Scientific Notation
Any quantity can be expressed using a power of ten. As you move the decimal
point, you multiply by 10 as many times as necessary to make the numbers
equal. Consider the following examples:
325
325
325
325
=
=
=
=
325 x 1
32.5 x 10
3.25 x 10 x 10
0.325 x 10 x 10 x 10
= 325 x 100
= 32.5 x 101
= 3.25 x 102 Written in scientific notation
= 0.325 x 103
A number in scientific notation has two parts. The number in front of the “x 10” is
called the coefficient. The power to which 10 is raised is called the exponent.
The coefficient must have one and only one digit in front of the decimal point.
The exponent will always be a whole number in chemistry, although the exponent
can be fractional number in other disciplines.
There are three rules for using scientific notation:
Rule 1: To express a number in scientific notation, you move the decimal
point to the position such that there is one nonzero digit to the left of the
decimal point.
Rule 2: If the decimal point is moved to the left, the exponent is positive.
Rule 3: If the decimal point is moved to the right, the exponent is negative.
A mnemonic that my help you remember how to keep the signs straight is
“Registered Nurses Love Patients”, or RN, LP, which stands for right --- negative,
left --- positive. If you move the decimal point to the right, the exponent is
negative. If you move the decimal point to the left, the exponent is positive.
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Scientific Notation Example 1:
Express 0.0003821 in scientific notation.
Solution:
0.0003821 = 3.821 x 10-4
The decimal point was moved to the right four spaces. If the decimal point moves
to the right, the exponent is negative. Since it moved four spaces, the exponent is a
–4. Notice that the decimal point was moved until one and only one nonzero digit
was in front of the decimal point.
Scientific Notation Example 2:
Express 56,873 in scientific notation.
Solution:
5.6873 x 104
The decimal point was moved to the left four spaces. If the decimal point moves to
the left, the exponent is positive. Since it moved four spaces, the exponent is a +4.
Notice that the decimal point was moved until one and only one nonzero digit was in
front of the decimal point.
Sometimes it is necessary to change a number from one power of ten to another power of
ten. When changing to a new power of ten, we must consider the sizes of both the
coefficient and the power of ten. In the number 3.2 x 105, the coefficient is 3.2 and the
power of ten is 5. If we wish to express this as another power of 10 that is equal in value
to 3.2 x 105, we must increase the power of ten if the coefficient decreases, and decrease
the power of ten if the coefficient increases. This is shown in the illustration below.
Scientific Notation Example 3:
Fill in the blank: 6.79 x 104 = ________ x 106
Solution:
The exponent has gotten larger. This means that we have multiplied the coefficient
by 10 two more times. Because the exponent has gotten larger by 2 places, the
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coefficient must get smaller by two places. We must change the 6.79 to 0.0679,
which is the value that goes in the blank.
6.79 x 104 = 0.0679 x 106
Although you will probably use your calculator to perform arithmetic operations on
numbers written in scientific notation, you should know the rules that govern these
processes. The rules are presented below.
Addition and Subtraction
1. Express the numbers as the same power of 10.
2. Perform the mathematical operation on the coefficients (add or subtract).
3. Bring down the power of 10 unchanged.
4. Express the answer in proper scientific notation.
Scientific Notation Example 4:
Subtract 4.72 x 102 from 5.32 x 103
Solution:
Make the exponents equal before we subtract the numbers. We can make both of
the exponents 102 or we can make them both 103. The result will be the same no
matter which common exponent we choose. We’ll make them both 102. Therefore
5.32 x 103 = 53.2 x 1024. Now we can subtract the two numbers. Since the final
answer has more than one nonzero digit in the coefficient, we need to rewrite the
number in proper scientific notation.
53.20 x 102
- 4.72 x 102
48.48 x 102 or 4.848 x 103
Multiplication and Division
To multiply two numbers expressed in scientific notation, multiply the coefficients and add
the exponents on the powers of 10.
To divide two numbers expressed in scientific notation, divide the coefficients and subtract
the exponents on the powers of 10.
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Scientific Notation Example 5:
What is the product of 5.1 x 10-4 and 3.2 x 10-2?
Solution:
Multiply 5.1 by 3.2 to get 16. Add the exponents to get –6. The answer is 16 x 10-6.
Change this number to proper scientific notation. The answer is 1.6 x 10-5.
Hints for Using Scientific Notation:
Is it necessary to use scientific notation?
Not every calculation requires scientific notation, but it is a very convenient system for
handling the very large and very small numbers commonly encountered in chemistry.
Scientific notation shows up very frequently in chemistry, so mastering it now will pay off in
the future.
Can I use a calculator to solve problems with number in scientific notation?
Yes. You will need to find out how to input scientific notation numbers into your calculator.
Check your calculator manual for directions. The most common mistake in this chapter
is entering a number in scientific notation incorrectly into a calculator!
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Rounding
It is often necessary to round numbers after performing calculations. This is usually
necessary when the calculated answer has more significant figures than are allowed. The
rules for rounding numbers are presented below.
1. If the last digit to be kept is followed by a number less than 5, round by dropping
all digits to the right of it.
2. If the last digit to be kept is followed by a number equal to or greater than five,
round up by adding 1 to the last digit to be kept and drop the rest.
Rounding Example 1:
Round the following number to the indicated number of significant digits.
54.674873
0.8765
5.875
rounded to four digits
rounded to two digits
rounded to three digits
Solution:
The last digit to be kept in 54.674873 is a seven. The seven is followed by a
number less than five, so drop all of the digits after the seven in blue. The answer
is 54.67.
The last digit to be kept in 0.8765 is a seven. The seven is followed by a number
greater than or equal to 5, so round up. The answer is 0.88.
The last digit to be kept in 5.875 is a seven. The seven is followed by a number
greater than or equal to 5, so round up. The answer is 5.88.
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Conversion Factors
Conversion factors are ratios of one object to another object. A ratio is a way of comparing
two quantities. The quantities can be compared in three different ways: a to b, a:b, or a/b.
At the local grocery store, a case of soda contains 24 cans. We can express the ratio in
three forms:
(a to b) 24 cans of soda to each case of soda
(a:b)
24 cans of soda: 1 case of soda
(a/b)
24 cans of soda/1 case of soda
This chapter uses conversion factors frequently to solve problems. Conversion factors are
ratios written in the fraction form (a/b).
Some of the conversion factors you will see in this chapter include metric-to-metric
conversion factors such as:
1 gram of silver is 1000 mg of silver
1 liter of water is 1 x 106 µL of water
The most common way that chemists represent ratios like these is in a fraction form. The
conversion factors would look like this in the fraction form:
1 gram silver
1000 mg silver
1 liter water
1 x 106 µL water
Each of these conversion factors can be written in the inverse form. The first ratio we
examined involved cans of soda. We expressed that ratio as 24 cans of soda in 1 case of
soda. The inverse form of this ratio is 1 case of soda contains 24 cans of soda. The key
idea here is that either the original ratio or its inverse form is true. It is true that one
case of soda contains 24 cans and it is true that there are 24 cans of soda in one case.
Conversion factors in chemistry can be written as shown above or in the inverse form.
Regardless of which form is used, the comparison the conversion factor makes is true.
For example, one gram of silver weigh 1000 mg and 1000 mg of silver weighs one gram.
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Here are the conversion factors written in the inverse form:
1000 mg silver
1 gram silver
1 x 106 µL water
1 liter water
The obvious question at this point is: Which form of the conversion factor should you use
in a problem? You use the form that allows the units to cancel. Units cancel when the
unit that appears on the top also appears on the bottom of a fraction somewhere in the
solution. This style of problem solving, where you arrange the conversion factors so that
the units cancel, is called unit analysis.
Conversion Factor Example 1
Which is the correct solution for the question: How many cases of soda can be made from
1200 cans of soda?
1200 cans of soda x
24 cans of soda = 28800 cans of soda2/case of soda
1 case of soda
1200 cans of soda x
1 case of soda = 50 cases of soda
24 cans of soda
Solution:
In which solution do the units cancel? In the second solution, the unit “cans of
soda” cancels. The correct solution will use conversion factors in a way that units
will cancel. Blue lines are drawn through units that cancel.
Conversion Factor Example 2
How many milligrams of silver are in 2.76 grams of silver
Solution
This question uses the metric-to-metric conversion factor: 1 gram = 1000 mg. Set
up the solution so that the units cancel. Blue lines are drawn through units that
cancel.
2.76 grams silver x 1000 mg silver = 2760 mg silver
1 gram silver
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Conversion Factor Example 3
If 8.35 x 109 µL of water was collected in a flask, how many liters of water was collected?
Solution
You don’t need to concern yourself with whether the conversion factor is inverted or
not. Just make sure that the units cancel. Blue lines are drawn through units that
cancel.
8.35 x 109 µL water
x
1 liter water = 8.35 x 103 liters water
1 x 106 µL
Hints for using conversion factors:
How do you know what quantity to start with when solving problems?
Start with the quantity given in the problem and not with the conversion factor. In general,
the first quantity in your unit analysis will not have a denominator. In the last example, we
started with the number of microliters of water:
8.35 x 109 µL water
x
1 liter water = 8.35 x 103 liters water
1 x 106 µL
Notice that the first quantity is a whole number and not a fraction. If you start with the
conversion factor, it is much harder to decide whether to use the original or inverse form of
the conversion factor.
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Should I enter all of the numerator numbers into the calculator before I enter the
denominator numbers, or should I enter each conversion factor into the calculator
before I move on to the next conversion factor?
Either style of entering the numerical quantities will work. You should make a decision to
always use one style of entering numbers into a calculator. This helps you avoid skipping
a number when you are calculating an answer.
In the example calculation below, you could enter the numbers in either of these orders
and obtain the same answer.
16 x 2.54 x 50 = 75.2
3
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Enter the numbers into your calculator as:
16 x 2.54 ÷ 3 x 50 ÷ 9 =
or
16 x 2.54 x 50 ÷ 3 ÷ 9 =
You don’t need to press the ENTER key after each mathematical.
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Fractions, Decimals and Percents
Fractions, decimals, and percentages are all related to each other. Each is used to
indicate some part of a whole quantity. For example, ¾, 0.75, and 75% are all used to
express a quantity representing three parts out of four. The fraction represents the
number of parts of the whole (3 parts of the whole, which is divided into 4 equal sized
parts).
The decimal represents the part of one whole (0.75 of the whole).
The percentage represents the number of parts of a whole that has been divided into 100
parts (75 parts of the whole which has been divided into 100 equal parts).
Divide pie up into
100 segments
The fraction, 3/4, is converted to a decimal by dividing the numerator (the number on top of
the fraction) by the denominator (the number on bottom of the fraction). Three divided by
four is 0.75.
The decimal is converted to the percentage by moving the decimal point two places to the
right. This is equivalent to multiplying the decimal by 100% in order to convert it to the
percentage. In this conversion, 0.75 becomes 75%
To convert a percent to a decimal, we move the decimal point two places to the left. The
decimal point in 75% is understood to be after the five in the percentage. In this
conversion, 75% becomes 0.75.
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In this chapter, we use percentages more often than decimals or fractions. The term
percent means “per cent” or “per 100.” When we say that 10% of people prefer Brand A,
we mean that 10 people out of 100 people prefer Brand A. When someone has 71% of
the votes in an election, it means that 71 people out of 100 people voted for that person. A
chemist might say that a substance is 64% nitrogen. The chemist means that 64 atoms
out of every 100 atoms are nitrogen.
A percentage is another conversion factor. It can be written as a fraction and used in
problem solving (also called unit analysis). Here are the examples of percentages we
used above written as conversion factors.
10 people prefer Brand A
100 people expressed an opinion
71 people voted for candidate
100 people voted
64 atoms of nitrogen
100 atoms present
The secret to using percentages correctly is to write the units correctly. It is very
important that the units for the “per 100” part indicate what substance is being measured or
counted. The numerator or top of the fraction should clearly indicate what small portion of
the whole quantity is being measured or counted. In other words, a percentage should be
written as:
The Part
The Whole
Percentage Example 1:
How would you present each percentage as a fraction?
47% of applicants were women
15% of atoms were sulfur
Solution
47 female applicants (the Part)
100 total applicants (the Whole)
15 atoms sulfur (the Part)
100 atoms total (the Whole)
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Sometimes we are asked to calculate percentages from actual measurements. In this type
of problem, a percent is not given in the problem.
Percentage Example 2:
What percent of people in the class are women if 30 of the 50 students are female?
Solution:
Remember to divide the part by the whole. What is the “whole” class number? 50.
We divide the “part” (which is 30 female students) by the “whole” (which is 50 male
and female students). Then, we multiply by 100 to create the percentage.
30 women x 100 = 60%
50 students
Chemists frequently use percentages in calculations. Remember, it is important that you
include units in your calculations to avoid mistakes.
Percentage Example 3:
A total of 476 people were surveyed. When asked to choose the best tasting pizza, 34.2%
of them chose Brand B. How many people selected Brand B as the tastiest pizza?
Solution:
First, convert the percentage into a conversion factor. Remember that the
denominator is always 100 (from the “per cent” part of the question).
The conversion factor we would write would be:
34.2 of people selected Brand B
100 people surveyed
The total number of people we surveyed was MORE than 100. It was 476. How do
we include this information in our calculations?
476 people surveyed x
34.2 of people selected Brand B
100 people surveyed
= 163 people selected Brand B
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Notice how the units cancel! We can check the answer to see if it is reasonable. If 476
people were surveyed, is it reasonable that less than 476 chose Brand B? Yes. The
percentage of people selecting Brand B was less than 50%. Is 163 less than half of 476?
Yes.
Percentage Example 4:
If 54% of the mass of a mixture of iron and copper is iron, how much iron is there in a
sample that has a mass of 25 grams?
Solution:
What is the conversion factor we would write for the percentage?
54 grams of iron
100 grams mixture
How did we choose the units “grams?” The unit in which the measurement was
done was given in the problem. Now, let’s use the conversion factor that we
obtained from the percentage in the solution.
25 grams mixture x
54 g iron
100 gram mixture
= 14 grams iron
Again, check to see if the answer is reasonable. If about half of the mixture was
iron and the mixture weighed 25 grams, we would expect an answer of about 12.5
grams. Our estimate of the answer agrees with the calculated answer.
Percentage Example 5:
If a mixture of iron and copper was 54% iron, how much of the mixture would you need to
use to have 200. grams of iron (plus the copper impurity in the mixture)?
Solution:
This question uses the same conversion factor as the previous example. But how
should you arrange the conversion factor so that the units cancel?
200. grams iron x 100 grams mixture = 370 g mixture
54 grams iron
Notice that we used the inverse form of the percentage. We know to do this so that
the units cancel. If you set up the problem so that the units cancel, you will be on
the right track to get the correct answer!
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Hints for using percentages:
What units do you choose when turning the percentage into a conversion factor?
Look at the question to see what unit of measurement is being used or what substance
was being counted. If the question describes centimeters of metal tubing, then both
numerator and the denominator will be in centimeters. If the question involved milliliters of
solution, then both the numerator and the denominator will be in milliliters. Add other
words to the units to help you distinguish “The Part” from “The Whole.”
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