Download Scientific Notation

Scientific Notation Information Page Astronomy Session 5
What Is Scientific Notation?
Scientific notation is a shorthand method for writing very large or very small numbers.
• An astronomer uses scientific notation to express very large numbers, such as distances between
planets or stars. For example, the distance from our Sun to the nearest star (Proxima Centauri) is
almost 25.8 trillion miles. This number can be written in two ways:
Standard number form: 25,800,000,000,000 mi
•
Scientific notation: 2.58 x 1013 mi
A physicist or chemist uses scientific notation to express very small numbers, such as the sizes
of atoms or molecules. For example, a hydrogen atom measures 12 ten-billionths of a meter
across. This number can also be written in two ways:
Standard number form: .00000000012 m
Scientific notation: 1.2 x 10-10 m
Writing a Number in Scientific Notation
A number written in scientific notation has two parts: the base* and the power of ten.
• The base is obtained by moving the decimal point until you have a number between one and 10.
Ž Move the decimal point 13 places to the left to obtain a base of 2.58.
Ž Move the decimal point 10 places to the right to obtain a base of 1.2.
•
The power of ten is the number of times you must multiply the base by 10 to return to the
original number. It consists of the number 10 and an exponent – a small number above and to the
right of the 10. The exponent is obtained by counting how many places you had to move the
decimal point to get a base between one and 10.
Ž For 25,800,000,000,000, the exponent is +13. Because the number is larger than one, you
moved the decimal point to the left; therefore, the exponent is positive.
Ž For .00000000012, the exponent is -10. Because the number is less than one, you moved the
decimal point to the right; therefore, the exponent is negative.
Orders of Magnitude
Sometimes powers of 10 are called orders of magnitude. This phrase is usually used to indicate whether
a particular calculation or estimate is within the correct range of values. For example:
Estimate of a person’s height:
6 feet
60 feet
(correct estimate)
(high by one order of magnitude, or one power of ten)
Estimate of the distance from Earth to the Sun: 93,000,000 miles
93,000 miles
(correct estimate)
(low by three orders of magnitude)
* Note: If no base is given, the base is assumed to be one; for example, 106 = 1 x 106.
© 2001‐2009 Pitsco Education 1 MO•0501•0409•04 Scientific Notation Information Page Astronomy Session 5
Table of Selected Powers of Ten
Power
of Ten
12
10
1011
109
106
103
102
101
100
10-1
10-2
10-3
10-4
10-6
10-9
10-10
10-12
10-14
Long Number
1,000,000,000,000
100,000,000,000
1,000,000,000
1,000,000
1,000
100
10
1
0.1
0.01
0.001
0.0001
0.000001
0.000000001
0.0000000001
0.000000000001
0.00000000000001
Place Designation
trillions
hundred billions
billions
millions
thousands
hundreds
tens
ones
tenths
hundredths
thousandths
ten thousandths
millionths
billionths
ten billionths
trillionths
hundred trillionths
Prefix
Example of Unit
Using Prefix
Real-World Example*
teraradius of solar system
gigamegakilohectodeka-
kilometer
decicentimilli-
decimeter
centimeter
millimeter
micronano-
micrometer
nanometer
pico-
picometer
radius of Earth
height planes fly
height of building
height of child
paper length
paper thickness
radius of atom
radius of nucleus
* For measurements of length in meters, these measurements of objects or distances are within one
power of ten (one order of magnitude) of the power described in the given row of the table.
Calculating with Scientific Notation
Multiplication:
• Multiply the bases and add the exponents.
• Adjust the resulting number so that the base is between one and 10.
• Example: (4 x 104) x (8 x 1010) = 32 x 1014 = 3.2 x 1015
Division:
• Divide the bases and subtract the exponents.
• Adjust the resulting number so that the base is between one and 10.
• Example: 20 x 109 = 4 x 104
5 x 105
Addition or Subtraction:
• Write both numbers in the same power of 10.
• Add or subtract the base numbers.
• Example: 5.4 x 106 + 3.2 x 109 = .0054 x 109 + 3.2 x 109 = 3.2054 x 109
© 2001‐2009 Pitsco Education 2 MO•0501•0409•04