## Scientific Notation In astronomy, as in Congress, we often deal with very big numbers. Andromeda, one of the closest galaxies, is about 2,000,000,000,000,000,000,000,000 centimeters away. Unlike politicians, astronomers also have to deal with very small numbers. The mass of the electron is 0.00000000000000000000000000091 grams. If we had to list all these zeroes all the time, we would use up so much paper that we would rapidly deforest the world, melt the polar icecaps, and drown. Scientific notation is therefore not only convenient, but also ecologically sound.

There is, needless to say, a better way. It's called scientific notation or exponential notation. Each time we add another zero after a number and before the decimal place, we are really multiplying the original number by 10. We could, instead, just save the relevant part of the number (the 2 in the case of Andromeda) and count the number of times we have multiplied by 10 (24 for Andromeda). In mathematical terms, this counting is called "raising" 10 to the power counted. In scientific notation, the distance to the Andromeda galaxy would be given as 2 x 1024 (two times ten to the twenty-four) centimeters. The 24 is the exponent that gives the notation its alternative name.

Here are some examples:

1 = 1 x 100

10 = 1 x 101

1000 = 1 x 103

1,000,000 = 1 x 106

1,000,000,000,000 = 1 x 1012

We express numbers less than 1 similarly:

1/10 = 0.1 = 10-1

1/100 = 0.01 = 10-2

1/100,000 = 0.00001 = 10-5

where the last example is pronounced "ten to the minus five."

Suppose you want to express a number like 2,345,000,000 in scientific notation. To do this, you move the decimal point to the left, counting as you go, until it is just to the right of the left-most number (the 2, so you should have counted to 9). You now rewrite the number with the decimal point at its new place and drop any zeroes. The left-shift of the decimal point is the equivalent of dividing a number by 10, so to get back to the original number, we must now multiply by 10x where x is the number of left-shifts you counted (in this case, 9). In scientific notation, the above number comes out as 2.345 x 109.

Small numbers can be expressed in a very similar way. You just move the decimal point to the right, counting as you go, until you reach the first non-zero number. This count now tells you the number of times the whole number has been divided by 10. To express the number 0.00045 in scientific notation, we move the decimal point 4 times to the right. This is the equivalent of multiplying by 104. To get the original number back, we now must divide by 104, or multiply by 10-4, so the final result is expressed as 4.5 x 10-4 (four point five times ten to the minus four).

Other examples:

6,855,000 = 6.855 x 106

3.191 = 3.191 x 100

0.0000212 = 2.12 x 10-5

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## Arithmetic and Scientific Notation

To add and subtract numbers, you multiply or divide one number by 10 until its exponent is the same as that of the other one. You then add or subtract in the normal way. Examples:

4.1 x 102 + 3.8 x 103 = (4.1 + 38) x 102 = 42.1 x 102 = 4.21 x 103

6.8 x 1099 - 5 x 1097 = (680 - 5) x 1097 = 6.75 x 1099

Multiplying and Dividing:

Here is where scientific notation really pays off. To multiply or divide, you separate numbers from the exponents. The numbers you multiply or divide in the normal way. You add together the powers of 10 you are multiplying and subtract the powers of 10 you are dividing to produce the answer. Examples:

2 x 103 x 3 x 104 = 6 x 107

1.5 x 105 x 4 x 10-2 = 6 x 103

8 x 1010/4 x 103 = 2 x 107

6 x 102/2 x 105 = 3 x 10-3

1 x 1099/1 x 1050 = 1 x 1049

Try doing the last example without scientific notation if you still need convincing!

Raising to Powers or Extracting Roots:

This is another task that can be greatly facilitated by scientific notation. To raise a number to a power, we raise the number to that power and multiply the exponent by that power. For example, the square of 400 is done as follows:

4002 = (4 x 102)2 = 42 x 102x2 = 16 x 104 = 1.6 x 105.

For a root, we take the appropriate root of the number and divide the exponent by the order of the root. For example, the square root of 400 is

sqrt(400) = sqrt(4 x 102) = sqrt(4) x 102/2 = 2 x 101 = 20.

Note that we often express roots as fractional exponents. For example, the square root of a number (a) can be written as sqrt(a) or a0.5. Here is an example of this usage:

(9 x 10-4)0.5 = 90.5 x (10-4)0.5 = 3 x 10-2

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## Appendix 4

### USEFUL INFORMATION ON UNITS, PHYSICAL CONSTANTS

Basic Units of Metric System

 (cgs) mass: gram (gm) distance: centimeter (cm) time: second (s) energy: erg = gm cm2 s-2 (MKS) kilogram (kg) = 103 gm meter (m) = 102 cm second (s) Joule (J) = 107 ergs

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Other Units of Distance

 1 kilometer = 103 meter 1 centimeter = 10-2 meter 1 millimeter = 10-3 meter 1 micrometer (micron) = 10-6 meter 1 Angstrom = 10-8 centimeter = 10-10 meter 1 AU = 1 astronomical unit = 1.5 x 1013 cm 1 pc = 1 parsec = 205206 AU = 3.1 x 1018 cm 3 x 1018 cm 1 ly = 1 light year = c x 3.16 x 107 s yr-1 9.5 x 1017 cm 1018 cm

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Units of Frequency

 1 Hertz = 1 Hz = one oscillation per second 1 kilohertz = 1 kHz = 103 Hz 1 Megahertz = 1 MHz = 106 Hz 1 Gigahertz = 1 GHz = 109 Hz

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Units of Time

 1 nanosecond = 10-9 second 1 microsecond = 10-6 second 1 millisecond = 10-3 second 1 year = 3.16 x 107seconds

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Units of Temperature

 1 kelvin = 1 K K = degrees C + 273 degrees C = 5/9(degrees F - 32) 0 K = absolute zero 273 K = water freezes 293 K 300 K room temperature 373 K = water boils

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Units of Power (Luminosity)

 1 Watt = 1 Joule s-1 = 107 ergs s-1 1 Solar Luminosity = L = 4 x 1033 ergs s-1

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Units of Mass

 1 solar mass = M = 2 x 1033 gm Mass of proton = Mp = 1.66 x 10-24 gm Mass of electron = Me = 9.1 x 10-28 gm

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Physical Constants

 Speed of light = c = 3 x 1010 cm s-1 Planck's constant = h = 6.6 x 10-27 gm cm2 s-1 Boltzmann's constant = k = 1.38 x 10-16 ergs K-1 