The cross I have to bear…

… Far too many ideas, far too little time. And so I have to carefully plan and allocate resources and spend a lot of time culling the ones that are less practical and putting the others into various categories. On the upside: never ever bored. And hey, I have “setting up a WordPress” site checked off the list!

Anyways, the project page is up. Expect some filling to happen at some point in the future.

The fine art of approximation

It has been said that, during the first atomic bomb test, Enrico Fermi wanted a quick estimate of the energy of the blast. So, as the shock wave hit, he tossed a handful of paper scraps in to the air and watched how far they were carried. He estimated that the energy was about ten kilotons – remarkably close to the measured value of twenty.

Whether this actually happened or not, is a subject for historians to debate, but it makes for a good story nonetheless. And it serves to illustrate how a quick and dirty estimate can aid in decision making. In science classrooms around the world, these sorts of approximation problems are used to teach “science thinking” without getting bogged down in math. And, like all good science tools, it’s partly a matter of convenience and partly a matter of laziness. Some approximation exercises that I remember from my own schooling:

  • How fast would you have to stir your coffee in order to make it boil?
  • How many gas stations are there in the United States?
  • If everyone in China faced west and sneezed at the same time, how would the earth’s rotation change?
  • How fast would you have to drive a car through a hard rain in order to meet a wall of water?

With these types of problems, it isn’t the answer that is interesting but how one arrives at it. And once you get the hang of this you can get a surprising level of accuracy, particularly if you know which way to fudge the numbers. Of course, there are a few tricks to this kind of “Fermi estimate”. Trick number one, don’t care too much about what the answer actually is. If you’re attached to an outcome, you may subconsciously pick numbers that steer toward it.

Trick two, go fast the first time around and then fix it later. The first pass-through is just to get the process right. In subsequent estimations, you can try to get better numbers or to include things that you hadn’t previously thought of.

Trick three, round to the nearest whatever. Some numbers are easier to work with than others. You can work with powers of ten just by moving the decimal point. Computer engineers and programmers know the powers of two better than their own phone numbers. Once you get a feel for how numbers themselves work, then calculation becomes a snap. And because you don’t care about the end number, you can feel free to round off a bit.

Here’s an interesting example…

Global air-conditioning

One morning, I was driving up to a nowhere spot in central California to meet with a client. Radio coverage was essentially non-existent and so I ended up listening to someone on AM talk radio. This someone made the claim that if global warming is man-made, it’s probably from everyone running their air conditioners. Let’s look at this claim and construct a very simple model.

There are approximately four hundred million people in the US right now (and we’re going to ignore Alaska – they probably don’t do too much air conditioning). We’ll assume that each and every person has a five thousand square foot home, five thousand square foot office, and ninety thousand square feet representing their share of communal space (public buildings, malls, etc.). So every man, woman, and child has their very own air-conditioned area of one hundred-thousand square feet. Further, we shall assume that their ceilings are ten feet high, giving each person a million cubic feet of air-conditioned bliss. Four hundred trillion cubic feet in total (notice all of the powers of ten that I’m using).

How cold do they like it? Let’s further assume that every person in the US is currently trying to fight one hundred degree weather and cool their space down to seventy degrees. If air conditioners were one hundred percent efficient (spoiler: they are not) then we’d have to warm a like mass of air by thirty degrees. For our initial model, we’ll assume that air conditioners are only twenty percent efficient (probably they’re a bit better than this, but this is closer to the truth) and so we will have to warm five times that volume (five is almost as easy to use as two and ten).

So, in our hypothetical model, we’re warming up two quadrillion cubic feet of air by thirty degrees. That’s a lot of air. Let’s convert to cubic miles, for sake of readability:

(2,000,000,000,000,000) / (5280 x 5280 x 5280)

So about thirteen thousand cubic miles – a much more manageable number.

How much air is there in the continental United States? According to Wiki, there are about three million square miles of surface area. The atmosphere extends upward to about sixty miles, but most of the action takes place within three miles of the surface, so let’s just use that and approximate that there are about ten million cubic miles of air.

Divide the one in to the other, and the air-conditioned-warmed air represents only one tenth of a percent (note that I’m doing a lot of rounding here) of the volume of air in the United States. All of our air conditioning would warm that mass by three one-hundredths of a degree.

But (and this is a big “but”), our model assumes that all cooling and all air is evenly distributed all over the country by the same amount, everywhere. This simply isn’t true. Additionally, it isn’t true that every person has that much cooled volume. And finally, it isn’t true that every person in the country requires cooling by thirty degrees, all at once. Cities like Phoenix may require more cooling all in one spot; and places like Seattle may not require any. So we can see some ways to begin to refine our model.

I’m not going to argue that air conditioning causes or doesn’t cause warming. It may actually have a measurable (though tiny) effect in some places. The point of this exercise is to show both the power and the peril of making a casual model.

My homework assignment to you: play with this (either on paper or in your head)! See if you can think of ways to refine it. See if you can think of wrenches to throw in to the works. See if you can find some better numbers to use. Feel free to cheat and use the internet if you get lazy (but please give it a go, first).

How can infinity have different sizes? It’s just infinite, right?

Well, not so much. You see, from a mathematical standpoint, the term “infinity” is defined as being the size of the set of integers. So is it possible to have something bigger than this? Yepers, and it is best illustrated by trying to map one thing to another.

By “mapping” you can think of placing marbles into an egg carton, one per slot. If you’ve marbles left over, then the number of marbles is larger than the number of slots. So let’s start off with an infinitely long egg carton, each slot numbered 1, 2, 3, etc. Now well take an infinite number of marbles, with the same numbers on them. None left over, obviously.

Suppose we’ve twice as many marbles, this time labeled 1, 2, 3, etc.; and then -1, -2, -3, etc. We’ll even throw in an extra marble labeled zero. Is this the same size? Yes it is, because you can alternate marbles. First put the zeroth marble into slot one, then the number one marble into slot two, then the negative one marble into three, the two marble into four, the negative two marble into five, and so on. You never run out of marbles, of course, but you never run out of slots either. And the important point: for any specific marble, you know exactly which numbered slot it goes into. So the set of all integers (positive, negative and zero) is the same size as the set of all positive integers.

Let’s throw a curve: all rational numbers. For those of you whose math escapes you, a rational number can be expressed as the ratio of two integers. In other words, any particular rational number x can be expressed as ( a / b ) in which a and b are both (non-zero) integers. So how big is the set of rational numbers? It turns out that it’s still the same size of infinity. Here’s how:

Let’s make a table of all possible a’s and b’s:

egg_crate_empty

The grid is infinite in both the a and b directions as it contains all positive and negative integers along those axes.  So how do we map this onto the line of positive integers?  Just start in the “middle” (pick a point, any point) and spiral out.  Yep, it sounds like cheating, but again, you can compute where any marble on the a-b grid would be placed into the infinite egg carton.

egg_crate_full

So again, the set of all rational numbers is the same size as the set of all positive integers — regular ol’ infinity.  So how about something larger than infinity?  Next we’ll try the set of all irrational numbers.

Again, in case you’ve forgotten, an irrational number is one that cannot be described as the ratio of two integers.  Things like pi and e (base of the natural logarithm) and the square root of two are decimal numbers that never end and never repeat.  So it would take an infinite number of integers (regular infinity) just to represent each one.  And guess what?  In between every rational number, there are an infinite number of irrational numbers.  If you take pi as an example, you could change any one of its infinite digits and come up with an entirely different number.

So guess what?  We just ran out of slots in our egg carton.  Since we can change any one or all of the digits in any particular irrational number (even an infinite number of them) to get a new number; and since there are an infinite number of irrational numbers, even between each rational number; we have waaay too few slots in our infinite egg carton.

The size of the set of all irrational numbers is the first transinfinite number, aleph-one (aleph-zero is another term for regular infinity).  We can even get bigger than this, defining in similar ways aleph-two, aleph-three, etc, even on to aleph-infinity.

So there you go: a quick and dirty (and honestly, not terribly rigorous) introduction on numbers that are actually larger than infinity.  To summarize (again, not very rigorously): infinity plus infinity equals infinity (all positive and all negative integers); infinity times infinity also equals infinity (all rational numbers); but infinity to the power of infinity is bigger (all irrational numbers).

Hola!

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