A LITTLE ARITHMETIC

Very large and small numbers abound in astronomy, for example:

A convenient way of writing these numbers which enables them to be manipulated easily is as follows.

1 x 100 = 1
1 x 101 =10 1 x 10-1 = 1/10 = 0.1
1 x 102 =100 1 x 10-2 = 1/100 = 0.01
1 x 106 =1,000,000 1 x 10-6 = 1/1,000,000 = 0.000001

etc.

etc.

The rule is to move the decimal point to the right the number of places equal to the positive exponent, or to the left the number of places indicated by the negative exponent. So:

Now suppose you want to multiply 3500 by 20:

Similarly, suppose you want to divide 3500 by 20:

A couple of examples: How long did it take a TV signal to travel from the Astronauts on the moon back to the earth? The TV signal is basically the same kind of radiation as light and travles the 240,000 miles from the moon to the earth with the same velocity as light, 186,000 miles/sec. Thus the

Another example:What is the volume of the earth? The earth is very nearly a perfect sphere for which the volume is given by the formula. V = 4/3pi R3 where pi = 3.14 and R = the earth's radius, about 4,000 miles = 4 x 103 miles.

ANGULAR MEASURE

Go around a circle once and you will have gone through an angle of 360 degrees (=360o) regardless of how many feet or miles you actually traveled. Each degree is divided into 60 minutes of arc (= 60'); each minute is divided into 60 seconds of arc (= 60"). Thus in 360o there are: (360) x (60) x (60)

The angular diameter of the sun as seen from the distance of the earth is about 30' or 1/2o; were the sun nearer to the earth, at S1 instead of at S, its angular diameter would, of course, be greater than 1/2o See figure 2.

Now it so happens that the moon's diameter (M in Figure 2) as seen from earth is also about 1/2o. It is because of this fortuitous circumstance that total eclipses of the sun can take place, i.e., as seen from the earth, the moon is just big enough in angular size to be able to cover the sun , to eclipse it momentarily.

The system of dividing things up by 60s, as in 60' = 1o, 60" = 1', seems little strange to us because our number system is based on 10.

That is, 3.452 = 3 + 4/10 + 5/(10)2 + 2/(10)3 . Our system of angular measure, however, comes to us from the Babylonians and their base number was 60, not 10. Thus 5o 39' 4" = 5 + 39/60 + 46/(60)2.

UNITS

The metric system is in world-wide use for scientific work and is in fact in general use nearly everywhere except in the U.S. We are now slowly converting to it (you've probably noticed highway signs with distances given in miles and kilometers both) and we will use the metric system in this course. In the English system the units of length, mass, and time are the yard, the pound, and the second, respectively, whereas the corresponding units in the metric system are the centimeter (cm), the gram (gm) and the second (the same in both systems). The larger and smaller units of the English system are clumsy; e.g., the mile is 1760 yards and the inch - 1/36 yard. In the metric system, however, all related units are multoples of 10 of each other; e.g., 1 meter (m) = 100 cm = 1000 millimeters (mm), and 1 kilogram (kg) = 1000 gm.

Examples: Suppose you are 5 feet 6 inches tall and weigh 140 pounds. In the metric system your height would be 5.5 feet x 30.5 cm/feet = 168 cm or 1.68 m, and your waight would be 140 pounds / 2.2 pounds/kg = 63.6 kg = 63,600 gm = 6.36 x 104 gm.