Dealing with Large and Small Numbers

In astronomy, we encounter both very large and very small numbers. There are several approaches that can be taken when the magnitude of a quantity becomes very large or very small.

• Continue to plow through, writing the numbers in "regular" notation. this is almost never done in practice, since the numbers become cumbersome and the probability of copying errors increases dramatically.
• Switch to mixing words and numbers. This is frequently done in common speech and in the media, as in "the national debt is over $5 trillion."
• Make use of the metric prefixes to switch to a multiple of the base unit that is closer to the relevant measurement. You measure the storage capacity of your hard drive in gigabytes (though last year you measured it in megabytes, and next year it will be in terabytes).
• Switch to using special units that disguise the true magnitude of the numbers involved. Chemists deal with moles instead of counting atoms, wavelengths of light are often measured in angstroms rather than meters, and the size of a star is frequently expressed as a multiple of the radius of our own sun. In a pinch, we can combine special units with metric prefixes to extend the scale even further.
• With the exception of the first item, all of the above are essentially aimed at avoiding large numbers in favor of words. But there is one other approach at simplifying large (or small) numbers that nevertheless still deals only with numbers. That is scientific notation. Scientific notation expresses any number as the product of a number between 1 and 10 and a power of 10.

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For illustration, the following table gives a few awkward numbers with some alternate representations. Notice that for extreme cases, scientific notation provides the only reasonable means for handling the numbers in any meaningful way.

Figures that are constantly changing, like the national debt, are current as of Sep. 23, 1998. Interesting to note that Bill Gates is worth almost exactly 1 percent of the national debt.

National debt	$5,653,000,000,000	$5.7 trillion	5,653 gigabucks	$5.653 x 1012
Bill Gates' net worth	$57,030,000,000	$57 billion	57,030 megabucks	$5.703 x 1010
1 mole (Avogadro's number)	602,257,000,000,000,000,000,000,000 molecules	602 sextillion molecules	602.257 zettamolecules	6.02257 x 1023 molecules
Distance to Alpha Centauri	40,120,000,000,000,000 m	4.24 lightyears	40.12 petameters	4.012 x 1016 molecules
Distance to Andromeda Galaxy	21,800,000,000,000,000,000,000 m	705 kiloparsecs	22 zettameters	2.18 x 1022 m
Mass of Sun	1,990,000,000,000,000,000,000,000,000,000,000 g	1 solar mass	199 billion yottagrams	1.99 x 1033 g
Bohr radius (radius of hydrogen atom)	0.000000000052918 m	0.52918 angstrom	52.918 picometers	5.2918 x 10-11 m
Mass of Hydrogen atom	0.0000000000000000000000016733 g	1673 octillionths g	1.6733 yoctograms	1.6733 x 10-24 g
Mass of electron	0.00000000000000000000000000091096 g	549 millionths amu	0.00091096 yoctograms	9.1096 x 10-28 g

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