# EARTH SCIENCE LAB Metric System and Scientific Notation

### Metric Units

The basic units of measure in the Metric System are:

 Length Meter (m) Capacity (liquid measure) Liter (l) Weight Gram (g) Pressure Bar (b) Temperature Celsius (°C), Kelvin (K) Force Newton

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

 deka- 10 Example: dekagram 10 grams hecto- 100 Example: hectoliter 100 liters kilo- 1000 Example: kilometer 1000 meters

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

 deci- 0.1 Example: decimeter 0.1 meters centi- 0.01 Example: centigram 0.01 grams milli- 0.001 Example: millibar 0.001 bars

### Metric System - English System Conversions

Changing metric units is as simple as multiplying or dividing by 10 or 1000. The diagram below shows you how to convert one unit into another by a series of multiplication or division. Changing English units isn't as simple, but can still be accomplished by a series of multiplications of divisions.

km - kilometer, m - meter, dm - decimeter, cm - centimeter, mm - millimeter

### Metric-English System Standard 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The United States is one of the few countries left in the world that still insists on using an archaic and outdated measurement system.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 standard conversions:
Enter all values without commas - example 5,678 would be entered as 5678.
Enter decimals less than 1 with a zero before the decimal point - example 0.25 or 0.0033.
Enter repeating decimals with an "r" designation - example 0.15733333333333 would be entered as 0.1573r.

English System Standard Conversions
1 inch = feet
1 inch = yards
1 inch = miles (report to 8 decimal places)
1 foot = inches
1 foot = yards
1 foot = miles (report to 8 decimal places)
1 yard = inches
1 yard = feet
1 yard = miles (report to 8 decimal places)
1 mile = inches
1 mile = feet
1 mile = yards

Metric System Standard Conversions
1 mm = cm
1 mm = m
1 mm = km
1 cm = mm
1 cm = m
1 cm = km
1 m = mm
1 m = cm
1 m = km
1 km = mm
1 km = cm
1 km = m

### English System Conversions

Convert the following English to English values.
Enter all values without commas - example 5,678 would be entered as 5678.
Enter decimals less than 1 with a zero before the decimal point - example 0.25 or 0.0033.

7 ft = in
3,450 in = ft
45 yd = ft
10,912 yd = mi
18,300 ft = yd
75.3 mi = yd
63.9 ft = in
4 yd = ft
63 mi = yd
19 mi = in
7 mi = in
1,938 mi = ft
5809 yd = ft
48 ft = yd
0.09 in = ft
82.368 in = mi
4,443 ft = yd
0.318 ft = yd
84.48 ft = mi
3,172 yd = in
0.78 yd = ft
4.4352 in = mi
0.75 mi = ft
1,531.2 ft = mi
91 mi = yd
0.34 yd = ft
0.44 ft = in
443.52 in = mi
54 in = ft
7389 in = yd

### Metric System Conversions

Convert the following metric to metric values.
Enter all values without commas - example 5,678 would be entered as 5678.
Enter decimals less than 1 with a zero before the decimal point - example 0.25 or 0.0033.

7 cm = mm
3,450 mm = cm
45 m = cm
10,946 m = km
18,300 cm = m
75.3 km = m
63.9 cm = mm
4 m = cm
63 km = m
19 km = mm
7 km = mm
1,938 km = cm
5809 m = cm
48 cm = m
0.09 mm = cm
83 mm = km
4,443 cm = m
0.32 cm = m
88 cm = km
3,172 m = mm
0.79 m = cm
5 mm = km
0.75 km = cm
1,544 cm = km
91 km = m
0.34 m = cm
0.44 cm = mm
452 mm = km
54 mm = cm
7389 mm = m

### English to Metric System Conversions

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 the appropriate value by 2.54.

To convert inches to centimeters, multiply the inches by 2.54.

Example 1: To convert 5 inches to centimeters, multiply 5 by 2.54.
Example 2: 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 the following English to Metric values.
Enter all values without commas - example 5,678 would be entered as 5678.
Enter decimals less than 1 with a zero before the decimal point - example 0.25 or 0.0033.

9 in = mm
850 ft = cm
76 yd = cm
252 mi = km (report to 2 decimal places)
36 in = m
393 yd = m
65 mi = mm
844 ft = cm
59 ft = m
90 mi = mm
652 yd = mm
710 in = cm
4 in = cm
612 ft = m
100 yd = cm
363 yd = km (report to 2 decimal places)
48 in = m
432 mi = m
96 ft = km
333 in = mm
12 yd = cm
489 mi = km (report to 3 decimal places)
5 ft = cm
199 in = km
23 mi = m
271 in = cm
43 ft = mm
200 mi = km
40 in = cm
411 in = m

### Metric to English System Conversions

To convert Metric to English we can use the same conversion factor as used for English to Metric conversions.

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 dividing the appropriate value by 2.54.

To convert centimeters to inches, divide the centimeters by 2.54.

Example 1: To convert 12 cm to centimeters, divide 12 by 2.54.
Example 2: To convert 7 meters to yards requires some additional steps. First convert 7 meters to centimeters (7 x 100 = 700). Then divide 700 by 2.54. This results in an answer of 275.5905512 inches. Then convert inches to yard (divide by 36). The result is 7 meters is equal to 7.65529 yards.

Convert the following Metric to English values.
Enter all values without commas - example 5,678 would be entered as 5678.
Enter decimals less than 1 with a zero before the decimal point - example 0.25 or 0.0033.

Report all answers in this section to 2 decimal places, unless otherwise directed.

8.6 km = mi
840 cm = yd
87 mm = ft
301 m = ft
19 km = yd
2.8 cm = in
49 mm = in
420 m = ft
0.44 km = yd
2302 mm = yd
36 m = in
99 km = ft
2.4 m = in
3.401 mm = yd (report to 4 decimal places)
0.2 cm = in
27 cm = ft
85 m = ft
0.62 km = mi
40 m = yd
770 m = mi
59 cm = yd
16 mm = ft
0.9211 m = ft
84 km = yd
161 cm = in
75 m = yd
30 km = mi
2 mm = in
57010 m = mi
0.5498592 km = ft

### 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

### Entering Scientific Notation into a Calculator

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 or EXP key. Check your calculator manual for detailed directions if the EE or EXP 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.

### Calculator Scientific Notation Display

Scientific calculators will be capable of displaying large and small numbers in scientific notation. There are several common ways in which the calculator will display these numbers. Some will be able to display the notation as discussed above, such as 10.5 x 1015. The other two common methods are shown below. 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.

### Standard to Scientific Notation

Write the following standard notation number using scientific notation. Enter the number and the exponent of 10 in the appropriate boxes.

 1. 1,000,000 = x 10
 2. 1,000,000,000 = x 10
 3. 1,000,000,000,000 = x 10
 4. 1,000,000,000,000,000 = x 10
 5. 0.001 = x 10
 6. 0.000001 = x 10
 7. 0.000000001 = x 10
 8. 0.000000000001 = x 10
 9. 0.00000000000000000000000000000000000000000000000765 = x 10
 10. 17,650,000,000,000 = x 10
 11. 864,760,000,000,000,000,000,000 = x 10
 12. 0.0000000000000000643 = x 10
 13. 1,872,900,000,000,000,000,000,000,000,000,000 = x 10
 14. 0.000007633 = x 10
 15. 8,980,000 = x 10
 16. 0.000000000852 = x 10
 17. 109,876,000,000,000,000 = x 10
 18. 0.0000000000000000000000000523 = x 10
 19. 423,000,000,000,000 = x 10
 20. 0.0087 = x 10
 21. 3,900,000,000 = x 10
 22. 0.000000000000000000000000000000000000641 = x 10
 23. 9,999,000,000,000,000,000 = x 10
 24. 0.0000006543 = x 10
 25. 83,970,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 = x 10
 26. 687,890 = x 10
 27. 0.9 = x 10
 28. 5,768,000,000,000 = x 10
 29. 0.0000000000000000000000000000000000000000000000000000000000000000000982 = x 10
 30. 75,870,000,000,000,000,000,000,000,000 = x 10

### Scientific to Standard Notation

Write the following scientific notation number using standard notation.

3.48 x 1027 =
1.85 x 10-39 =
4.732 x 1035 =
3.9 x 10-16 =
9.6157 x 1019 =
7.58 x 102 =
4.81 x 1023 =
3.833 x 10-9 =
4.8 x 1050 =
9.31 x 1010 =
4.872 x 10-5 =
5.61 x 100 =
1.31 x 1012 =
8.067 x 10-4 =
7.0221 x 10-10 =
4.82 x 1015 =
9.588 x 1012 =
7.053 x 10-18 =
7.775 x 1020 =
3.01 x 101 =
1.59 x 10-3 =
5.758213 x 103 =
6.155 x 1021 =
7.421 x 10-2 =
7.325 x 102 =
2.45 x 1025 =
5.25 x 10-25 =
9.673 x 10-11 =
6.74 x 107 =
4.61 x 1010 =

### Scientific Notation Calculations

Complete the following calculations. Write the answer using standard notation.

(7.1 x 1020)(2.54 x 102) =
( 8.01 x 1030) ÷ (9.0 x 1018)=
(2.62 x 1021) ÷ ((1.31 x 109)(2.0 x 1015)) =
(3.6 x 1036)(7.2 x 101)(5.9 x 10-6) =
(3.755 x 107) + (9.78 x 104)(4.69 x 102) =
(3.93 x 1015) ÷ (9.6 x 109) =
(1.08 x 104) ÷ (4.5 x 108) =
(9.66 x 1016) - (5.45 x 109)(9.23 x 106) =
(4.5 x 108)(7.8 x 105)(8.2 x 10-8)(6.3 x 105) =
(9.7 x 109)(2.8 x 104) ÷ ((1.4 x 105)(1.6 x 108)) =
(1.92 x 1012)(4.5 x 106) - (9.2 x 1018) =
(1.99 x 105) + (1.114 x 105) =
(9.2 x 105) ÷ (1.15 x 107) =
(1.812 x 1040)(6.555 x 109) =
(8 x 104)4 =

### Scientific Notation Calculations

Complete the following calculations. Write the answer using scientific notation. Enter the number (round to 2 decimals) and the exponent of 10 in the appropriate boxes.

 1. (6.482 x 104)(1.42 x 108) = x 10
 2. (4.85 x 1012)/(3.4 x 10-3) = x 10
 3. (2.76 x 1016)(8.6 x 10-17) = x 10
 4. (6.26 x 101)(5.678 x 100) = x 10
 5. (3 x 104)(6.3 x 105) = x 10
 6. (7 x 10-3)(6.1 x 10-8) = x 10
 7. (4.4 x 105)/(9.6 x 1012) = x 10
 8. (7.3 x 104)/(2.8 x 10-7) = x 10
 9. (3.5 x 1027) + (4.73 x 1027) = x 10
 10. (7.58 x 102) - (3.833 x 103) = x 10
 11. (4.8 x 103)3 = x 10
 12. (8.067 x 10-4)-7 = x 10
 13. sin(7.325 x 106) = x 10
 14. log(2.45 x 1025) = x 10
 15. 1/((1.81 x 1040)(1.59 x 10-3)) = x 10