Download A Quick Review of the Metric System and Scientific Notation

A Quick Review of the
Metric System and Scientific Notation
The Metric System
The United States of America is the only industrialized nation in the world that uses its
own system to weigh and measure things. Since the fundamental purpose of having defined
units of weight and measure is to standardize communication and understanding, Americans
are at a distinct disadvantage with the US Customary System of Measurement, a derivative
of the British Imperial System. All other countries, except Liberia (Myanmar began switching
in 2013), use Le Systeme International d’Unites, known also as the SI system or Metric
System, as their official system of weights and measures. Even the United Kingdom adopted
the SI system for official purposes in 1965.
What does this mean to you? When you want to travel the world or communicate in
science, you have more thinking to do. But the good thing is that you are already familiar
with the system that is harder and more confusing. The US Customary System has over 300
different units to measure various quantities, and converting from one unit to another for the
same physical property follows no logical pattern. For example, for length there are 12
inches in one foot, 3 feet in one yard, and 1760 yards to a mile. For volume there are 16
tablespoons in one cup, 2 cups in one pint, 2 pints in one quart, and 4 quarts in one gallon.
And for mass there are 16 ounces in one pound and 2000 pounds in one ton. And don’t
confuse the 16 weight ounces in one pound with the 8 fluid ounces in one cup! If you were
measuring water, 2 cups of water would be about one pound, but 2 cups of flour will be only
about half a pound. The metric system is far easier to understand as it is based on units of
10. Conversion between units for the same physical property (length, volume, or mass)
simply involves shifting the decimal place.
The base units in the metric system are the meter/metre (m) for length, the liter/litre (L)
for volume, and the gram (g) for mass. Greek or Latin prefixes are then used on these base
units to denote powers of ten:
Prefix
pico
nano
micro
milli
centi
deci
base unit
deka
hecto
kilo
mega
giga
tera
Amy Warenda Czura, Ph.D.
Symbol
p
n
µ
m
c
d
da
h
k
M
G
T
Factor
Equivalent
0.000000000001
0.000000001
0.000001
0.001
0.01
0.1
1
10
100
1,000
1,000,000
1,000,000,000
1,000,000,000,000
-12
10
10-9
10-6
10-3
10-2
10-1
100
101
102
103
106
109
1012
1
Name
trillionth
billionth
millionth
thousandth
hundredth
tenth
one
ten
hundred
thousand
million
billion
trillion
SCCC BIO130 Metric Review
The prefix indicates the factor, the number of places to the decimal, as compared to the base
unit. For example there are 1,000 millimeters (mm) in a meter (m) and 1,000 meters (m) in a
kilometer (km). And since the same prefixes apply to meters, liters and grams, there is only
one scheme to remember for three different forms of measure. Note that when writing out
the symbols for metric units, the abbreviations are not punctuated, and there must be a space
between the number and unit abbreviation. Also the abbreviations are case-sensitive, for
example: g is gram, but G is giga. When writing quantities where the number is less than
one, always use a zero before the decimal; 0.2 mL for example.
Although the liter is the base unit for volume in the metric system, the volume of an
object can also be measured as a distance cubed (length X width X height). 1 cm3 is equal to
1 mL. Cubic centimeters (cc or cm3) is a unit of measure that is commonly used for injectable
drugs. If a doctor prescribed injecting a patient with 5 cc of a certain medication, this would
be equivalent to 5 mL (which is also equivalent to 1 teaspoon).
Temperature in the metric system is measured in degrees Celsius (°C). On this scale,
water freezes at 0°C and boils at 100°C. On the Fahrenheit scale, these changes in the state
of water, the most important substance for life, are far more inconvenient numbers (32°F and
212°F, respectively).
As many details in the biology class will be discussed in metric terms, it can be
confusing to comprehend and put into context in your US Customary System of
Measurement mind. For example, if you were on vacation in Australia and the weather report
indicated a high of 35, would you be thinking you needed to dress in shorts or a winter coat?
(35°C = 95°F so hopefully you are in shorts!) And then you go to the petrol station (gas
pump) and the price says 1.43/L; how does this compare to home where the price was $2.06
per gallon when you got on the plane? (Two issues here, first they are measuring the gas in
liters, 3.79 liters per gallon, and the price is in AU$ (Aussie dollars) not USD (American
dollars). With the exchange rate it will be $1.01 USD/liter X 3.79L/gal = $3.83 per gallon. So
the gas is a lot more expensive and you will have to remember to drive on the wrong (left)
side of the road! If you had gone to France instead, the pump would have said 1.34/L. With
the currency in Euros, that comes to $5.46 per gallon, even more expensive, but at least you
would be driving on the correct (right) side of the road.)
To help you put some of the metric into perspective, here are a few common items that
you are likely familiar with:
Item
Can of soda pop
Family size bottle of soda pop
Family size jug of milk
Pint of beer
Shot of whiskey
Box of four sticks of butter
Bag of flour or sugar
5 K run
Marathon
Human body temperature
A fever of 104
A room temperature incubation
Your refrigerator
Amy Warenda Czura, Ph.D.
US Customary
12 fluid ounces
2.1 quarts / 67.6 fluid ounces
1 gallon
1 pint / 16 fluid ounces
1.5 fluid ounce
1 pound / 16 ounces
5 pounds
3.1 miles
26.2 miles
98.6°F
104°F
77°F
39°F
2
Metric
355 milliliters
2 liters
3.8 liters
473 milliliters
44 milliliters
454 grams
2.3 kilograms
5 kilometers
42.2 kilometers
37°C
40°C
25°C
4°C
SCCC BIO130 Metric Review
And although not entirely accurate due to rounding for simplicity sake, here are a few
equivalents you can use to try to understand the scope and scale of some of the units of
measure:
Length
1 inch = 2.5 cm = 25 mm
1 foot = 30.4 cm
1 yard = 0.9 m
1 mile = 1.6 km = 1,600 m
39.4 inches = 1 m = 100 cm = 1,000 mm
Mass
1 ounce = 28.3 g
1 pound = 454 g
2.2 pounds = 1 kg = 1,000 g
Temperature
°F = °C X 9/5 + 32
°C = 5/9 X (°F – 32)
Volume
1 teaspoon = 5 mL
1 fluid ounce = 30 mL
1 pint = 16 fluid ounces = 473 mL
1 quart = 2 pints = 32 fluid ounces = 0.95 L
1 gallon = 4 quarts = 8 pints = 128 fluid ounces = 3.79 L
Scientific Notation
Scientists frequently have data with extremely large or extremely small numbers
that can be awkward to write out and work with in mathematical equations. A way to
make these numbers more manageable is to express them in scientific notation. In
scientific notation the numbers are composed of the coefficient, the base, and the
exponent. The coefficient has to be greater than or equal to 1 but less than 10, the
base is always 10, and the exponent is the number of places the decimal has to be
moved to change the number to its standard notation.
Let us consider the average number of red blood cells that could be in an
average adult male. There are approximately 5.2 million RBCs per µL of blood and
approximately 5 L of blood in the body.
The equation would look like this:
5,200,000 RBC/µL X 1,000,000 µl/L X 5 L = 26,000,000,000,000 RBCs
If the numbers were expressed in scientific notation, the equation would look like this:
5.2 X 106 RBC/µL X 1 X 106 µL/L X 5 L = 2.6 X 1013 RBCs
Notice how 5,200,000 (5.2 million) was converted: the coefficient is 5.2, the base is 10
and the exponent is 6. You had to move the decimal 6 places to the left to create a
coefficient number that is greater than one and less than ten.
If we were working with very small numbers, the decimal would have to move to
the right and we would note that with a negative exponent. For example, if we assume
Amy Warenda Czura, Ph.D.
3
SCCC BIO130 Metric Review
the average red blood cell weighs 27 picograms (27 trillionths of a gram), and express
this in grams, the number would be 0.000000000027 g (27 X 10-12). And in standard
-11
scientific notation: 2.7 X 10 g. We had to move the decimal 11 places to the right to
create a coefficient number that is greater than one and less than ten.
To multiply numbers written in scientific notation, multiply the coefficients, and
add the exponents. To divide numbers written in scientific notation, divide the
coefficients and subtract the exponent of the divisor/denominator from the exponent of
the dividend/numerator. Always be sure to express the resulting answer in standard
scientific notation, and round your final answer to have the same number of significant
digits as the least number of significant digits in any of the starting quantities.
9
-2
Multiplication Example:
(5.3 X 10 ) X (8.0 X 10 ) =
Multiply the coefficients: 5.3 X 8.0 = 42.4
Add the exponents: 9 + (-2) = 7
7
Formulate the answer: 42.4 X 10
8
Express in proper standard scientific notation: 4.24 X 10
8
Round to significant digits: 4.2 X 10
5
-4
Division Example:
2.5 X 10 ÷ 6.0 X 10 =
Divide the coefficients: 2.5 ÷ 6.0 = 0.4167
Subtract the exponents: 5 - (-4) = 9
9
Formulate the answer: 0.4167 X 10
8
Express in proper standard scientific notation: 4.167 X 10
8
Round to significant digits: 4.2 X 10
When adding or subtracting numbers written in scientific notation, all of the
numbers should be converted to the same exponent value prior to performing the
calculations. This means that the decimal place of coefficient may have to move
outside of the rule of a value of more than one and less than 10 for the purpose of the
calculation. However, once the calculation is complete, always express the resulting
answer in standard scientific notation, and round your final answer to the same number
of decimal places as the least number of decimal places in any of the starting quantities.
9
7
Addition Example:
7.54 X 10 + 9.01 X 10 =
9
9
Convert to the same exponent: 7.54 X 10 + 0.091 X 10 =
Add the coefficients: 7.54 + 0.091 = 7.631
9
Formulate the answer: 7.631 X 10
9
Round to significant digits: 7.63 X 10
Amy Warenda Czura, Ph.D.
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SCCC BIO130 Metric Review