## Scientific Notation

Astronomy deals with big numbers. Really big numbers. It’s impossible to talk about the distance to the Sun or the speed of light without thinking about the tremendously huge. But big numbers intrude on all aspects of our lives and as responsible citizens, we ought to have a way of dealing with them. You need only drive past a McDonald’s restaurant to see that numbers larger than a billion (or however many they’ve served by now) are commonplace in our society. On a more serious note, a person must have an understanding of the word "trillions" in order to comprehend the size of the National Debt.

Mathematicians and scientists use scientific notation to handle large numbers. Scientific notation is necessary when discussing astronomical quantities, unless we want to write out a lot of zeros ("Excuse me, Dr. Sagan, did you say the Universe began 15,000,000,000 years ago or 150,000,000,000 years ago?"). Scientific notation is also essential for dealing with extremely small numbers, such as the mass of a Hydrogen atom or the wavelength of visible light.

The following table and tips are intended to be used as a reference:

Words

Decimal Representation

Scientific Notation

Metric Prefix

Symbol

one billionth 0.000000001 1 x 10-9

nano

n
one hundred-millionth 0.00000001 1 x 10-8
one ten-millionth 0.0000001 1 x 10-7
one millionth 0.000001 1 x 10-6

micro

m
one hundred-thousandth 0.00001 1 x 10-5
one ten-thousandth 0.0001 1 x 10-4
one thousandth 0.001 1 x 10-3 milli m
one hundredth 0.01 1 x 10-2 centi c
one tenth 0.1 1 x 10-1 deci d
one 1 1 x 100
ten 10 1 x 101
one hundred 100 1 x 102
one thousand 1,000 1 x 103 kilo k
ten thousand 10,000 1 x 104
one hundred thousand 100,000 1 x 105
one million 1,000,000 1 x 106 Mega M
ten million 10,000,000 1 x 107
one hundred million 100,000,000 1 x 108
one billion 1,000,000,000 1 x 109 Giga G
ten billion 10,000,000,000 1 x 1010
one hundred billion 100,000,000,000 1 x 1011
one trillion 1,000,000,000,000 1 x 1012 Tera T

### Hints

• A number written in scientific notation consists of a coefficient (the part before the times sign) and an exponent (the power of 10 by which the coefficient is multiplied. For example, in 4.3 x 106 (which equals 4,300,000; four million three hundred thousand), 4.3 is the coefficient and 6 is the exponent. Sometimes the "times" symbol "x" is replaced by a dot, for example 4.3.106.
• When you multiply two numbers, you multiply the coefficients and add the exponents. For example,
4.3 x 106 x 2 x 102  = 8.6 x 108
4.3 x 106 x 2 x 10-2 = 8.6 x 104
• When you divide two numbers, you divide the coefficients and subtract the exponents. For example,
4.2 x 106 / 2 x 102 = 2.1 x 104
4.2 x 106 / 2 x 10-2 = 2.1 x 108
• When you move the decimal place in the coefficient one position to the left (i.e. you divide the coefficient by 10), you add one to the exponent. For example,
42 x 106 = 4.2 x 107
4200 x 106 = 4.2 x 109
42 x 10-6 = 4.2 x 10-5
• When you move the decimal place in the coefficient one position to the right (i.e. you multiply the coefficient by 10), you subtract one from the exponent. For example,
0.42 x 106 = 4.2 x 105
0.000043 x 106 = 4.3 x 101
0.42 x 10-6 = 4.2 x 10-7
Always adjust the decimal place in the coefficient so that the coefficient is always greater than one but less than ten. Mathematically it doesn't make any difference, but that is the standard practice, and it does make a number easier to read.
• When you add two numbers, you need to make their exponents equal. Take the number with the smaller exponent and move the decimal point to the left until its exponent matches the larger. Then add the coefficients and keep the (matching) exponent. For example,
4.2 x 106 + 6.4 x 105 = 4.2 x 106 + 0.64 x 106 = 4.84 x 106
4.2 x 10-6 + 6.4 x 10-5 = 0.42 x 10-5 + 6.4 x 10-5 = 6.82 x 10-5
9.2 x 1011 + 9.4 x 1010 = 9.2 x 1011 + 0.94 x 1011 = 10.14 x 1011 = 1.014 x 1012
• When you subtract two numbers, you again need to make their exponents equal. Take the number with the smaller exponent and move the decimal point to the left until its exponent matches the larger. Then subtract the coefficients and keep the (matching) exponent. Note that you might have to adjust the exponent when you are done to get into "standard form." For example,
4.2 x 106 - 6.4 x 105 = 4.2 x 106 - 0.64 x 106 = 3.56 x 106
4.2 x 10-6 - 6.4 x 10-5 = 0.42 x 10-5 - 6.4 x 10-5 = -6.38 x 10-5
1.2 x 1011 - 9.4 x 1010 = 1.2 x 1011 + 0.94 x 1011 = 0.26 x 1011 = 2.6 x 1010
• WARNING TO PEOPLE WHO USE CALCULATORS: Many calculators handle scientific notation. The exponent is usually displayed all the way on the right, with a space between it and the coefficient. To enter a number in scientific notation, you enter the coefficient, press the EXP key (one some calculators it is labeled EE) and enter the exponent. For example, 4. 05 means 4 x 105 . To enter the number 103, you have to enter 1. EXP 03. DO NOT enter 10. EXP 03, since that equals 10 x 103 or 1 x 104. This is a common mistake. See your calculator instruction booklet for more help.