Chemistry 30S
Math Skills
Name:
1
Exponents
An exponent is a shorthand notation for the number of times a number
is multiplied by itself. Examples:
n4 = n x n x n x n
a6 = a x a x a x a x a x a
24 = 2 x 2 x 2 x 2 = 16
32 = 3 x 3 = 9
Negative exponents are shorthand for the inverse of the corresponding
positive exponents. Examples:
n-4 = 1/n4 = 1/(n x n x n x n)
2 -4 = ½ x ½ x ½ x ½ = 1/16
3-2 = 1/3 x 1/3 = 1/9
Powers of Zero and One
Any nonzero number raised by the exponent 0 is 1. Examples:
n0 = 1
20 =1
9990=1
Any number raised by the exponent 1 is the number itself. Examples:
n1 = n
21 = 2
9991 = 999
Powers of Ten
In science, numbers are often written in powers of ten. For numbers
greater or equal to one, the exponent on the base 10 equals the
number of spaces to the right of the “1” that you place the decimal
point. Examples:
100 = 1
101 = 10
102 = 10 x 10 = 100
103= 10 x 10 x 10 = 1 000
2
Negative Powers of Ten
Negative exponents equal the inverse of the corresponding positive
exponents. For numbers less than one, a negative exponent on the
base 10 equals the number of spaces to the left of the “1” that you
place the decimal point
10-1 = 1/10 = 0.1
“one tenth”
-2
10 = 1/100 = 0.01 “one hundredth”
10-3= 1/1000 = 0.001 “one thousandth”
Practice: Evaluate the following
1) 28 =
2)
2-8 =
3)
34 =
4)
105 =
5)
10 -8 =
6)
100=
7)
51 =
8)
60 =
9)
10 -5 =
10)
10-1=
3
Scientific Notation
Do you know this number 300,000,000 m/sec? It's the speed of light!
Do you recognize this number 0.000 000 000 753 kg? This is the mass of
a dust particle!
Scientists have developed a shorter method to express very large and
very small numbers. This method is called scientific notation. Scientific
Notation is based on powers of 10.
Distance from Earth to the Sun: 149 600 000 km
In scientific notation this number is written as 1.496 x 108 km
Charge of one proton: 0.000 000 000 000 000 000 1602 Coulombs
In scientific notation this number is written as 1.602 x 10-19 Coulombs
A positive exponent indicates that the decimal point was shifted that
number of places to the left. A negative exponent indicates that the
decimal point was shifted that number of places to the right.
Writing numbers in Scientific Notation
Scientific Notation is a method of writing numbers in terms of decimal
numbers between one and 10 multiplied by a power of ten. There is
only one digit to the left of the decimal place, a digit of 1-9. Placeholder
zeros are dropped.
0.0005 all zeros are placeholders therefore it is written 5 x 10-4
35,000 all zeros are placeholders therefore it is written 3.5 x 104
1054 this zero is NOT a placeholder therefore it is written 1.054 x 103
Zeros between non zero digits (1-9) are not placeholder zeros. Zeros at
the beginning of a number are placeholder zeros. Zeros at the end of a
number are placeholder zeros only if they do not follow a decimal.
4
The number 123,000,000,000 in scientific notation is written as:
The first number 1.23 is called the coefficient. It must be greater than
or equal to 1 and less than 10. The second number is called the base. It
must always be 10 in scientific notation. The base number 10 is always
written in exponent form. In the number 1.23 x 1011 the number 11 is
referred to as the exponent or power of ten.
To write a number in scientific notation:
1. Put the decimal after the first digit and drop the non-significant
zeroes (placeholder zeros). In the number 123,000,000,000 the
coefficient will be 1.23
2. To find the exponent count the number of places the decimal was
moved. In 123,000,000,000 the decimal is moved from the end of
the last zero on the right 11 places to the left and stops between
the 1 and the 2. Therefore we write 123,000,000,000 as: 1.23x1011
Exponents are often expressed using other notations. The number
123,000,000,000 can be found written on your calculator display as:
1.23E+11 or 1.23 X 10^11
For small numbers we use a similar approach. Numbers less than 1 will
have a negative exponent. A millionth of a second is:
0.000001 sec.
or
1.0E-6
or
Examples: Write the following in scientific notation:
1.25 =1.25x100
35 000 = 3.5 x 104
25.78 = 2.578x101
0.0035 = 3.5 x 10-3
100 410 = 1.0041x105
35 000 000 =3.5 x 107
35 = 3.5 x 101
0.35 =3.5 x 10-1
5
Examples 1: Write the following numbers in scientific notation:
a) 2 232 000 =
b) -0.000 146 =
Examples 2: Write the following in standard notation:
a) 1.142 × 105 =
b) 1.485 × 10-6 =
Practice 1:
Write the following numbers in scientific notation:
a) 1001 =
b) 53 =
c) 6 926 300 000 =
d) -392 =
e) 0.003 61 =
f) 0.135 92 =
g) -0.003 8 =
h) 0.000 000 13 =
i) -0.567 =
j) 0.0137 =
Practice 2: Write the following scientific notations in standard notation:
a) 1.92 × 103 =
b) 3.051 × 101 =
c) -4.29 × 102 =
d) 6.251 × 109 =
e) 8.317 × 106 =
f) 1.03 × 10-2 =
g) 8.862 × 10-1 =
h) 9.512 × 10-8 =
i) -6.5 × 10-3 =
j) 3.159 × 102 =
6
Using Scientific Notation in Arithmetic Operations
Calculate the following:
a) 3.12 × 106 * 4.50 × 10-4 ÷ 8.13 × 10-2 =
b) (4.1357 × 10-15) (5.4 × 102) =
c)
d)
4.1357 × 10−15
5.4 × 102
=
(3.5 𝑥 102 )(1.5 𝑥102 )
(2.0 𝑥 103 )(5.0 𝑥 101 )
=
Practice: Calculate the following:
a. 1.695 × 104 ÷ 1.395 × 1015 =
b. (4.367 × 105) (1.96 × 1011) =
c. 6.97 × 103 * 2.34 × 10-6 + 3.2 × 10-2 =
d. 5.16 × 10-4 ÷ 8.65 × 10-8 + 9.68 × 104 =
e.
f.
3.0 𝑥 105
6.0 𝑥 102
=
(3.0 𝑥 102 )(2.0 𝑥102 )
(2.0 𝑥 102 )(4.0 𝑥 104 )
7
SI Measurement
The International System of Units (abbreviated SI from French: Système
international d'unités) is the world's most widely used system of
measurement and is the measurement system used in science.
The SI units we will use in Chemistry 30S include:
 Second (s) for measuring time
 Metre (m) for measuring length
 The mole (mol) for the amount of
Metric Prefixes
a substance
Text Symbol
Factor
Prefixes used in the SI system are
tera
T
1 000 000 000 000
shown in the table.
giga
G
1 000 000 000
Using the meaning of the prefixes, we
mega
M
1 000 000
can see that:
kilo
k
1 000
hecto
h
100
 1 kilometre = 1000 metres
deca
da
10
 1 kilogram = 1000 grams
(none) (none)
1
 1 centimetre = 0.01 metres
1
deci
d
0.1
or
of a metre
100
centi
c
0.01
 1 milligram = 0.001 grams
milli
m
0.001
1
micro
μ
0.000 001
or
of a gram
1000
nano
n
0.000 000 001
 1 Liter = 1000 milliliters
pico
p
0.000 000 000 001
8
Metric Prefixes
_____________________________________________________________________
Prefix
Symbol
Meaning
Number
Power of Ten
______________________________________________________________________
Prefixes that increase the size of the unit
megaM
one million
1 000 000
106
kilo-
1 000
103
Prefixes that decrease the size of the unit
decid
one tenth
0.1
10-1
centi-
c
one hundredth
0.01
10-2
milli
m
one thousandth
0.001
10-3
micro-

one millionth
0.000 001
10-6
nano-
n
one billionth
0.000 000 001
10-9
k
one thousand
___________________________________________________________________________________________________________
What are the equivalent measurements for the following?
1. 1 km = ____________m
2. 1 m = _____________cm
3. 1 m= _____________nm
4. 1 cm = ____________mm
5. 1 L = ______________mL
6. 1kg = ______________g
7. 1 hour = _________ min
8. 1 min = _________ s
9. 1 day =__________ hour
10. 1 year = _________ days
9
Conversions of Units
Conversion factors are a ratio of equivalent measurements. For
example since 1 m = 100 cm we can write the ratio in two ways:
100 𝑐𝑚
1𝑚
and
1𝑚
100 𝑐𝑚
Notice that in a conversion factor the measurement in the numerator
(on the top) is equivalent to the measurement in the denominator (on
the bottom). Conversion factors are useful in solving problems in which
a given measurement must be expressed in some other unit of
measure. When a measurement is multiplied by a conversion factor,
the numerical value is generally changed, but the actual size of the
quantity measured remains the same.
Some commonly used conversion factors include:
1𝑚
60 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
1 𝑘𝑔
60 𝑚𝑖𝑛𝑢𝑡𝑒𝑠
100 𝑐𝑚
1 𝑚𝑖𝑛𝑢𝑡𝑒
1000 𝑔
1 ℎ𝑜𝑢𝑟
Practice: Write conversion factors for the following measurements.
a) mg and g
b) m and km
c) day and hours
d) L and mL
e) Kg and g
f) µg and ng
10
Converting Between Units
In Chemistry you often need to express a measurement in a unit
different from the one given or measured initially. Problems in which a
measurement with one unit is converted to an equivalent
measurement with another unit are easily solved using conversion
factors.
Example 1: Convert 12.5 m to its equivalent in cm.
Example 2: Write 455 250 mm as its equivalent in kilometers.
Example 3: Determine how many minutes there are in 2 days.
Example 4: Convert 80 m/s to km/hr
11
Practice: Conversion Factor Calculations
1. Convert 14 mm to its equivalent in m.
2. How many kg is 3 500 grams?
3. Write 57 mg as its equivalent in kilograms.
4. Convert a speed of 20 m/s to its equivalent in km/h.
5. Express 50 km/h in m/s.
12
Directions: Perform the following conversions as indicated.
Length
1. 70 cm to m =
2. 49 cm to mm =
3. 8 m to mm =
4. 14.76 m to cm =
5. 8500 cm to m =
6. 250 mm to m =
7. 68.9 cm to mm =
8. 3.25 cm to mm =
9. 59.8 mm to cm =
10. 3.542 mm to cm =
11. 5.3 km to m =
12. 9.24 km to m =
13. 27.500 m to km =
14. 14.592 m to km =
15. 2.4 km to cm =
16. 1.95 km to cm =
Mass
17. 6 kg to mg =
18. 4.1 g to ng =
19. 8.7 g to kg =
20. 12.5 g to µg =
21. 925 mg to g =
22. 412 mg to g =
23. 8974 mg to g =
24. 5639 mg to cg =
25. 8.4 g to mg =
26. 2.79 g to mg =
27. 8.64 kg to g =
28. 4.53 mg to g =
Volume
17. 6 L to ml =
18. 4.1 L to ml =
19. 8.7 L to ml =
20. 12.5 L to ml =
21. 925 ml to L =
22. 412 ml to L =
23. 8974 ml to L =
24. 5639 ml to L =
25. 8.4 L to ml =
26. 2.79 L to ml =
27. 8.64 ml to L =
28. 4.53 ml to L =
13
Solving Literal Equations
Rearrange the equation to solve each formula for the indicated
variable.
1. 𝒚 = 𝒙 + 𝒃 (𝑓𝑜𝑟 𝑥)
2. 𝑷 = 𝒂 + 𝒃 – 𝒄 (𝑓𝑜𝑟 𝑏)
3. 𝒚 = 𝟓𝒙 – 𝒃 (𝑓𝑜𝑟 𝑏)
4. 𝑷𝑽 = 𝒏𝑹𝑻 (𝑓𝑜𝑟 𝑇)
5. 𝑷𝑽 = 𝒏𝑹𝑻 (𝑓𝑜𝑟 𝑅)
6. 𝑷𝑽 = 𝒏𝑹𝑻 (𝑓𝑜𝑟 𝑃)
7. 𝑨 =
8. 𝒂 =
9. 𝑺 =
𝒃
( 𝑓𝑜𝑟 𝑐)
𝒄
(𝒃 – 𝒄)
(𝑓𝑜𝑟 𝑏)
𝟐
𝒑−𝒌
(𝑓𝑜𝑟 𝑝)
𝒎
10. °𝑪 = 𝑲 − 𝟐𝟕𝟑
𝟓
11. °C = (°F -32)( )
𝟗
(𝑓𝑜𝑟 𝐾)
(for °F)
14
Rules for Determining Significant Figures in Measurements
1. All nonzero numbers are significant.
2. Zeros may or may not be significant depending on their position in
the number. Examples are shown on the table below.
Rules
Examples # of sig figs.
1. A number is a significant figure if it is
a) A nonzero digit
6.5 g
132.34 m
b) A zero between significant figures
305 m
2.056 kg
20.0
c) A zero at the end of a decimal number
50. L
28.0 cm
18.00 g
2
5
3
4
3
2
3
4
d) Any digit in a number written in scientific notation
4.0 x 105 m
2
-3
6.70 x 10 g 3
2. A number is not significant if it is
a) A zero at the beginning of a decimal number (between a
decimal point and a nonzero digit)
0.0008 kg
1
0.0953 m
3
b) A zero used as a placeholder in a large number without a
decimal point
750 000 km
2
1 430 000 mm
3
15
Practice: Determine the number of significant figures in the numbers
below.
Number
a) 500
b) 5.00
c) 0.500
d) 5.0 x 102
e) 5
# of sig. figs.
Number
f) 500.00
g) 0.0005
h) 5 0005
i) 55 000
j) 0.5
# of sig. figs
Rules for Rounding Off
When doing calculations from measured numbers using a calculator,
you get answers that give several digits. However, your answer cannot
be more precise than your actual measurements. For example if you
are calculating the area of a piece of cloth that measures 6.3 m by 3.4
m, the answer is 21.42 m2. Your measurements only have 2 significant
figures; so all four digits cannot be significant. You must round off your
final answer to two significant figures giving 21 m2.
There are two rules to remember when rounding off numbers:
1. If the first digit to be dropped is 4 or less, it and the following
digits are just dropped.
Ex. Rounding off 5.3132 to 3 significant figures = 5.31
2. If the first digit to be dropped is 5 or greater, the last retained
digit is increased by 1.
Ex. Rounding off 15.684 to 3 significant figures = 15.7
16
Practice 1: Round to 1 significant figures.
Number
Rounded
number
a) 444.4
b) 6.666
c) 83.241
Number
Rounded
number
d) 65 281
e) 0.12345
f) 0.0002361
Practice 2: Round to 2 significant figures.
Number
Rounded
number
a) 444.4
b) 6.666
c) 83.241
Number
Rounded
number
d) 65 281
e) 0.12345
f) 0.0002361
Practice 3: Round to 3 significant figures.
Number
Rounded
number
a) 444.4
b) 6.666
c) 83.241
Number
Rounded
number
d) 65 281
e) 0.12345
f) 0.0002361
Practice 4: Round to 4 significant figures.
Number
a) 444.44
b) 6.6666
c) 83.241
Rounded
number
Number
Rounded
number
d) 65 281
e) 0.12345
f) 0.00023610
17
Rules for Rounding using Significant Figures
When multiplying and dividing, round the answer to the lowest
number of significant figures found in the question. For example:
2.2 x 3.25 = 7.15 rounding to 2 sig. figs gives an answer of 7.2
30.5/ 5 = 6.1 rounding to 1 sig fig. gives an answer of 6
Note: Exact numbers, such as the number of people in a room, have an
infinite number of significant figures. Exact numbers are made by
counting up how many of something are present, they are not
measurements made with instruments. Another example of this are
defined numbers, such as 1 foot = 12 inches. There are exactly 12
inches in one foot. Therefore, if a number is exact, it DOES NOT affect
the accuracy of a calculation nor the precision of the expression.
When adding and subtracting, round the answer to the lowest number
of decimal places in the question. For example:
2.2 + 3.25 = 5.45 would round to one decimal place 5.5
36.2 – 0.06 = 36.14 would round to one decimal place 36.1
36.222 – 0.06 = 36.14 would round to two decimal places of 36.1
18
Practice: Round your answer accordingly to the rules.
1) 7846 X 92437 X 235.649 X 3300=
2) 583.00 ÷ 83=
3) (57.6 X 3) ÷ (34 X 3.00 X 87.507)=
4) 78.00 + 45.6 + 0.00467 + 39.45 + 276.999=
5) 567.000 - 12=
6) 8597 - 0.l=
7) (3.50 X 105) X [2.8 ÷ (5.4 - 4.09)]=
8) (6.10 X 107 ) + (3 X 107 )=
9) 787 X 3.0=
10) 2.34 x 33.5 =
11) 3.461728 + 14.91 + 0.980001 + 5.2631 =
12) 23.1 + 4.77 + 125.39 + 3.581 =
13) 22.101 - 0.9307 =
14) 0.04216 - 0.0004134 =
15) 564 321 – 264 321 =
16) (3.4617 x 107) ÷ (5.61 x 10¯4) =
17) [(9.714 x 105) (2.1482 x 10¯9)] ÷ [(4.1212) (3.7792 x 10¯5)] =
18) (4.7620 x 10¯15) ÷ [(3.8529 x 1012) (2.813 x 10¯7) (9.50)] =
19) [(561.0) (34 908) (23.0)] ÷ [(21.888) (75.2) (120.00)] =
20) (2.5 x 104)2 =
19
Appendix A: Solutions to Practice Questions
Page 3 Answers:
1) 256
2) 0.00390625 3) 81
4) 100 000
5) 0.000 000 01 6) 1 7) 5 8) 1 9) 0.00001
10) 0.1
Page 6 Answers:
1 a) 1.001 X 103 b) 5.3 x 101 c) 6.9263 x 109
d) -3.92 x 102
e) 3.61 x 10-3 f) 1.3592 x 10-1
g) -3.8 x 10-3 h) 1.3 x 10-7 i) -5.67 x 10-1 j) 1.37 x 10-2
2a) 1 920 b) 30.51 c) -429 d) 6 251 000 000 e) 8 317 000 f) 0.010 3
g) 0.886 2 h) 0.000 000 095 12 i)-0.0065 j) 315.9
Page 7 Answers:
a) 1.215 × 10-11
c) 4.83 × 10-2
e) 5.0 x 102
b) 8.56 × 1016
d) 1.03 × 105
f) 7.5 x 10-3
Page 9 Answers:
1.
1000 m
2.
100 cm
3.
1 000 000 000 nm
4.
10 mm
5.
1000 ml
6.
1000 g
7.
60 min
8.
60 s
9.
24 hours
10. 365 days
20
Page 10 Answers:
1000 𝑚𝑔
a)
1𝑔
b)
c)
d)
e)
f)
1000 𝑚
1 𝑘𝑚
1 𝑑𝑎𝑦
24 ℎ𝑜𝑢𝑟𝑠
1000 𝑚𝑙
1𝐿
1000 𝑔
1 𝑘𝑔
1 µ𝑔
1000 𝑛𝑔
and
and
and
and
and
and
Page 12 Answers:
1. 0.014 m 2. 3.5 kg
1𝑔
1 000 𝑚𝑔
1 𝑘𝑚
1000 𝑚
24 ℎ𝑜𝑢𝑟𝑠
1 𝑑𝑎𝑦
1𝐿
1 000 𝑚𝑙
1 𝑘𝑔
1 000 𝑔
1000 𝑛𝑔
1 µ𝑔
3. 0. 000057 kg
4. 72 km/h
5. 14 m/s
21
Page 13 Answers
Length
1. 70 cm to m = 0.7 m
2. 49 cm to mm = 490 mm
3. 8 m to mm = 8 000 mm
4. 14.76 m to cm = 1 476 cm
5. 8500 cm to m = 8.5 m
6. 250 mm to m = 0.25 m
7. 68.9 cm to mm = 689 mm
8. 3.25 cm to mm = 32.5 mm
9. 59.8 mm to cm = 5.98 cm
10. 3.542 mm to cm = 0.3542 cm
11. 5.3 km to m = 5 300 m
12. 9.24 km to m = 9 240 m
13. 27.500 m to km = 0.027500 km
14. 14.592 m to km = 0.014592 km
15. 2.4 km to cm = 240 000 cm
16. 1.95 km to cm = 195 000 cm
Mass
17. 6 kg to mg = 6 000 000 mg
18. 4.1 g to ng = 4 100 000 000 ng
19. 8.7 g to kg = 0.0087 kg
20. 12.5 g to µg = 12 500 000 µg
21. 925 mg to g = 0.925 g
22. 412 mg to g = 0.412 g
23. 8974 mg to g = 8. 974 g
24. 5639 mg to cg = 563.9 cg
25. 8.4 g to mg = 8 400 mg
26. 2.79 g to mg = 2 790 mg
27. 8.64 kg to g = 8 640 g
28. 4.53 mg to g = 0.00453
Volume
17. 6 L to ml = 6 000 ml
18. 4.1 L to ml = 4 100 ml
19. 8.7 L to ml = 8 700 ml
20. 12.5 L to ml = 12 500 ml
21. 925 ml to L = 0.925 L
22. 412 ml to L = 0.412 L
23. 8974 ml to L = 8.974 L
24. 5639 ml to L = 5.639 L
25. 8.4 L to ml = 8 400 ml
26. 2.79 L to ml = 2790 ml
27. 8.64 ml to L = 0.00864 L
28. 4.53 ml to L = 0.00453 L
22
Page 14 Answers:
1. 𝑥 = 𝑦 – 𝑏
2. 𝑏 = 𝑃 – 𝑎 + 𝑐
𝑃𝑉
4. 𝑇 =
𝑛𝑅
𝑃𝑉
𝑛𝑅𝑇
5. 𝑅 =
6. 𝑃 =
𝑛𝑇
𝑉
8. 𝑏 = 2𝑎 + 𝑐 9. 𝑝 = 𝑆𝑚 + 𝑘
9
11. °F= °C( )+ 32
3. 𝑏 = 5𝑥 – 𝑦
𝑏
7. 𝑐 =
𝐴
10.K = °C + 273
5
Page 16 Answers:
Number
a) 500
b) 5.00
c) 0.500
d) 5.0 x 102
e) 5
# of sig. figs.
1
3
3
2
1
Number
f) 500.00
g) 0.0005
h) 5 0005
i) 55 000
j) 0.5
# of sig. figs
5
1
5
2
1
Page 17 Answers:
Practice 1: Round to 1 significant figures.
Number
Rounded
Number
number
a) 444.4
400
d) 65 281
b) 6.666
7
e) 0.12345
c) 83.241
80
f) 0.0002361
Practice 2: Round to 2 significant figures.
Number
a) 444.4
b) 6.666
c) 83.241
Rounded
number
440
6.7
83
Rounded
number
70 000
0.1
0.0002
Number
Rounded
number
d) 65 281
65 000
e) 0.12345
0.12
f) 0.0002361 0.000 24
23
Practice 3: Round to 3 significant figures.
Number
Rounded
Number
number
a) 444.4
444
d) 65 281
b) 6.666
6.67
e) 0.12345
c) 83.241
83.2
f) 0.0002361
Practice 4: Round to 4 significant figures.
Number
a) 444.44
b) 6.6666
c) 83.241
Page 19 Answers:
1) 5.6 X 1014
2) 7.0
3) 0.02
4) 440.1
5) 555
6) 8597
7) 750000
8) 9 X 107
Rounded
number
444.4
6.667
83.24
Rounded
number
65 300
0.123
0.000236
Number
Rounded
number
d) 65 281
65 280
e) 0.12345
0.1235
f) 0.0002361 0.0002361
9) 2400
10)
78.4
11)
24.61
12)
156.8
13)
21.170
14)
0.04175
15)
3.00000 x 105
16)
6.17 x 1010
17)
13.40
18)
4.62 x 10¯22
19)
2.28 x 103
20)
6.3 x 108
24