Prealgebra, Chapter 1 - Whole Numbers
1.1 Whole Numbers and Place Value
Number Systems
The most basic reason that we need numbers:
Examples
Throughout history, numbers have been used
Egyptian hieroglyphs
Roman numerals
Numbers have been in use as long as civilization has been around.
Our number system:
This number system is in widespread use today, largely because it is superior to other number
systems that have developed. In particular, mathematical operations are far easier to do when
using our number system than they are when using something such as hieroglyphs or Roman
numerals.
Place Value
The key to understanding our number system lies in understanding the fact that it is a place value
system.
Each digit in a number sits in a certain place, and each place has a certain value.
Example:
374
Understanding the concept of place value is the key to understanding our number system and how
it works.
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Expanded Form
If we understand place value, we can take a number, such as 297, and expand it.
297 =
This is called writing a number in expanded form.
standard form.
Examples:
The more usual “297" is referred to as
Write in expanded form.
815
4,372
The Place Values
Consider this very large number:
8,994,893,694,829
Each place has a name. Starting at the far right, we have
9 ones, 2 tens, 8 hundreds, 4 thousands,
9
6
3
9
8
4
9
9
8
Numbers this large do in fact show up in the real world. The number above represents the US
Government debt, in dollars, at the time this was written.
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Identifying Place Value
You should be able to look at a number and identify the place value of each digit.
Example:
27,803
2
7
8
0
3
Example:
For the number 41,827,936
What is the place value of the 8?
What is the place value of the 3?
What is the place value of the 4?
What is the place value of the 1?
Writing Numbers with Words
Commas are used to make large numbers easier to read. The commas group the digits into
groups of three, starting at the right. Each group has a name.
4,378,810,924
Instead of expanded form, where we expand each digit according to its place, we can expand each
group.
Thinking of the number in terms of groups is the key to writing it in words.
Example:
5,302,811
Note: We do not use the word “and”
We do not write the name of the ones group at the end.
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Why is writing numbers in words important? There are plenty of cases where numbers are written
by hand. Sometimes, handwriting is messy and difficult to read.
308,215
The 0, when written by hand, can sometimes look like a 6. This, of course, changes the meaning
of the number. Other handwriting mistakes can also be make. But it is usually important to get the
number correct.
Example:
Writing a check.
Example:
17, 821
Example:
Three trillion, two hundred seventy two billion, eight hundred twelve million, six
hundred forty two thousand, nine hundred fifteen.
‘ Practice Problems
‘ HW 1A
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1.1 Practice
Write these numbers in expanded form.
1.
729
2.
5,634
Give the place value for the indicated digits.
3.
The 8 in 819
4.
The 5 in 1,075
Which digit tells the number of tens?
5.
45,432
6.
4,890
Which digit tells the number of ten thousands?
7.
96,584,273
8.
5,374,190
Write these numbers in words.
9.
76,540
10.
6,008,529
Give the numerical form of these numbers.
11.
Seventeen thousand, two hundred seventy-two
12.
Five billion, one hundred six million, seven thousand, nine hundred
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Write the whole number in these sentences in standard form.
13.
The earth is approximately ninety-three million miles from the sun.
14.
Some new cars cost one hundred twenty-five thousand dollars.
Country
Rank
Size
(1000's of square miles)
Russia
Canada
United States
China
Brazil
1
2
3
4
5
6591
3854
3718
3704
3286
Write answers in standard form using the data in the table.
15.
What is the size of China?
16.
What is the size of the United States?
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1.2 Adding Whole Numbers
When we are counting, we generally start from 1.
1, 2, 3, 4, ...
This group of numbers is sometimes called the counting numbers or the natural numbers, simply
because this is the way that we naturally count things. We start at 1 and we count up.
But the number zero is important, too. If we want to do more with the numbers than just counting,
that it, if we want to add, subtract, multiply, and divide, the zero is necessary.
All of the counting numbers and zero
0, 1, 2, 3, ...
are referred to as whole numbers.
In this section, we will discuss the rules and techniques for adding whole numbers.
Adding things that are of like kind
The following point is very important. We can add numbers:
We can also add numbers that represent things
4 cats + 2 cats =
3 birds + 8 birds =
And this works for anything, as long as they are the same kind
20 dollars + 8 dollars =
7 miles + 9 miles =
But if they are not the same kind, we cannot add them
5 dollars + 3 cats =
2 birds + 20 miles =
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Adding numbers
Definitions
3 + 8
the “plus” sign, or addition symbol
read, “three plus eight”
3 and 8 are referred to as addends.
3 + 8 = 11
11 is called the sum.
Adding numbers is often visualized with objects:
qqq + qq = qqqqq
Adding things in this way is fairly intuitive, and fits with our everyday experience.
Addition can also be represented on a number line. This is a little more abstract, but is just as
important.
The number line has certain characteristics:
Evenly spaced markings
Larger numbers to the right
The zero point is called the origin.
Addition can be pictured as movement along the number line.
Example:
5+2
Example:
7+1+4
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Basic Addition Facts
A table of the basic addition facts looks something like this:
0
1
2
3
4
0
0
1
2
3
4
1
1
2
3
4
5
2
2
3
4
5
6
3
3
4
5
6
7
4
4
5
6
7
8
And the values for the numbers at least up through 10 rows and 10 columns should be memorized.
You should be able to recall use these facts quickly and easily.
Properties of Addition
The Commutative Property
If you think about the basic addition facts, you will see that
3+4 = 7
and also that
4+3 = 7
The same pattern holds for all of the numbers. If we are adding two numbers, it does not matter
which one comes first. We say that addition is commutative. This simply means that the order of
the numbers does not affect the answer. This fact is referred to as the commutative property of
addition.
Examples:
8+2 =
3,479 + 12,892 =
The Associative Property
Sometimes it is useful to group numbers together. This is generally the case when we are adding
more than two numbers.
Think about adding these three numbers:
8+4+3
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The associative property of addition says that the way in which the numbers are grouped does not
affect the sum. In other words, we could associate the 4 with the 8:
(8 + 4) + 3
or we could associate the 4 with the 3
8 + (4 + 3)
and we see that the sum is the same in each case.
Associating, or grouping, numbers, can come in handy when adding long lists of numbers. It is
often possible to find pairs or groups of numbers that add up to 10 or multiples of 10, which can
allow the sum to be computed quickly and easily in your head.
Examples:
7 + 12 + 3 + 8
15 + 3 + 1 + 11 + 3 + 5 + 6
38 + 1 + 4 + 2 + 5 + 16 + 7 + 4
2 + 7 + 6 + 8 + 18 + 3 + 4 + 22 + 5 + 2 + 6
The Additive Identity Property
Adding zero to a number is the same as not adding anything at all.
5+0 =
0 + 213 =
The number 0 is called the identity for addition.
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Adding Numbers with Multiple Digits
When add numbers that have multiple digits, we line up the columns so that all the digits of one
column have the same place value.
We add the numbers in each column. We can do this because the numbers in the ones column
represent “ones”, and the numbers in the tens column represent “tens”. In other words, all the
numbers in a given column represent the same thing.
This is similar to adding items that are of the same type. We can add apples and apples, or inches
and inches. The things we are adding have to be of the same kind. By the same reasoning, we
can add ones and ones, or tens and tens.
Examples:
Carrying
In many cases, the digits add up to a number that has more than one digit, and we have to carry
a digit into the next column.
We always start at the right, in the ones place, and we work our way to the left, toward the higher
place values. In this example, when we add the 7 and the 8, we get 15, but we recognize that 15
is really 1 ten plus 5 ones. The “1" has to go in the tens column because it does in fact represent
tens. So we put it there, usually be writing a “1" at the top of the tens column.
Sometimes we have to carry in multiple steps.
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Examples:
Recognizing words that indicate addition
Often we encounter spoken or written statements of problems that have to be solved
mathematically. Someone might say, “That will cost $300 for the new sink and $125 for the mirror
and $48 for the faucet.” Situations like this are common. This is an English statement, but at its
core there is a mathematical problem to be solved.
We would write
We knew to add because of the use of the word and. There are other words and phrases that are
commonly used to indicate addition.
sum, plus, add to, more, more than, increased by, total, in addition to, additional
Example:
In July, there were 12 students signed up for the class. During the month of
August, 6 more students signed up. What was the total number of students in the
class.
Application: The Perimeter of a Figure
The perimeter of a closed figure is defined as the distance around the figure. Finding the perimeter
of a figure is usually a matter of adding up the lengths of all the sides.
Example:
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Example:
Think about the perimeter of a rectangle:
Every rectangle has four sides, two that we call the width and two that we call the length. Instead
of referring to a specific rectangle, such as the one shown above, we can refer to a general
rectangle.
This could be any rectangle, not just the one shown. Whatever its actual length and width are, we
know that the perimeter will be
Perimeter =
And we usually write this with a more convenient notation:
This is an example of a ________________. The letters P, W, and L are _________________.
If we know the value of W and L, we can plug them into the formula and do the addition to find P.
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Example:
A rectangle has a width of 6 yd and a length of 20 yd. What is its perimeter?
Example:
A field is 160 feet by 220 feet. You want to build a fence around the field. What is
the total length of fence that you need to enclose the field?
Using a Calculator
You need to be able to add numbers both by hand and by using a calculator. Different calculators
work in different ways, but virtually all of them use the “+” key for addition. On some calculators,
you may use the equals key, “=”, to compute the sum. Others may have an “Enter” key. Be sure
that you know how to use your particular calculator.
‘ Practice Problems
‘ HW 1B
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1.2 Practice
1.
Identify the addends and the sum in the equation: 3 + 7 = 10
Addends:
Sum:
Identify which property of addition is shown in the equations below.
2.
(5 + 3) + 2 = 5 + (3 + 2)
3.
6+2+3 = 2+6+3
Add these numbers.
4.
5.
6.
8.
Which number is 216 more
than 3,771?
10.
What is the sum of 56, 234, 1120, and 221?
11.
Find the perimeter of the figure shown.
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9.
7.
What is the total of 123 and 7,421?
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12.
Find the perimeter of the figure shown.
13.
A picture frame has the dimensions 7 in. by 5 in. What is its perimeter?
14.
A clothing store sold 1,010 dresses in April, 965 dresses in May, and 982 dresses in June.
How many dresses did the store sell?
15.
Bob’s Burgers sold the following food items from December to February. Compute the
totals and fill in the information missing in the chart.
Food
January
February
March
Hot dogs
1,721
1,888
1,714
Hamburgers
1,432
1,516
1,499
French Fries
2,004
1,989
1,990
Sodas
2,138
2,210
2,511
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Totals
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1.3 Subtracting Whole Numbers
Subtraction is the opposite of addition. If addition means adding more, subtraction means taking
some away so that you have less.
The minus sign: -
used to indicate subtraction
Terminology
Consider this example:
9-2
The number 9 is called the _______________________
The number 2 is called the _______________________
9-2 = 7
The number 7 is called the _______________________
The term difference fits with our everyday understanding of the term. If we want to know
the difference between two numbers, we subtract the smaller from the larger.
Subtracting Numbers with Multiple Digits
When working with numbers with multiple digits, we arrange them in a manner similar that used
for addition, with the place values lined up in the same column. We subtract the ones from the
ones, the tens from the tens, etc.
Examples:
Because subtraction is the opposite of addition, we can use addition to check our work and make
sure our answer is correct.
We get an answer of 532. Therefore, 532 and 441 should add up to 973.
Check:
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Recognizing words that indicate subtraction
Many situations in the real world require subtraction. When trying to figure something out or solve
a problem, we need to be able to recognize when we need to subtract rather than add.
In a written or spoken problem, certain words or phrases often serve as clues that subtraction will
be involved.
subtract, minus, difference, less than, decreased by
Example:
4250 gallons of water were in the storage tank. Irrigating the garden decreased the
water supply by 120 gallons. How much water was left in the storage tank?
Sometimes the question “How much more...?” can indicate the need to subtract.
Example:
Doug wants to purchase a new mountain bike for $475. He has saved up $320.
How much more money does he need to save before he can purchase the bike?
Borrowing
When adding, we sometimes need to carry. When subtracting, we sometimes need to borrow.
You have probably done subtraction problems that involve borrowing. Rather than simply following
a mechanical process called borrowing, you should try to understand why this process works.
Look at this problem:
We cannot subtract 5 from 3, because it is bigger than 3. We solve this problem by borrowing.
The eight in the above problem represents 8 tens. We can take one of these tens and add it to the
3, making the 3 a 13. We still have 8 tens. One of them has simply been borrowed by the ones
column.
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Again, we can check our work:
Check:
Sometime we need to borrow multiple times.
Examples:
Applications
Plenty of real world situations involve subtraction.
Example:
Three students are selling books to raise money for a charity. They need to sell a
total of 2500 books. John has sold 733 books. Mary has sold 815 books. Sue has
sold 392. How many more books need to be sold for them to reach their goal?
Example:
Joe has $852 in the bank. He withdraws $230 to purchase a new bike and $176 to
purchase a new computer monitor. How much does he have left in the bank?
‘ Practice Problems for section 1.3
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1.3 Practice
1.
In the statement 6 – 5 = 1
6 is called the
5 is called the
1 is called the
Write the related addition statement.
Do the indicated subtraction.
2.
3.
4.
5.
6.
7.
8.
What number is 34 less than 89?
10.
Jake is driving his Jeep over mountainous terrain near the Ocomat Gorge. Starting at an
elevation of 1,435 ft, his elevation then decreases by 306 ft, increases by 152 ft, then
decreases of 75 ft. Find Jake’s final elevation.
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9.
What is the difference between 718
and 291?
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11.
Thomas has saved $297. He wants to buy a TV that costs $459. How much more money
does he need to save?
12.
The difference between two numbers is 142. The greater number is 411. What is the
smaller number?
13.
A certain high school basketball team scored 72, 66, 69, and 70 points in its first four
games. That same team scored 81, 82, 75, and 94 points in its last four games. How many
more points did the team score in its last four games than in its first four games?
14.
Fredrick scored 97 points on a history test. Sara scored 19 fewer points on the same test.
How many points did Sara score on the test?
15.
John bought a desk chair for $89. Greg bought an identical desk chair to the one John
bought for $125. How much more did Greg pay?
16.
The land area of Georgia is 59,425 sq. mi. The land area for Florida is 65,755 sq. mi. How
much larger is Florida than Georgia?
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17.
Evaluate these expressions:
(a)
12 – (8 – 1)
(b)
(12 – 8) – 1
Do they give the same answer? What can you conclude about subtraction and the
associative property?
18.
This problem involves a bank account and money withdrawn (subtracted) from the previous
amount. Fill in the blanks.
Original balance
1st withdrawal
2nd withdrawal
3rd withdrawal
$ 729
$ 234
$ 167
$ 94
a) Find the balance after
the first withdrawal.
b) Find the balance after
the second withdrawal.
c) Find the balance after
the third withdrawal.
Work the following problems on your calculator.
19.
22.
21.
156,705 – 78,892 =
22.
Subtract 743 from the sum of 365 and 458.
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1.4 Rounding, Estimating, and Ordering Numbers
Rounding
In most cases, mathematics is exact. 2 + 2 = 4. The answer is exactly 4. Not just some number
close to 4, but exactly 4.
In many cases, however, we don’t need exact numbers, only estimates. Sometimes, in fact,
estimates make more sense than exact numbers.
Think about the distance between two cities:
Atlanta and Jacksonville. If you want to drive from
Atlanta to Jacksonville, you will drive approximately
310 miles. The exact distance depends on where
in Atlanta you start, and where in Jacksonville you
finish. To find the exact distance, we would need
two specific points, such as the distance from the
boundary line of one city to the boundary line of the
other, or the distance from the courthouse of one
city to the courthouse of another. Or maybe you
need to know the distance from your house in
Atlanta to your Grandmother’s house in
Jacksonville. Without specifying two specific
points, trying to find the exact distance is
impossible. It makes more sense to speak of the
approximate distance between the cities, which is
about 310 miles.
The distance 310 miles is the distance between the cities, rounded to the nearest 10 miles. It may
also make sense to approximate the distance to the nearest hundred miles, in which case we would
say that the distance from Atlanta to Jacksonville is 300 miles. When you say something like this,
people generally understand that you are giving an approximate number.
Approximating a number by using a simpler number that is close to the original number is called
rounding.
We can visualize the process of rounding by using a number line.
The number line shown here is marked off in increments of 10. The number 84 is also shown. It
is apparent that 84 is closer to 80 than it is to 90. We say that 84 rounded to the nearest 10 is 80.
Shown here is a number line calibrated in increments of 100. The number 527 is seen to be closer
to 500 than it is to 600, so we say that 527 rounded to the nearest hundred is 500.
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The number line shown here is marked in increments of 1000. The number 52,775 is closer to
53,000 than it is to 52,000. We say that 52,775 rounded to the nearest thousand is 53,000.
Rules for Rounding
The rules for rounding are simple, and should make sense if you understand what rounding is.
1. Find the place you want to round to.
2. If the next digit is a 5 or higher, round up. Otherwise round down.
That’s it. Some practice, of course, will help.
Examples:
Round to the nearest 10.
51
52
53
54
Round to the nearest 10
621
625
628
Round to the nearest 100
628
684
98
34
Round to the nearest 1000
27,811
27,500
27,499
Rounding up a 9
Examples:
Round to the nearest 100
2921
2973
43,984
29,978
192
198
Round to the nearest 10
497
2398
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Estimating
In some cases, exact numbers are not needed and an estimate is sufficient.
Suppose, for example, that you need to know the number of people seated in a baseball stadium.
You do not need to know the exact number, just an estimate that is accurate to the nearest
hundred. You have the following information from the ticket sales:
In section A:
In section B:
In section C:
In section D:
In section E:
In section F:
128 people
188 people
417 people
530 people
312 people
694 people
It would take some time to add all the numbers up. There would be a lot of carrying. Mistakes
might be made. Even typing all the numbers into the calculator could take a little while, and there
may be typographical errors.
Instead, you can get a quick estimate of the total to the nearest 100 by rounding each number to
the nearest 100 and then adding.
A:
B:
C:
D:
E:
F:
And you can probably compute the sum in your head without too much difficulty.
A warning:
Note that this method works well, and gives reasonably accurate results when some of the
numbers get rounded up and some get rounded down. Rounding the numbers make the final
answer inaccurate, but if some of the numbers are rounded up and some are rounded down, then
these inaccuracies tend to cancel each other out. This is usually the case. If we have a random
sample of numbers, and we are trying to round, say, to the nearest hundred, some of the numbers
will probably be a little above an even hundred, and will round down while other numbers will be
a little below an even hundred and will round up. The rounding errors tend to cancel each other
out. Realize that it is possible, however, for most or all of the errors to be in the same direction.
That is, most of the numbers could, by chance, end up rounding up (or down), and the errors could
accumulate to a large total. This method, therefore, while reasonably good, is not perfect.
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A Bad Example:
Estimate the sum of the following numbers to the nearest hundred by rounding each
number to the nearest hundred and then adding.
472
251
160
278
354
155
770
When we add the rounded numbers, we get 500 + 300 + 200 + 300 + 400 + 200 + 800,
which equals 2700.
The actual value is 2440. Rounded to the nearest hundred we should have ended up with
2400. We are off by 300! This is because all of the numbers ended up rounding up, so our
result was skewed to high side. This is usually not the case. Realize, however, that it is
possible for this method of rapid estimating to produce incorrect results.
Example:
Suppose you go to the hardware store and select the following items
light bulbs:
batteries:
screwdriver:
tape:
bolts:
boards:
$15.95
$3.95
$2.95
$6.95
$8.15
$4.25
You have $50 in your wallet. Round each price to the nearest dollar and make a
quick estimate of the total. Do you have enough money to purchase all of the
items?
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Ordering Numbers on the Number Line
The number line is typically drawn with smaller numbers to the left and larger numbers to the right.
If we plot two numbers on the number line, 3 and 7 for example. We can see that
7 is to the right of 3
3 is to the left of 7
We can write these facts using the greater than and less than symbols.
Example:
Plot the numbers 6 and 8 on the number line. Then write two mathematical
statements about the numbers, with one statement using the greater than symbol
and one using the less than symbol.
Examples:
Complete the following statements by writing < or > so that each statement is
true.
4 _____ 8
6000 _____ 5000
75 _____ 57
0 _____ 4
‘ Practice Problems for section 1.4
‘ HW 1C
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1.4 Practice
Round each number to the indicated place.
1.
89 to the nearest ten
2.
412 to the nearest hundred
3.
1,283 to the nearest ten
4.
6,120 to the nearest hundred
5.
48,973 to the nearest hundred
6.
26,099 to the nearest thousand
7.
34,500 to the nearest thousand
8.
149,992 to the nearest ten thousand
9.
34,499,238 to the nearest million
Estimate the answers by rounding to the indicated place. Then, check the reasonableness of the
estimate by working the problem.
The nearest ten:
10.
11.
The nearest hundred:
12.
13.
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Estimate the answers by rounding to the indicated place. Then, check the reasonableness of the
estimate by working the problem.
The nearest thousand:
14.
15.
16. Complete the following statements with greater than (>) or less than (<) signs.
200 ___
70
19 ___
56
900 ___ 899
17. Michael went to a bookstore and bought these items: Science textbook, $39.88; Math textbook,
$45.99; History book, $19.98; Gift card, $10.00. Estimate the total cost by rounding the numbers
to the nearest dollar.
18. Lisa went to the grocery store and bought these items: Milk, $3.59; Eggs, $2.79; Bread, $3.49,
Cookies, $3.09. Estimate the total cost by rounding the numbers to the nearest dollar.
19. A number rounded to the nearest hundred is 500. What is the largest possible number it could
be? What is the smallest possible number it could be?
20. A certain number rounded to the nearest ten is 800. What is the largest possible number it
could be? What is the smallest possible number it could be?
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1.5 Multiplying Whole Numbers
Multiplication can be thought of as repeated addition.
The × symbol is used to indicate multiplication. However, as we get into algebra, we use variables,
and the letter x is commonly used as a variable. To avoid confusion, the × symbol is not often used
to represent multiplication. Instead, a dot is used:
When we multiply, as in the example above, 5 and 3 are called the ___________________
and 15 is called the ______________________.
The Multiplication Table, or Basic Multiplication Facts.
0
1
2
3
4
0
0
0
0
0
0
1
0
1
2
3
4
2
0
2
4
6
8
3
0
3
6
9
12
4
0
4
8
12
16
You should know the multiplication table out through at least 10. These should be memorized, and
you should be able to recall and use these facts quickly.
Example: Use multiplication to find the number of parking spaces in the parking area shown here.
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Multiplying by using the Distributive Property
Sometimes relatively large numbers can be multiplied quickly and easily without a calculator.
Knowing the distributive property is the key.
Example:
Example:
Multiplication using the Short Method
You should already be familiar with the procedure for multiplying large numbers. Just as the
procedures for addition, it is based on placed value and involved carrying.
Example:
Example:
Multiplication with 0 and 1
Any number multiplied by 1 is simply that same number.
= __________
= ___________
This concept can be stated with a variable:
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Any number multiplied by 0 is simply 0.
= _____
= _____
= _____
= _____
= _____
This concept can also be stated with a variable:
Multiplying with Units
If we multiply a number that has units, the untis remain in the answer.
Example:
= ____________
= ____________
Multiplying numbers with two or more digits
We have a procedure for multiplying large numbers.
Examples:
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Sometimes a zero in a number will allow us to speed up the process:
Example:
A repeated digit can also allow us to speed up the process:
Example
Examples:
Application
Multiplication, of course, shows up in many places in the real world.
Example:
Suppose we own a car rental company. We have 47 cars in the lot. We need to
put 14 gallons of gas in each car. How much gas will we need?
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Multiplying with the number 10
Because we have a base 10 number system (ten digits), multiplication with the number 10 is
particularly easy. Multiplying a number by 10, by 100, by 1000, or by any other power of 10 is
simply a matter of adding zeros onto the end of the number.
= ____________
= ____________
= ____________
Our procedure for multiplying larger numbers can also be simplified when working with any number
that ends in 0.
Examples:
Try these:
378 x 4000
420 x 9000
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270 x 5200
Multiplication and Estimating
Sometimes we don’t need exact numbers. Instead, an estimate may be sufficient.
Example:
The campground has 215 campsites. Each campsite needs about 4¼ gallons of
water per day. What is the total daily water usage for the campground?
Note that in this case it makes sense to round and use estimated values. The exact
numbers aren’t really meaningful. The campground might not be 100% occupied, and even
if it was, every campsite would not necessarily be using exactly 4¼ gallons of water each
day. “Exact” numbers would necessarily be any better than rounded numbers in this case.
Note also that we can get large errors at times if both factors round up or both round down a
significant amount.
A Bad Example:
There will be 150 people at the picnic. Each person will eat about 2½ hot
dogs. How many hot dogs do we need to buy to get ready for the picnic?
The answer we get from estimating, in this case, is pretty far from the actual result of 375
from multiplying the numbers without estimating. This is because both factors rounded up,
and did so significantly.
A Bad Example:
There are 143 kids at the summer camp. We will buy each one a t-shirt for
$13.50. Round the numbers to get a quick estimate of the total cost.
Again, our answer is pretty far from the actual calculated value of $1930.50. The point is
that we have to be careful when rounding and estimating. We should be aware of the
possibility of large error when both numbers round in the same direction, and when the
rounding is significant.
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Multiplication using the Associative and Commutative Properties
When multiplying, the ordering and the grouping does not affect the answer.
We can sometimes use these concepts to multiply numbers more quickly and easily.
Examples:
Try these:
Area
Multiplication is useful when we need to calculate area. Area is used to measure the amount of
surface.
Example:
The area of a table top tells us how much useful surface there is on the top of the table.
Area is typically measured in square units.
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Square inches,
, would be used to measure relatively small objects, such as a piece of paper.
Fabric is typically measured in square yards,
. In the metric system, a square meter,
, is
just a little bigger than a square yard. The area of a large city or lake might be measure in square
kilometers,
, or square miles,
. The city of Atlanta is a little over 130 square miles.
Think about the top of a table that is 2 feet wide and 4 feet long.
In the diagram, we can count the square feet. There are 8.
We can get the same result by multiplying:
the answer.
Example:
. Note the units on the factors and
A driveway is 12 feet wide and 70 feet long. What is the area?
In general, the area of a rectangular region will always be found by the equation
This is a formula, which can be written concisely using variables:
If the rectangle is a square, we can write:
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Areas of odd-shaped figures
Example:
Find the area of the figure shown
Three ways to solve this:
Example:
Find the area of the figure shown
Application
You are buying tiles to tile your kitchen floor, shown in the diagram. The tiles cost $7.60
per square foot. How much will the tiles for the project cost.
‘ Practice Problems for section 1.5
‘ HW 1D
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1.5 Practice
1.
If
, we call 3 and 4 the ______________ of 12, and we call 12
the _______________ of 3 and 4.
2.
Find the number of one square foot tiles in a floor that is 13 feet long and 8 feet wide.
Multiply.
2.
3.
4.
5.
6.
Name each property that is illustrated.
7.
A painter can paint 740 square feet per hour. How many square feet can he paint in six
hours?
8.
Katerina decides to buy a motorcycle. She pays $4,500 as a down payment, plus she
makes monthly payments of $141 dollars per month for three years. What is the total cost
of the motorcycle?
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9.
You buy the New York Yankees baseball team and need to hire a new staff. The table
shows how much each person will need to paid.
Job
Hourly Cost
Bookkeeper
$40
Secretary
$42
Vice-President
$51
Accountant
$49
Custodian
$30
For a game day, you will need 17 custodians and one person for each of the other
positions. You will need everyone there for nine hours. What is the total cost of your staff
for a game day?
Estimate by rounding each factor to the nearest hundred and then multiplying.
10.
11.
Multiply and write your answer with the proper units.
12.
13.
14.
True or false?
15.
True or false?
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16.
Find the area of the figure shown.
17.
Find the area of the figure shown
18.
Find the area of the figure shown.
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1.6 Division
Division is most easily understood with an example.
Example:
24 marbles are divided evenly among 4 children.
We can see that each child will get 6 marbles.
Notation:
Division can be thought of as repeated subtraction.
We subtract 4 a total of 6 times, so 24 ÷ 4 = 6.
In the above example, 24 is the ________________, 4 is the ________________,
and 6 is the ________________.
Units
When dividing a number that has units (a denominate number) by another number, the units
remain.
Example:
600 ft ÷ 12 =
If both numbers have units, then the answer has units accordingly.
Example:
A car drives 100 miles in 2 hours. We can find the average speed by dividing the
distance by the time.
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Recognizing words that indicate division
In addition to the obvious mathematical terms, some other words are commonly used to indicate
division:
divide, divided evenly, split evenly, how much . . . each, how many . . . each
Example:
28 tons of rice is divided among 4 trucks. How much rich goes into each truck?
Checking division with multiplication
The answer to a division problem can be checked by multiplying.
63 ÷ 7 =
Dividing and leaving a remainder
Numbers are not always exactly divisible by the divisor. For example, 22 ÷ 5.
We have a remainder, a part that is left over after all the repeated subtraction.
When we check our division by multiplying, we have to add the remainder back in.
Example:
43 ÷ 8
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Division and Zero
0 divided by any number is simply 0.
0 ÷ 8 = ________
0 ÷ 1000 = ________
0 ÷ 2,834,296,401 = ________
Division by 0 is undefined.
5 ÷ 0 = ________________
Long Division
The process of long division should also already be familiar to you. Right now we will look at how
the process works as a shortcut for repeated subtraction.
Example:
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Example:
Long division with a two digit divisor
Example:
Example:
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Division, Estimation, and Rounding
People rarely do long division by hand any more because the process is cumbersome, time
consuming, and error prone. These days, calculators are readily available and are generally
preferred.
Estimating answers to division problems, however, is much easier and faster and can be very
useful. We can round numbers so that they are more easily divided and get an estimate of an
answer that will hopefully be reasonably close.
Example:
A flight from New York to London travels 3471 miles. A typical commercial airliner
travels at about 530 miles per hour. How many hours should the trip take?
When we divide 3000 by 500, we get 6, which is reasonably close to the value of
6.55 that would result from dividing the numbers.
Note once again that estimated answers are reasonable in this case. The “exact” answers
would not really be exact. Different planes travel at different speeds. There are different
routes that have different distances. Wind affects the speed of the aircraft. To estimate
the answer instead of trying to compute it exactly is a very reasonable approach.
Note also that the usual warnings about errors arising from rounding also apply in this case.
Division on a Calculator
Try these division problems on your calculator:
357 ÷ 17 = ____________
357 ÷ 0 = ____________
0 ÷ 20 = ____________
50 ÷ 7 = _____________
Note that when there is a remainder, the calculator expresses it as a decimal.
‘ Practice Problems for section 1.6
‘ HW 1E, HW 1F
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1.6 Practice
1.
In the equation 144 ÷ 16 = 9, the number 144 is the _______________, the number
16 is the _______________ , and the number 9 is the _______________ .
2.
Jim is driving 9 different people to various places, one person at a time. Each of the 9
people spends an equal amount of time in the car. If the total time spent on the drives is
45 hours, how much time is each person in the car.
Divide. Write your answers with the correct units.
3.
21 days ÷ 3
4.
5280 feet ÷ 15
5.
24 miles ÷ 6 hours
6.
108 dollars ÷ 9
Divide using long division.
7.
48 ÷ 7
8.
59 ÷ 6
9
79 ÷ 4
10.
0 ÷ 12
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Divide using long division.
11.
12.
13.
14.
15.
John jogged around a square city park, finishing exactly where he started. If he jogged a
total of 3600 feet, how long was each side of the park?
16.
In his race car, Dave must complete 7 laps in 11 minutes 19 seconds. How much time
does he have for each lap?
17.
Eighty major league baseball players are contributing to build a new wing for a hospital.
The project will cost 1.4 million dollars. How much should each player contribute?
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18.
At the end of the year, Susan’s company has $10,989 left over. She decides to distribute
the money equally to her 37 employees. How much will each employee receive?
19.
The fuel tank on Jake’s boat holds 104 gallons of fuel. If he drives 2392 miles on a full
tank, what is the gas mileage of his boat in miles per gallon?
Estimate the answers to the following problems by rounding the numbers first and then dividing.
20.
901 ÷ 18
21.
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7192 ÷ 53
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1.7 Exponents and Order of Operations
Exponents
Multiplication can be thought of repeated addition. But what if we have repeated multiplication?
Exponents are a way to express repeated multiplication.
can be written as ______
In the above expression, the 5 is called the ________________
and the 4 is called the ____________________________________________
This is read as “five to the power of four" or “five to the fourth power.”
Examples:
Write the following in exponential notation:
= ________
(7)(7)(7)(7) = ________
Evaluate the following:
= ________
= ________
= ________
Remember that exponents are not the same thing as ordinary multiplication.
thing as
= ________
is not the same
= ________
Powers of 0 and 1
Any number raised to the power of 1 is simply that same number.
= ________
= ________
This concept can be stated with a variable:
This should make sense. The exponent tells us the number of times that the base is used as a
factor. If it is used just one time, then we simply have that number.
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We also need to know what happens when the exponent is 0. The result is not necessarily
intuitive, but mathematicians have defined any base to the power of 0 is equal to 1.
= ________
= ________
or, the concept can be stated in general using a variable:
Again, this doesn’t necessarily make intuitive sense, but it is true by definition.
Powers of 10
Powers of 10 are important because we have a base 10 number system.
=
=
=
=
=
...
=
One particular power of 10 has a special name
Order of Operations
Consider the following expression:
12 - 6 ÷ 2
How do we evaluate this? Should we do the subtraction first, or the division?
( 12 - 6 ) ÷ 2
or
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12 - ( 6 ÷ 2 )
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The rules for the order of operations:
Do multiplication and division, left to right, and then addition and subtraction, left to right.
Examples:
=
=
=
÷3 =
Order of Operations with Exponents
There is an additional rule when exponents are involved:
Exponents are evaluated before multiplication and division.
We need this rule because of situations like this:
Without a known rule, we have an ambiguous expression. Is this
or
The Order of Operations:
1. Evaluate exponents first
2. Then do multiplication and division, left to right
3. Then do addition and subtraction, left to right
Examples
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Parentheses
Operations within parentheses are done first. For example,
Within parentheses, we obey the order of operations.
Examples:
÷2
÷2)
÷2
20 ÷
40 ÷
Order of Operations on a Calculator
Most calculators today are capable of dealing with large expressions that contain many parts. Most
also understand the correct order of operations.
Enter
on your calculator. Examine the result. If it is 17, then the calculator is performing
the order of operations correctly (multiplication before addition).
Most calculators also have parentheses.
Enter the following on your calculator. Examine the results.
‘ Practice Problems for section 1.7
‘ HW 1G, HW 1H
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1.7 Practice
Evaluate each of the following.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
12.
13.
9÷3+8
14.
18 - 16 ÷ 2
15.
(18 - 16) ÷ 2
16.
14 + 6 ÷ 3
17.
60 ÷ 4 × 5
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18.
42 ÷ 7 - 15 ÷ 5
19.
20.
21.
22.
23.
60 ÷ (12 - 7 + 1)
24.
Are the numbers 1, 2, and 3 a Pythagorean Triple? Why or why not?
25.
Are the numbers 7, 24, and 25 a Pythagorean Triple? Why or why not?
26.
Are the numbers 1, 1, and
27.
Is
equal to
a Pythagorean Triple? Why or why not?
?
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1.8 An Introduction to Equations
An expression is either a number or a collection of numbers together with arithmetic operators such
as +, -, ×, and ÷.
Examples:
÷2
An equation will have an equals sign in it.
Examples:
An equation is a mathematical statement. It states something. That is, it tells us something.
Just like a statement or sentence in English, a mathematical statement can be either true or false.
Examples:
Translating Sentences into Equations
Math is used to solve problems. Many problems in the real world can be solved mathematically.
Usually, we will have a statement of the problem in English, and we need to translate the English
statement into a mathematical equation.
Right now we will practice translating very simply English statements into simple equations. Later
we will translate more complicated, and more useful, statements.
Examples: Translate the following sentences into mathematical equations.
Five squared minus ten is fifteen.
Twenty four divided by eight is equal to twelve minus nine.
Six less than nineteen is ten plus three.
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1.8 Practice
1.
Label each of the following as an equation or an expression.
a)
b)
c)
d)
2.
Determine whether each of the following statement is true or false.
a)
b)
c)
d)
e)
3.
Write each of the following as a mathematical expression or equation..
a)
Four times three squared
b)
Six less than twenty-five.
c)
Five squared plus three is twenty eight.
d)
Seventeen is equivalent to the cube of three subtracted from the product of four and
eleven.
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