Scientific Notation
The numbers encountered in astronomy are often very large or very small when expressed in units of everyday experience. In other words, expressing values in feet, miles, or years often results in numbers that are cumbersome to write. In order to handle these numbers efficiently scientists often employ a shorthand system of notation called scientific notation.
We usually write numbers using decimals and zeroes, such as 648 or 3.908, but there is a more convenient method using factors and exponents. The factors of a given number consist of two or more numbers that when multiplied together, give us the original number. An exponent is a number placed at the upper right of another number called the base. The exponent, or power, indicates the number of times that the base must be written out and multiplied to get a particular number. For example, 125 can be factored into the product of three 5s and written as
125 = 5 x 5 x 5 = 53.
The distance to the Sun's nearest neighbor Alpha Centauri is greater than 10,000,000,000,000,000 m. In the shorthand notation of exponents and bases we can more conveniently write this distance as
10,000,000,000,000,000 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10
x
10 x 10 x 10 x 10 x 10 x 10 = 1016.
The base 10 raised to the 16th power is equivalent to the original number. Not only does this make the number much easier to write, it also makes it much easier to multiply and divide numbers in your head. When multiplying numbers written as exponents with the same base, simply add the exponents together. Similarly, when dividing numbers written as exponents with the same base, subtract the exponent in the denominator (on the bottom) from the exponent in the numerator (on top).
Thus,
1019 x 107 = 1019 + 7 = 1026
and
1019 ÷ 107 = 1019 – 7 = 1012.
A similar technique may be used for numbers that are very small. For example, writing the size of an atom in meters becomes
0.0000000001 m = 0.1 x 0.1 x 0.1 x 0.1 x 0.1 x 0.1 x 0.1 x 0.1 x 0.1 x
0.1
= (0.1)10 m.
If we express 0.1 as 1/10 and recognize that 1/10 is simply 1/101 or 10-1, then we may write our previous answer as
(0.1)10 = 10-10.
Thus, the negative exponent indicates how many times we are multiplying the base (0.1).
Not all numbers are simply expressed as powers of 10 or (0.1), however. The speed of light—300,000,000 m/s—must be written using other factors as well as an exponent of 10. Thus
c = 300,000,000 m/s = 3 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 3 x 108 m/s.
Numbers with decimal values or others not quite so simply written can be expressed in scientific notation as shown in the following example:
137 = 1.37 x 10 x 10 = 1.37 x 102.
Likewise, a number smaller than one, such as 1/137 would be expressed as
1/137 = 0.730 x (0.1) x (0.1) = 0.730 x 10-1 x 10-1 = 0.730 x 10-2.
It is accepted practice to always write numbers in scientific notation so that only one digit is placed before the decimal point, however. Thus, we should write the number above as
7.30 x 10-3
since we have moved the decimal point one place to the right. We would then solve the following calculation—where parentheses are added simply to clarify what is being demonstrated—thusly
(3.4 x 105) x (7.2 x 103) = 24.48 x 108 = 2.448 x 10 x 108 = 2.448 x 109.
Note that not only do you add exponents together when multiplying numbers expressed in scientific notation, you also multiply the factors in in front of the base. Since we moved the decimal point one place to the left, we had to add one extra factor of 10 to the expression and the exponent changed from the 8th power to the 9th power.
When dividing numbers expressed in scientific notation, you divide the factors in front of the exponents:
7.45 x 107 ÷ 6.33 x 10-4 = 1.18 x 10(7 – (-4)) = 1.18 x 1011.
It is also wise to pay attention to the sign of numbers when working with exponents and remember the rules of adding and subtracting negative values!
Significant figures
Although I will not strictly enforce the rules of signficance when grading,
you should be cognizant of the importance of the accuracy and precision
when dealing with numbers in measurement and calculation (accuracy,
by the way, refers to how close a number approaches an accepted or theoretical
value whereas precision refers to the number of places to which
a number can be determined.)
Basically, you should restrict the number of digits in your answer to the number of digits in your least precise number. Ideally, this should be done at each intermediate step a series of calculations. For example
45.678 x 6.54 = 299
since I rounded the calculated value (298.73412 on my calculator) to the number of digits in my least accurate number (6.54). Likewise,
(3.567 + 3.05) ÷ 1.2105 = 6.62 ÷ 1.2105 = 5.47
if we round properly at each step in the calculation.