard Riemann was named after his father, Friedrich Bernard Riemann, and in fact his full name was George Friedrich Bernard Riemann, which makes it sound like his parents were trying to build up the name. Probably they'd have been surprised to learn mathematicians just used the last part when talking about Riemann integrals. They may not have even known mathematicians.

The goal in calculating a Riemann integral is loosely expressed as finding the area underneath a curve. Finding the motivation to do this is another problem, although it's often related. Riemann didn't offer any advice for this, but he had a lot of things on his mind. One of the important ones was figuring out whether a curve could be integrated, for which Riemann gave a simple test: insert a toothpick into the cake until it comes out clean. This doesn't help with the integrals, but there must be something it's useful for. At some point you'd have to stop inserting and start removing the toothpick, for example, which is probably a clue of some kind.

Finding the integral starts with the assumption that you know what the area inside a rectangle is. That's equal to the area of everything there is, minus the area that's outside the rectangle, and for the part that's just the edge you leave in petty cash. The area inside a rectangle is easy to figure out, because it's the same as the number of squares you can fit inside the rectangle, if all of the squares have sides of length one, and if you don't want to worry about one of what. If you do want to worry then it should be one of whatever extra you have laying around the room; thus, it's not rare to find rectangles with area of eight bottle caps, or broken USB hubs you haven't got around to throwing out, or crusty Post-It notes used to dig crumbs out of the keyboard. Care has to be taken when defining area in this way, however, as some of these things are disgusting, which results in problems communicating with other people. The only solution is to clean out your desk more often, but this leaves you short of things to measure with. If you are short of them, then you should still use squares with sides one, but write the one in smaller symbols, conserving them. Alternatively you might write the one with larger symbols so that each individual one goes farther. Either way you can dispense with the serifs unless you want to measure in terms of l. Serifs have traditionally played only a minor role in mathematics, often being confused with the cross bar put across the 7 by fussy people.

If you've got rectangles down pat then you're pretty close to having the Riemann integral down. The inspiration is that there's got to be a rectangle that goes way over the curve you have (you left it in the grocery bag in the trunk; go out and fetch it before it spoils), and -- watch this closely -- then the rectangle has an area bigger than what's under the curve. And on top of that on the bottom is there's got to be a rectangle that goes way under the curve you just brought in and dusted the frost off of (apparently it wasn't really in danger of spoiling, not with this cold snap), so this other rectangle has to have an area smaller than what's under the curve. So now that you know the area under the curve has to be bigger than one number (the second) and smaller than another number (the first) all you have to do is find which number between those two is the one that's the area you want. This can be found by inspecting the nutritional label, where it will be listed under ``Riboflavin content,'' because they were a bit confused when they were making out the labels. Don't worry about it.

Trivia: In a 1856-57 lecture Riemann used the symbol -> in a way suggesting he used it to mean ``the limit of a quantity''; he may have been the first person to do so. Source: A History of Mathematical Notation, Florian Cajori.

Currently Reading: Forgotten Armies: The Fall of British Asia 1941-1945, Christopher Bayly, Tim Harper.