EARTH SCIENCE LAB
Metric System and Scientific Notation

Metric System

Metric Units

The basic units of measure in the Metric System are:

Multiples of the basic units are designated by the following prefix: (the "c" is pronounced like a "k")

Fractions of the basic units are designated by the following prefix: (the "c" is pronounced like an "s")

Metric System Conversions

Changing metric units is as simple as multiplying or dividing by 10, 100 or 1000. The diagram below shows you how to convert one unit into another by a series of multiplication or division.

Click here to see the English Conversion diagram.

Convert the following values. (enter all values without commas - example 5,678 would be entered as 5678)
1. 7 cm = mm
2. 3,450 mm = cm
3. 45 m = cm
4. 10,946 m = km
5. 18,300 cm = m
6. 75.3 km = m
7. 63.9 cm = mm
8. 4 m = cm
9. 63 km = m
10. 19 km = mm

Metric-English System Conversions

The United States is one of the few countries left in the world that still insists on using an archaic and outdated measurement system. The English System of weights and measures is not based on a consistent system. For example, lengths are based on 12 or 36, 12 inches in a foot and 36 inches in a yard. Pounds are further subdivided based on 16: 16 ounces in a pound. It makes for a messy and inconsistent system of measure that results in a lot of confusion.

The Metric System is a base 10 system. This is much more intuitive for humans (we are born with 10 fingers and toes!) and the math system that we use is based on 10 as well. Units and subunits within the Metric System are multiples of 10.

Unfortunately, within the United States it has become necessary to learn two systems. The English System and the Metric System. We do use the Metric System in this country to a greater and greater extend as the years pass.

Conversion of Metric units to English and English to Metric requires the use of a conversion factor or equation. In this course we will primarily be concerned with measurements of length, so this exercise will only deal with those conversions.

Complete the following English-English and Metric-Metric conversions (enter all values without commas - example 5,678 would be entered as 5678):
11. 1 foot = inches
12. 1 yard = feet
13. 1 yard = inches
14. 1 mile = feet
15. 1 mile = yards
16. 1 mile = inches
17. 1 kilometer = meter
18. 1 meter = decimeters
19. 1 meter = centimeters
20. 1 meter = millimeters
21. 1 centimeter = millimeters
22. 1 km = millimeters

To convert English to Metric we need a conversion factor.

1 inch = 2.54 centimeters

With this one conversion factor it is possible to convert all units of length from one system to the other. Memorize this conversion factor. It is simply a matter of multiplying or dividing the appropriate value by 2.54. For example to convert 5 inches to centimeters, multiply 5 by 2.54. To convert 5 cm to inches, divide 5 by 2.54. To convert 5 feet to meters requires two additional steps. First convert 5 feet to inches (5 x 12 = 60). Then multiply 60 by 2.54. This results in an answer of 152.4 centimeters. Then convert centimeters to meters. The result is 5 feet is equal to 1.524 meters.

Convert English to Metric units. (enter all values without commas - example 5,678 would be entered as 5678)(enter all decimal values less than one with a zero before the decimal - example 0.53, 0.32)(round all decimals to 2 decimal places)
23. 5 inches = cm
24. 14 inches = m
25. 5 foot, 8 inches = mm
26. 3.6 yards = m
27. 1 mile = km

Convert Metric to English units. (enter all values without commas - example 5,678 would be entered as 5678)(enter all decimal values less than one with a zero before the decimal - example 0.53, 0.32)(round all decimals to 2 decimal places)
28. 5.08 cm = inches
29. 7.112 m = inches
30. 3.048 mm = feet
31. 1.83 m = yards
32. 1 km = mile

Scientific Notation

The mass of the Earth is 5,979,000,000,000,000,000,000,000,000 kilograms. The molecular diameter of ammonia is 0.0000000297 centimeters. Very large and very small numbers are prone to errors when writing these numbers or typing them. To prevent these types of errors, these numbers can be written in a form called scientific notation. This allows numbers to be written as the product of a power of 10 and a number greater than or equal to 1, but less than 10. The mass of the Earth in scientific notation is 5.979 x 10²⁷ kg and the molecular diameter of ammonia is 2.97 x 10^-8 cm.

The simplest method for conversion of standard notation to scientific notation is move the decimal. For very large numbers, move the decimal to the left, until it is after the first numeral. Count the number of places the decimal has moved. This then becomes the exponent on the 10.

EXAMPLE
Write 4,567,000 using scientific notation.
4,567,000 becomes 4.567000
The decimal place has been moved 6 places to the left.
4,567,000 = 4.567 x 10⁶

For very small numbers, move the decimal to the right, until it is after the first non-zero numeral. Count the number of places the decimal has moved. This becomes a negative exponent on the 10.

EXAMPLE
Write 0.00000005436 using scientific notation.
0.00000005436 becomes 00000005.436
The decimal place has been moved 8 places to the right.
0.00000005436 = 5.436 x 10^-8

To convert scientific notation to standard notation, reverse the procedure given above. For very large numbers (positive exponent on the 10), move the decimal to the right the same number of places as given by the exponent. For very small numbers (negative exponent on the 10), move the decimal to the left the same number of places as given by the exponent.

EXAMPLE
Write 7.943 x 10⁶ using standard notation.
7.943 x 10⁶ = 7,943,000

EXAMPLE
Write 4.302 x10^-7 using standard notation.
4.302 x10^-7 = 0.0000004302

NOTE: Fractions which are less than 1 but greater than -1 should always be written with a zero before the decimal. This is to avoid confusion in cases where the decimal is not clearly marked or looks like a small 1 instead of a decimal.

EXAMPLE
.453 should be written as 0.453

EXAMPLE
-.234 should be written as -0.234

Scientific calculators will be capable of expressing large and small numbers in scientific notation. There are two common ways in which the calculator will display these numbers. The first uses a small letter E in place of the "x 10".

EXAMPLE
10.5 x 10¹⁵ will be displayed as
10.5 E 15
The 15 is the exponent of the 10.

EXAMPLE
3.463 x10^-24 will be displayed as
3.463 E -24
The -24 is the exponent of the 10.

The other common method of displaying scientific notation is to place a space between the number and the exponent.

EXAMPLES
10.5 x 10¹⁵ will be displayed as
10.5 15
The 15 is the exponent of the 10.

EXAMPLE
3.463 x10^-24 will be displayed as
3.463 -24
The -24 is the exponent of the 10.

Write the following using scientific notation. Use the "E" format for "x10": example 3.463 E -24 (the space before and after the E is important)
33. 7,456,000,000 =
34. 384,400 =
35. 7423 =
36. 543,332,000,000,000,000,000 =
37. 0.000032 =
38. 0.065 =
39. 0.000000000000000000000062 =

Write the following using standard notation. (enter all values without commas - example 5,678 would be entered as 5678)(enter all decimal values less than one with a zero before the decimal - example 0.53, 0.32)
40. 6.482 x 10⁴ =
41. 1.42 x 10^-8 =
42. 4.85 x 10¹² =
43. 3.4 x 10^-3 =
44. 2.76 x 10¹⁶ =
45. 8.6 x 10^-17 =
46. 6.26 x 10¹ =

Entering scientific notation into a calculator is accomplished in various manners, depending on the type of calculator you are using. A common method uses the EE key. Check your calculator manual for detailed directions if the EE key is not present or ask the instructor for help.

EXAMPLE
To enter the number 7.42 x 10²², begin by entering 7.42. Then press the EE key. Then enter 22. You may then proceed with normal calculations.

EXAMPLE
To enter the number 3.64 x 10^-13, begin by entering 3.64. Then press the EE key, then press the (+/-) key. Then enter 13. NOTE: The (+/-) key is not the same as the (-) function key. The (+/-) key changes the value from positive to negative, while the (-) function key performs subtraction.

Perform the indicated operations. Express all answers in scientific notation.
Use the "E" format for "x10": example 3.46 E -24 (round all decimals to 2 decimal places)(the space before and after the E is important)
47. (3 x 10⁴)(6.3 x 10⁵) =
48. (7 x 10^-3)(6.1 x 10^-8) =
49. (4.4 x 10⁵)/(9.6 x 10¹²) =
50. (7.3 x 10⁴)/(2.8 x 10^-7) =