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Math 300
Basic College Mathematics
Chapter 1
The Whole Numbers
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
Page 1 of 22
1.2 Place Value and Names for Numbers
Place Value
 A digit is a number 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 that names a
place-value location.
 For large numbers, digits are separated by commas into groups
of three called periods.
 Each period has a name: ones, thousands, millions, billions,
trillions, and so on.
 See the place value chart below.
Place-Value Chart for Whole Numbers
HundredTenThousands Thousands Thousands Hundreds
100,000
10,000
1,000
100
Tens
Ones
10
1
Beginning with the digit to your right, the place value goes as follows:
ones… tens… hundreds,… thousands… ten-thousands… hundredthousands,… millions… ten-millions… hundred-millions,… billions…
ten-billions... hundred-billions,… trillions… and so on.
You must learn the place value chart for whole numbers.
It will help you write the place value of a digit
as well as writing a number in words.
Example: Determine the place value of the digit 5 in the whole
number.
1. 45, 987
Write the whole number in words.
2. 45,987
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.3 Adding Whole Numbers and Perimeter
 Sum – The total when adding 2 or more numbers together.
 Addends – The numbers that are added together.
 Additive identity –The sum of zero and any number is that
number. Example: 7  0  7
 Associative law of addition –Changing the grouping of addends
does not change their sum. Example: 3  5  7  3  12  15 and
3  5  7  8  7  15
 Commutative law of addition –Changing the order of two
addends does not change their sum.
Example: 2  3  5 and 3  2  5
 Perimeter – The distance around an object. To find the
perimeter of an object, add all sides.
Example: Add.
1.
1521
+ 348
(Hint: Begin adding the digits in a column from right to left.)
2. 10,120 + 12,989 + 5738
(Hint: Set this problem up vertically by lining up the numbers from right to left)
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.4 Subtracting Whole Numbers
 Subtraction is finding the difference of two numbers. The first
number is called the minuend and the second is called the
subtrahend.
Subtraction Properties of Zero
(1) The difference of any number and that same number is equal to
zero. Example: 11  11  0
(2) The difference of any number and zero is equal to that same
number. Example: 45  0  45
Examples: Subtract.
1. 526
– 323
(Hint: Begin by subtracting in a column from right to left.)
Subtracting with Borrowing
 When subtracting vertically, if a digit in the second number
(bottom #) is larger than the corresponding digit in the first
number (top #), borrowing is needed.
 Sometimes we may have to borrow from more than one place.
 Let’s look at the examples below to help us practice borrowing.
Be sure to take notes and ask plenty of questions.
2. 7007
– 6349
3. 6000
– 3149
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
4. 9035
– 7489
Page 4 of 22
1.4 Subtracting Whole Numbers (cont)
Solving Problems by Subtracting
Key Words or Phrases
subtract
difference
less
less than
take away
decreased by
subtracted from
Examples
subtract 5 from 8
the difference of 10 and 2
17 less 3
2 less than 20
14 take away 9
7 decreased by 5
9 subtracted from 12
Symbols
8–5
10 – 2
17 – 3
20 – 2
14 – 9
7–5
12 – 9
Solve.
5. Subtract 9 from 21.
6. When Lou and Judy Zawislak began a trip, the odometer read
55,492. When the trip was over, the odometer read 59,320. How
many miles did they drive on their trip?
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.5 Rounding and Estimating
 Steps for rounding a number
1. Locate the digit in the given place value.
2. Look at the digit to the right of the given place value.
3. When looking at the digit to the right of the given place value,
determine one of the following:
a. If this digit is 5 or higher, add 1 to the digit in the given
place value and replace each digit to its right with zero. All
digits to the left will remain the same.
or
b. If this digit is less than 5, keep the digit in the given place
value the same and replace each digit to its right with zero.
All digits to the left will remain the same.
1. Round 467 to the nearest ten.
2. Round 4645 to the nearest hundred.
3. Round 7500 to the nearest thousand.
When estimating, you will need to round first, then perform the
operation:
4. Estimate the difference by first rounding to the nearest hundred,
the performing the operation:
6852
- 1748
_____
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.6 Multiplying Whole Numbers and Area
 The numbers that we multiply are called factors.
 The result of a multiplication problem is called a product.
Example:
3 x 5 = 15
{In this case, 3 and 5 are the factors and 15 is the product.}
 The product of zero and any whole number is equal to zero.
 The product of 1 and any whole number is equal to that whole
number. This also called the multiplicative identity.
 Commutative law of multiplication – You can multiply two
numbers in any order as long as you are multiplying
throughout the problem.
Rule: a  b  b  a (The answer will be the same)
 Associative law of multiplication – It does not matter how
you group the numbers, the answer will be the same as long
as you are multiplying throughout the problem.
Rule: a  b  c  a  b  c (The answer will be the same)
 The area of rectangle can be found by using the formula below:
Area = length  width or A  l  w
Multiply.
1. 6078
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.6 Multiplying Whole Numbers and Area (cont)
Multiply.
2. 432  375
Estimate the product by first rounding to the nearest hundred, then
multiplying:
3. 355 x 299
4. What is the area of this region?
129 yd
65 yd
Solving Problems by Multiplying
Key Words or Phrases
multiply
product
times
Example
multiply 5 by 7
the product of 3 and 2
10 times 13
Symbols
57
32
10  13
Solve.
5. One ounce of hulled sunflower seeds contains 14 grams of fat.
How many grams of fat are in 6 ounces of hulled sunflower seeds?
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.7 Dividing Whole Numbers

To divide, you should divide the dividend by the divisor to get
the quotient.
The dividend is the number to be divided.
The divisor is the by which the divided is divided.
The quotient is the answer.
Example:
20  5  4
20
4
5
4
5 20
In the above problems 20 is the dividend, 5 is the divisor, and 4
is the quotient.
Other rules to make note of when dividing:
 Division by 1 – Any number divided by 1 is equal to that
number.
Examples:
7
7
1
 Dividing a number by itself – Any number (nonzero) divided by
itself is equal to 1.
Examples:
55
1
55
 Dividends of 0 – Zero divided by any number (nonzero) is equal
to zero.
Examples:
0
0
11
 Excluding division by 0 – Division by 0 is undefined or not
defined.
Examples:
3
 undefined
0
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.7 Dividing Whole Numbers (cont)
 With larger numbers, you will need to use a process called
long division.
 When using long division you may run into a case where you
are left with a remainder. A remainder is the number that is
left over from the dividend. The remainder is to be less than
the divisor. The symbol for a remainder is R.
Divide.
1.
56
1
2. 699  3
3. 6 4846
4. 27 9724
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.7 Dividing Whole Numbers (cont)
Key Words or Phrases
divide
Examples
divide 10 by 5
quotient
the quotient of 64
and 4
9 divided by 3
divided by
divided or shared
equally among
$100 divided
equally among five
people
Symbols
10
5
64
64  4 or
4
10  5 or
9
3
100
100  5 or
5
9  3 or
Solve.
5. Find the quotient of 90 and 7.
6. Martin teaches American Sign Language classes for $55 per
student for a 7-week session. He collects $1430 from the group of
students. Find how many students are in the group.
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.8 An Introduction to Problem Solving
Problem-Solving Steps
1. UNDERSTAND the problem. (Read and reread the problem. Know
what you are looking for.)
2. TRANSLATE the problem. (Write the problem as an equation.)
3. SOLVE the problem. (Work out the problem and check to see if the
answer you get is reasonable.)
4. INTERPRET the result. (Check your work and state your answer.)
Key words or phrases use to help indicate which operation to use
Addition
 
sum
Subtraction Multiplication


difference
product
Division
Equality
quotient
equals


plus
minus
times
divide
is equal to
added to
subtract
multiply
is/was
more than
less than
multiply by
shared
equally
among
increased
by
total
decreased
by
less
of
divided by
double/triple
divided into
yields
Solve.
1. What is the product of 12 and 9?
2. 78 decreased by 12 is what number?
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.8 An Introduction to Problem Solving (cont)
Solve.
3. A parking lot in the shape of a rectangle measures 100 feet by 150
feet.
a. What is the perimeter of the lot?
b. What is the area of the parking lot?
4. Three people dream of equally sharing a $147 million lottery. How
much would each person receive if they have a winning ticket?
5. Find the total cost of 10 computers at $2100 each and 7 boxes of
diskettes at $12 each.
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.9 Exponents, Square Roots, and Order of Operations
Exponential Notation
 Exponential notation is a shorter notation for writing products
that occur often.
Example:
3  3  3  3 can be shorten to the exponential notation of 3 4
Note: The 3 is called the base number and the 4 is called the
exponent.
The exponent tells you how many times
to
multiply the base times itself
Writing Exponential Notation
Example: Write exponential notation for 10  10  10  10  10
5
Answer: 10
Example: Write exponential notation for 2  2  2
3
Answer: 2
Evaluating Exponential Notation
To evaluate exponential notation, we rewrite it as a product and
compute the product.
Example: Evaluate 10
3
3
Answer: 10 3 = 10  10  10 = 1,000 not 10 = 10  3 = 30
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.9 Exponents, Square Roots, and Order of Operations (cont)
4
Example: Evaluate 5
4
Answer: 5 = 5  5  5  5 = 625
Evaluating Square Roots
 A square root of a number is one of two identical factors of the
number.
Example: 7  7  49 , so a square root of 49 is 7.
 We use a radical sign
  for finding square roots.
Example: Find each square root.
1.
25  5 because 5  5  25
2.
81  9 because 9  9  81
3.
0  0 because 0  0  0
 Make sure to understand the difference between squaring a
number and finding the square root of a number.
Squaring a number: 9 2  9  9  81
Find the square root of a number:
9 3
* A list of perfect squares can be found on page 767. *
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.9 Exponents, Square Roots, and Order of Operations (cont)
Finding the Area of a Square
Formula for finding the area of a square
Area of a square = side 
2
or
Area of a square = side

side
Example: Find the area of a square whose side measures 5 inches.
Answer: A = side 
2
A = 5 
2
A = 25 square inches
Example: Find the area of this square.
8 cm
Answer: A = side 
2
A = 8 
2
A = 64 square cm
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.9 Exponents, Square Roots, and Order of Operations (cont)
“Please Excuse My Dear Aunt Sally”
P stands for parentheses…………………( ), [ ], { },
Always work the innermost set of parentheses first!
Think inside-out!!
2
4
E stands for exponents……………….… 2 , 3 , 45
Be sure to evaluate exponents correctly!
The exponent (small number) tell you how many times
to multiply the base (big number) to itself!!
M stands for multiplication………….… , , *,
D stands for division……………….….… /, 
Multiplication and division go together!
It must be worked from left to right!!
A stands for addition…………….……... +
S stands for subtraction……………..… 
Addition and subtraction go together!
It must be worked from left to right!!
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.9 Exponents, Square Roots, and Order of Operations (cont)
Write exponential notation.
1. 5  5  5
Evaluate.
2.
63
Find the square root.
3.
36
Simplify.
4. 42  7  6
5.
512  7   4
5 2  18
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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1.9 Exponents, Square Roots, and Order of Operations (cont)
Simplify.


2
2
6. 35  3  9  7  2  10  3
7. 3  25  2  81
8. 8 x 9 – (12 – 8)/4 – (10 – 7)
9. 72/6 – {2 x [9 – (4 x 2)]}
Find the area of each square.
10.
4 meters
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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Math 300 Chapter 1
Glossary
Digits - 0, 1, 2, 3, 4, 5, 6, 7. 8, 9
Whole Numbers - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
Place Value - Position of a number
Period - Beginning at the right of the number, groups of three digits separated by a
comma
Standard Form - Write the number using periods separated by a comma. For example,
1,234.
Expanded Form - Shows the number with each digit in its place value with zeros to the
right. For example the standard form number 1234 in expanded form is 1000+200+30+4
Table - Display that organizes and shows facts about a set of numbers.
Addend - Any number being added to another number
Sum - The result of an addition operation.
Polygon - Any flat figure that is created by joining straight lines.
Perimeter - The distance around a polygon.
Minuend - Number from which a number is subtracted.
Subtrahend - Number that is subtracted from another.
Difference - Result when one number is subtracted from another.
Rounding - Approximating a number to a certain accuracy.
Factor - When two number multiply to get a result, the numbers are called factors of the
result.
Product - When two number are multiplied together, the result is called the product.
Distribute - to disperse over an area. (See Distributive Property)
Area - Measure of a surface. (Result is expressed in square units such as sq ft or sq mi)
Dividend - The number begin divided.
Divisor - The number by which the dividend is divided.
Quotient - The result when the divisor is divided into the dividend
Average - A special application of division and addition. It is the result when a list of
number is added and the sum is divided by the number of numbers. For example, the
average of 3, 6, 9, 14 is
5+6+10+11 = 32 divided by 4 = 8. 8 is the average.
Exponent - a value that is placed above and after a base value to denote the power to
which the base is to be raised.
Base - the value that is to be raised to a power indicated by the exponent.
Exponential notation - This notation is shown with the exponent using a smaller font type
and it is raised above the base value.
Perfect squares - the product of any whole number multiplied by itself. For example,
1=1x1, 4=2x2, 144=12x12
Square root - A square root of a number is one of two identical factors. For example, the
square root of 144 is 12 since 12x12=144. The square root of 256 is 16 since 16x16 =
256.
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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Math 300 Chapter 1
Glossary (cont)
Properties
Additive property of 0 - The sum of 0 and any number is that number. 0 + 9 = 9
Commutative Property of Addition - Changing the order of two addends does not
change their sum. For example, 5 + 6 = 11 and 6 + 5 = 11
Associative Property of Addition - Changing the grouping of addends does not change
their sum. For example, 1 + ( 2 + 3 ) = 1 + 5 = 6 and
(1 + 2) + 3 = 3 + 3 = 6
Subtraction Properties of 0  The difference of any number and that same number is 0. For example 24 -24 = 0
 The difference of any number and 0 is that same number. For example 24 - 0 = 24
Multiplication Property of 0 - The product of 0 and any number is 0. For example, 5 x 0
=0
or 0 x 5
=0
Multiplication Property of 1 - The product of 1 and any number is the same number.
For example, 24 x 1 = 24 or 1 x 24 = 24
Commutative Property of Multiplication - Changing the order of two factors does not
change their product. For example 24 x 1 = 24 and 1 x 24 = 24
Associative Property of Multiplication - Changing the grouping of factors does not
change their product. For example, (5x6)x7= 5x(6x7)
Distributive Property - Multiplication distributes over addition. For example,
4(5+6) = 4x5 + 4x6 = 20+24 = 44
Division Properties of 1

The quotient of any number and that same number is 1. For example, 9/9 = 1

The quotient of any number and 1 is that same number. For example, 12/1=12
Division Properties of 0

The quotient of any number and 0 (except 0) is 0. For example, 0/12=0

The quotient of any number and 0 is not a number, we say it is undefined. For
example, 12/0 - undefined
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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Math 300 Chapter 1
Hints (cont)
A comma may or may not be inserted in a four-digit number. For example, 1234 and
1,234 are both acceptable ways of writing this whole number.
Rounding

Locate the digit to the right of the given place value

If the digit in that place value is 5 or greater, add 1 to the digit in the given place
value and replace each digit to the right with a 0

If the digit in the place value is less than 5, replace each digit to the right with a 0

Multiply by whole numbers ending in 0 - The product is found by adding the number of
ending 0’s to the other factor. For example, 23x100 = 2300 and 45x1,000,000 =
45,000,000
Area of a rectangle (or square) - Multiply the length times width with the answer specified
in square units. Area = length x width.
Opposite operations

Since addition and subtraction are opposite operations, you can check addition by
doing a subtraction. You can check a subtraction by doing an addition.

Since multiplication and division are opposite operations, you can check
multiplication by doing a division. You can check a division by doing a multiplication.
Problem solving steps

Understand the problem. Read the problem at least twice. Draw a sketch or graph
to be sure you know what is being asked. Look for “magic” keywords (see table).

Translate the problem from words into numbers and symbols that indicate what
math operation you need to perform.

Solve the problem created in the prior step

Interpret the result. Verify that the solution satisfies the stated problem. Be sure to
put your answer in the proper units of measure. The units of measure will multiply and
divide just like numbers so the units of your answer may give you a hint if you have
approached the problem properly.
Exponents

An exponent only applies to its base.

Evaluation of an exponent is calculated by multiplying the base times itself
“exponent” times. It is not base x exponent.
Order of operations

Perform all operations within grouping symbols first. The grouping symbols are
parenthesis (), brackets [] and other grouping symbols.

Evaluate any expressions with exponents.

Multiply or divide beginning at the left and work right across the expression

Add or subtract beginning at the left and work right across the expression
Math 300 M-G 4e Chapter 1; Rev: Mar 2011
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