Moondark for April: Trillions and Trillions

Carl Sagan’s famous line needs an upgrade: the trillion is the new billion. And with the media incessantly reporting  on trillion-dollar budgets and deficits, there’s no getting away from it. We are already bombarded with mega- as in millions (megapixels in your camera) and giga- as in billions (GB, a unit of disk space), and even now, terabyte hard drives (that is, a trillion bytes) are available at any office supply outlet. A trillion is a thousand billion, and to paraphrase Everett Dirksen, a trillion here and a trillion there and pretty soon you’re talking about a lot of anything. Professional astronomers are familiar with large numbers: after all, the Universe is the biggest and oldest thing there is. In astronomical quantities, a trillion is the number of stars in the Milky Way and Andromeda Galaxies combined, or the number of meters light travels in just an hour, yet neither of these is particularly astounding. Written out, one trillion is just 1,000,000,000,000 or 1 followed by twelve (12) zeros. It’s frequent to encounter quantities that are much larger or much smaller than one in any of the physical sciences. There’s a more compact way to write such numbers called exponential notation. Written this way, one trillion is 1012, as in “ten to the twelfth power,” where 12 is the number of zeros to write after a one. For small numbers smaller than one, the exponent is negative, for example, 0.1 = 10-1, 0.001 = 10-3 and so on. The exponent here is the number of places to move the decimal point, adding zeros as you go. This manner of writing numbers is used in many technical disciplines and thus is sometimes called “scientific notation.” In exponential notation, here are some numbers familiar to astronomers: 93.5 x 106 miles is 150 x 109 meters, 13.7 x 109 years old is 4 x 1017 seconds, and the diameter of the 200” mirror on Mt. Palomar is surprisingly close to 5 billion (109) nanometers (or 10-9 m) or almost 107 wavelengths of visible light across. The calculator functionality available at the Google search bar understands exponential notation and many common units. Just type in a number, formula or unit conversion, and the answer will appear above any search hits. The Wikipedia page on scientific notation gives many more examples and explains calculations in more detail. As these numbers show, you can be more precise by preceding with a number between 1 and 10 (along with as many decimal places as appropriate) that is multiplied by the power of ten. And computation with these numbers is also easy: to multiply two numbers, just add the exponents and attach that power of ten to the product of the numbers out front (mantissas). For division, divide the mantissas but just subtract the powers of ten. So products and quotients are easy as adding and subtracting, which is (not coincidentally) why a slide rule works, but that’s a whole other topic. In physics and astronomy (maybe now in economics though certainly not in my checking account) the mantissa, those numbers preceding the power of ten, can often be ignored. It’s the order of magnitude of the numbers that really determines any number’s size and for rough calculations, that’s all that matters. Such quick and dirty calculations are integral to making “back of the envelope” estimates, sometimes known as Fermi calculations. The classic example is estimating the number of piano tuners in New York City, and it can be done surprisingly easily and probably as accurately as anyone knows with information you already have. Simply estimating the order of magnitude of several relevant quantities, adding and subtracting the exponents gives the answer. In a similar way, can you estimate how many amateur astronomers there are in the US? Give it a try! However you write it, a trillion is a big number, yet exponential notation makes it easier to see and understand what “astronomical” numbers mean. It makes good sense to pay attention (puns intended) to these numbers whether we’re talking about the number of milliseconds in a generation or your tax dollars. Moondark is written by Douglas C. Miller, published at the Moondark web site, and printed in the Delmarva Star Gazers' Star Gazer News. This document was last revised on 29 March 2009. Text and graphics on this web page are free for non-commercial use with attribution under a Creative Commons Attribution-Non-commercial 3.0 License. Ask Doug about other uses.

From the Sloan Digital Sky Survey, a full 3-D map of over 2 x 105 galaxies reveals filamentary structure created by the pull of invisible matter and seed fluctuations from long, long ago …