12 January 2010


What do chemistry, music, astronomy, biology, physics, and engineering have in common? The answer is that adaptable, versatile tool -- math. Math is both a language and a process, with a clear and essential vocabulary, syntax and grammar. All of the above disciplines employ math to express complex concepts in universal terms.

If you think about learning to count on your fingers as a child, it becomes intuitively obvious why so much of the world uses math with the value ten as its basic unit of measure or counting. We call this "math to the base ten." (Had we been born with eight digits instead, we would probably use math the the base eight.)

In science, it is customary to employ the decimal system for expressing both values smaller than one, and also values of great size. This usage is referred to as scientific notation, and is a kind of shorthand for manipulating numbers more easily and quickly -- one can simply shift the decimal point, rather than working out complex fractions. This handy system is also called working in powers of ten (i.e., ten multiplied times itself a given number of times).

To illustrate: we know that there is only one ten contained in the number 10, so in scientific notation the value is written as 10 (the coefficient) with a superscript 1 (the exponent). It looks like this: 101. By the same token, we know that there are ten tens (ten times ten, or ten squared) contained in the number 100. So the coefficent remains 10, but the superscript now is written as 2. It looks like this: 102. Similarly, a thousand is written as 103, or ten cubed (ten times ten times ten). And so on.

So what's the big deal? Well, if we're talking, say about the distance from the earth to the sun, it's both easier to write and easier to do math with, if we write it as 9.3 x 107, than it is to write it as 93,000,000 miles. And that is a truly puny distance in astronomical terms. By the same token, a monumentally tiny quantity, so small that many zeros appear between the decimal point and any integers, can be conveniently written in scientific notation. The mass of a proton is easily written as 1.67 x 10-27, as opposed to longhand, 0.00000000000000000000000000167 kg.

Those of us who grew up using the metric system find scientific notation intuitive, even second nature. Those of us who (unfortunately) grew up with the more cumbersome English system of measurement may have to do a bit of unlearning and relearning, but the effort is worth it, especially when dealing with size relations between extremely small and extremely large objects or distances. Imagine having to convert between inches, feet, yards, miles and light years using cumbersome fractions and conversion tables, and then compare that to the ease of simply shifting a decimal point. In such shifts between orders of magnitude, the metric system and scientific notation come into their own. Just for fun, try to wrap your imagination around this illustration of orders of magnitude -- click on the image to enlarge.

No comments:

Post a Comment