Getting comfortable with some big numbers

kw: technical information, numbers, large numbers

When I was in college a classmate told me of something his Fourth-Grade teacher had done: She cut up about 20 sheets of "millimeter paper", the kind of graph paper with a millimeter grid that has 5- and 10-mm highlights, and taped them together into a 1,000x1,000 sheet, one meter square. This she hung on the wall with a sign above, "This is What a Million Looks Like."

I had occasion to remember this recently. It got me thinking. Most of us can't easily think of numbers such as a million or billion, or even several thousands. Yet we live in a world in which large numbers like that are bandied about: "93 million miles (or 150 million km) to the sun", "4 billion dollars" for such-and-such a system of highways, "7 billion people on Earth", "20 trillion dollar national debt", and so forth. What does a billion or a trillion even mean any more, when you can get a pocket-sized external hard drive with 1TB or 2TB of storage, or even more, for a hundred dollars or so? (Folks, a TB is a TeraByte, or a trillion 8-bit computer "characters").

Let's first be clear whose billion and trillion we mean. These days, even the English and other Europeans have pretty much surrendered to the American system of large numbers, in which a billion is 1,000 million, which is a 1 followed by 9 zeroes, and a trillion is a million million, or a 1 followed by 12 zeroes. But when I was young, the British and others still clung to an older system in which a billion had twelve zeroes and a trillion had eighteen. Some used the French term "milliard" for 1,000 million, the American billion. I remember reading a humorous article, "Why there will never be a British Billionaire", that made this vocabulary stick in my head.

Now we can start to think first of the humble Million. The King James Bible has 8/10 of a million words, or 783,137 if you don't count chapter headings and other auxiliary items. So consider the time it might take you to read the whole thing, add about 1/4, and that's the time you'd need to read a million words. I read novels at about 600 wpm, and nonfiction, if it is any good, at about half that speed. Thus I could read a million words of fiction in some 28 hours (so reading the Bible in a year isn't all that hard, no more than 4 minutes daily) and a million words of nonfiction in twice that time.

I once downloaded The Papers And Writings Of Abraham Lincoln, Complete from Project Gutenberg. In plain text (UTF-8) it comes to 3.1 MB, from which I infer about half a million words. That is Abraham Lincoln's lifetime out put of text. About half a million words, or some 64% of the King James Bible in volume. Now, you know how long reading that would take, but imagine writing those half million words longhand, with a quill pen. Writing with a good mechanical pencil I cannot exceed 20 wpm, and I am pretty sure that even a fast writer could seldom exceed half that using a quill. So Lincoln put a lot of time into his writing, perhaps the equivalent of a year or two of full time work. Several percent of all the minutes that he lived.

OK, let's talk about one billion. That teacher with her million tiny squares on the classroom wall would be hard put to show the children a billion tiny squares. All spread out, it would be larger than 30x30 meters. On some reasonable set of surfaces, such as a long stretch of 8-foot (2.4 m) wall, the paper would extend more than 415 meters, just a bit over a quarter mile. A stack of 1,000 1x1m sheets, had she the patience to make them, would be compact enough, about 10 cm thick (4 inches).

So let's consider something a bit easier to put in a bucket, such as sand. I have on hand some sand from Imperial Beach, California, that I collected about a year ago when I was visiting family there. It is from the southern end of the beach, near the Mexican border, where they don't dump a lot of dredged sand to replenish the beach; thus, it is the "natural" sand from that beach. After some examination with a low-power microscope, and counting the grains in a few milligrams of sand, I found that the average grain diameter is 1/3 millimeter and a gram of the sand would contain about 26,500 grains. That means that a billion grains would weigh 37.7 kilograms (about 83 pounds). The volume comes to about 20 liters (porosity is about 40% because the sand is rather angular and poorly sorted), or 5.3 gallons. That's about two buckets of sand; our household buckets are just under 3 gallons' capacity.

How about something smaller? I'd like to have a billion of something I can conveniently, and without strain, hold in one hand. Considering a weight of a kilogram or less, let's start by assuming a specific gravity similar to water and work backwards. A billionth of a kilogram is then a mass of one microgram, and a cube of ice with such a weight would be 0.1 millimeters on a side. In this size realm, the micron (micrometer for purists) is a convenient dimension. A cube 100 microns on a side is about the size of a mammalian fat cell, so a kilogram of fat contains, very approximately, a billion cells. The volume of that kg of fat is one liter (just over a quart).

Fat cells are larger than average. Another familiar cell type is the buccal cell, those you can gather by the hundreds by lightly scraping the inside of your cheek with a soup spoon. Their diameter is about 25 microns and their mass about one-eighth that of a fat cell, so a billion of them would weigh 125 g and fill 1/8 of a liter (about 4 fluid ounces). That's about the size of a golf ball.

For a big step into smallness let's burrow inside. We all have within us trillions of microbes. Most of them make up our "intestinal flora". They are called "flora" because something like a century ago bacteria were thought to be some kind of plant life. Now we know they are a kingdom of their own. But the term remains. What size are they?

They come in quite a range of sizes, because there are thousands of species. But the most common, the now-familiar Escherichia coli ("E coli" in the Press), also known as "coliforms", have a cell volume close to 2 cubic microns, and with a density just a little greater than that of water, a mass of about 2 trillionths of a gram. Whoa! We've already entered a realm in which it isn't hard to imagine a trillion of something. Two grams of E. coli bacteria contain a trillion cells! The volume would be about that of a thimble.

Now, bacteria are small, but viruses are smaller yet. Let's pick the "familiar" influenza virus. They have a modest range of size, but average 100 nanometers (nm). That is 1/1000th the size of the fat cells we mentioned above. The virus particles are flexible enough to pack together with little porosity, if you can gather a large number of them. So one billion of them, packed together, would have the same volume as one fat cell. And a trillion of them would have the volume of 1,000 fat cells; if packed into a little cube it would be one millimeter on a side. That same volume would hold half a million cells of E. coli.

I don't know how much this might help anyone think about the quantities million, billion or trillion. The meter-square piece of "millimeter paper" is easy enough to imagine, and not too hard to make. You could try holding a golf ball and thinking, "A billion of the cells that line my cheek would just fill this ball". Then, pluck a thimble from the nearest sewing kit and, holding it like a cup, say to yourself, "Fill 'er up with E. coli, and that's a trillion." I can't think of any convenient artifact that would hold "only" a trillion influenza virus particles. One cubic millimeter is pretty small!

Well, this was fun to write, and satisfies a "wild hair" I had a couple of hours ago.

Keeping my signals clear

kw: technical information, digital infrastructure

Are you ready for February's Digital Diversion? These days, it seems everyone has cable TV, to the tune of $50 monthly or so. Not me, I'm too cheap. I have an antenna in the attic, the largest one that would fit in the end room, with good cable between it and the set. The set has all the tuners needed to receive both digital and analog broadcasts and converts everything to a 1080i image. So I've been watching as much digital TV as the broadcasters have been making available.

In the suburbs some twenty miles southwest of Philadelphia, that's a lot, actually. I get five reasonably good analog signals and a handful of snowy messes. Digital? I get eight pretty solid signals and two spotty ones. Several of those channels have two or more feeds, so I can receive a total of sixteen to eighteen digital feeds, on a clear day (rain absorbs signal, and several drop out in bad weather).

I recently learned how to estimate the situation that is coming in February next year. Go to tvfool.com and enter your address. If you can, put the height of your antenna (If it is set-top 'rabbit ears', put zero). You'll get a nice graphic like the one below. Click on the image for a full-size GIF.


What does this show you? The radar circle on the left shows where to point an antenna for the strongest groups of signals in your area. Each circle is about a factor of ten weaker in strength. The colors give a good hint about what kind of antenna can receive a signal:

• Green = rabbit ears ought to be OK.
• Yellow = An attic antenna (like mine) can get these.
• Pink/Red = You need a roof-mounted antenna, the higher the better.
• Gray = Forget it, unless you can afford a really tall antenna tower plus a preamp.

I didn't show the gray ones on the screen clip here. It is for illustration anyway. I made this image for an antenna at 20 feet, about eight feet higher than the one I have now, to see how things would improve if I move the antenna I have to the top of the roof. It gained me a factor of five to ten in signal strength for the "fringier" signals, which ought to bring in several stations I don't get now. This is a great planning aid!

The list on the right is more detailed. The most important figure is NM(dB) under Signal. This is the Noise Margin, and needs to be greater than zero for your set to show you a picture. This needs to be adjusted; for example, typical preamps give you 10 or 20 dB more signal, while the wall of a house or the roof of an attic can absorb 5 to 10 (or more) dB. On the list I made with my real antenna height of 12 feet, I can estimate other losses. For example, I get channel 17.1, which has NM around 38, but not 12.1, at 26. I know I am losing about 10 dB in my roof, so there is at least another 16 dB loss in the system, probably from (1) the cable from antenna to set and (2) lots of trees in my yard. If I add a 20 dB preamp, it might pull in 12.1.

Going to roof mount, I find that these two channels have 57 dB and 47 dB, respectively, which is good news: about 20 more dB from height alone, plus I won't have the losses through the roof. Sounds like it is worth doing!

One other column worth noting is the Path under Signal:

• LOS = Line of Sight
• 1Edge = Diffracted once (adds 10 dB loss or more)
• 2Edge = Diffracted twice (adds 20-30 dB loss)
• Tropo = Scattered off the air in the troposphere. Basically well below the horizon!

Only the strongest broadcast signals can overcome the losses of the latter two categories. You'll note that in this image, all the Green signals are in the first two categories. Since I'm not yet pulling in all these, they're my target goal. If I manage to scrounge up any Pink ones in the bargain, that's gravy.