Chapter 3
Whole Numbers:
Operations and Properties
Addition of whole numbers
Set Model
Dora caught 2 stars, and Boots caught 3 stars. How
many stars did they catch all together?
Addition of whole numbers
Set Model
Let a and b be any two whole numbers. If A and B are
disjoint sets with a = n(A) and b = n(B), then
a + b = n(A B)
Tween Dora the Explorer
How many ducks did they find together?
How many newspaper did they collect together?
Examples that use Set Model to perform Addition
1. Odelia got 43 candies during “Trick or Treat”, her sister got
38. How many candies did they totally get at that night?
2. On one Wednesday, Kevin borrowed 3 books from the
school library, and later that day he borrowed another 4
books from the public library, how many books did he
totally borrow on that day?
Measurement Model for Addition
Measurement Model
In this model, the addition is performed on a number line.
For example, to calculate 3 + 4,
(1) draw an arrow with length 3 starting at 0.
(2) draw an arrow with length 4 starting from the end of the
previous arrow.
(3) the sum is then the length of the two arrows combined
together this way.
0
1
2
3
4
5
6
7
8
9
Examples that use Measurement Model to perform Addition
1. On one Saturday morning, Jonathon walked 2 kilometers
from home to his friend Derek’s house. They then walked 3
kilometers together to the beach. How many kilometers did
Jonathon walk from home to the beach?
2. Mr. Faber’s house is 23 feet tall, and he put an 8-foot tall flag
pole on top of the roof. How high is the top of the flag pole
from the ground?
Measurement model for addition
In this model, a bunny is sitting above the number 0
(which is its home base) facing right (which is the
positive direction), and will jump forward if it has to
perform an addition problem.
0
1
2
3
4
5
Click whenever you are ready.
6
7
Example : 2 + 4
1. Our rabbit starts at 0 facing right.
2. It then hops forward 2 units because it
sees 2 in the beginning.
(Click to see animation.)
0
1
2
3
4
5
6
7
Example : 2 + 4
1. Our rabbit starts at 0 facing right.
2. It then hops forward 2 units because it
sees 2 in the beginning.
0
1
2
3
4
5
6
7
Example : 2 + 4
1. Our rabbit starts at 0 facing right.
2. It then hops forward 2 units because it
sees 2 in the beginning.
0
1
2
3
4
5
6
7
Example : 2 + 4
1. Our rabbit starts at 0 facing right.
2. It then hops forward 2 units because it
sees 2 in the beginning.
3. After this the bunny has to jump 4 more
steps forward because it sees + 4.
(click when you are ready)
0
1
2
3
4
5
6
7
Example : 2 + 4
1. Our rabbit starts at 0 facing right.
2. It then hops forward 2 units because it
sees 2 in the beginning.
3. After this the bunny has to jump 4 more
steps forward because it sees + 4.
0
1
2
3
4
5
6
7
Example : 2 + 4
1. Our rabbit starts at 0 facing right.
2. It then hops forward 2 units because it
sees 2 in the beginning.
3. After this the bunny has to jump 4 more
steps forward because it sees + 4.
Since the bunny finally stops at 6, we know that 2 + 4 = 6.
0
1
2
3
4
5
6
7
Properties of Addition
1. Closure property
The sum of any two whole numbers is still a whole number.
2. Commutative property
For any two whole numbers a and b, a + b = b + a
(This can easily be demonstrated by the set model, see next slide.)
A closed Ecosystem is a self-replenishing system in which
life can be maintained without external factors or outside aid.
An ecosphere.
Properties of Addition
3. Associative property
For any whole numbers a, b, and c,
(a + b) + c = a + (b + c)
4. Identity property
For any whole number a, a + 0 = a = 0 + a
Exercise: What properties of addition are used below?
1. (27 + 89) + 13 = 27 + (89 + 13) associative property
_______________
2. 27 + (89 + 13) = 27 + (13 + 89)
commutative property
_______________
3. 27 + (13 + 89) = (27 + 13) + 89
associative property
_______________
4. 2000 + 382 = 2382
identity property
_______________
Subtraction of whole numbers
Take-away approach
5 baby bears are playing near the river. Later on 2 ran
away. How many baby are left behind?
Subtraction of whole numbers
Take-away approach
Let a and b be whole numbers. Let A be a set with a
elements and B a subset of A with b elements, then
a – b = n(A – B)
The number “a – b” is called the difference and is read
“a minus b”, where a is called the minuend and b the
subtrahend.
Measurement Model for Subtraction
Measurement model for Subtraction
1. There is a difference between addition
and subtraction.
2. To perform subtraction, the rabbit has
to turn around (180 deg) first.
0
1
2
3
4
5
Click whenever you are ready.
6
7
Example 4: 7 – 3
1. Our rabbit still starts at 0 facing right.
2. It then hops forward 7 units because it
sees 7 first.
(Click to see animation.)
0
1
2
3
4
5
6
7
Example 4: 7 – 3
1. Our rabbit still starts at 0 facing right.
2. It then hops forward 7 units because it
sees 7 first.
0
1
2
3
4
5
6
7
Example 4: 7 – 3
1. Our rabbit still starts at 0 facing right.
2. It then hops forward 7 units because it
sees 7 first.
3. Now the rabbit has to turn around
because it sees the subtraction sign.
(click to see animation)
0
1
2
3
4
5
6
7
Example 4: 7 – 3
1. Our rabbit still starts at 0 facing right.
2. It then hops forward 7 units because it
sees 7 first.
3. Now the rabbit has to turn around
because it sees the subtraction sign.
0
1
2
3
4
5
6
7
Example 4: 7 – 3
1. Our rabbit still starts at 0 facing right.
2. It then hops forward 7 units because it
sees 7 first.
3. Now the rabbit has to turn around
because it sees the subtraction sign.
4. Finally the rabbit has to jump forward 3
steps because it sees the number 3.
0
1
2
3
4
5
6
7
Example 4: 7 – 3
1. Our rabbit still starts at 0 facing right.
2. It then hops forward 7 units because it
sees 7 first.
3. Now the rabbit has to turn around
because it sees the subtraction sign.
4. Finally the rabbit has to jump forward 3
steps because it sees the number 3.
0
1
2
3
4
5
6
7
Example 4: 7 – 3
1. Our rabbit still starts at 0 facing right.
2. It then hops forward 7 units because it
sees 7 first.
3. Now the rabbit has to turn around
because it sees the subtraction sign.
4. Finally the rabbit has to jump forward 3
steps because it sees the number 3.
0
1
2
3
4
5
We now know that 7 – 3 is 4.
6
7
Missing addend approach
Let a and b be whole numbers. Then a – b is the number
that when added to b equals to a.
i.e. a – b = c
if and only if
c + b = a.
Example: 7 – 4 = 3 because 3 + 4 = 7.
Example:
Nicole wants to buy a $13 DVD. She has saved only $5. How much
more does she need to save?
This is obviously a subtraction problem: 13 – 5 = ?
But we can think this way: ? + 5 = 13
Comparison approach
is used to determine how many more or fewer when two
known quantities are compared.
Example: Victoria has 4 bunnies, and she bought them 7 carrots.
How many more carrots did she buy than the bunnies she had?
set A
set B
Each situation described next involves a subtraction problem. In each
case, determine what approach best describes the situation.
Typical answers may be take-away, measurement, missing addend,
comparison.
a.
Robby has accumulated a collection of 362 sports cards. Chris has a
collection of 200 cards. How many more cards does Robby have than
Chris?
b.
Jack is driving from St. Louis to Kansas City for a meeting, a total of
250 miles. After 2 hours he notices that he has traveled only 114miles.
How many more miles does he still have to drive?
c.
An elementary school library consists of 1095 books. As of May 8,
105 books were checked out of the library. How many books were still
available for check out on May 8?
d.
Sam drove 20 miles in the morning from home to his office. At noon,
he drove back 5 miles along the same road to see a client. How far was
he from home at that point?
Remarks
1.
2.
3.
4.
The set of whole numbers is not closed under subtraction.
Subtraction is not commutative.
Subtraction is not associative.
There is no two-sided identity for subtraction.
Multiplication of whole numbers
Repeated-addition approach
Let a and b be whole numbers where a ≠ 0, then
a×b = b + b + … + b
a addends
Example:
A dragonfly has 4 wings, how many wings
do 12 dragonflies have?
If a = 0,
then a×b = 0
Rectangular array approach
Let a and b be whole numbers, the product a×b is
defined to be the number of elements in a rectangular
array having a rows and b columns.
Example: 5×3 is equal to the number of stars in the
following array.
5 rows
3 columns
Another example of Rectangular Arrays
It is much faster to use multiplication to find out the number
of seats in a section of a baseball stadium.
Cartesian Product approach
Let a and b be whole numbers. Pick a set A with a
elements and a set B with b elements. Then a×b is the
number of elements in the set A×B, i.e. number of
ordered pairs whose first component is from A and whose
second component is from B.
This approach is best for counting combinations.
Example:
There are 3 kinds of meat:
ham, turkey, and roast beef.
And there are 2 kinds of bread:
white and wheat.
How many different types of sandwiches with only
one type of meat can we make?
Answer:
(ham, white) (turkey, white) (roast beef, white)
(ham, wheat) (turkey, wheat) (roast beef, wheat)
Determine which of the following approach is most appropriate
for each situation: repeated addition, rectangular array, or
Cartesian product.
1.
A store employee taking inventory counted 8 rows of cans of tomato soup.
Each row had seven cans of tomato soup. How many cans of soup were
there altogether?
2.
A can typically contains 12 ounces of soda. How many ounces are in 9
cans?
3.
A certain package contains 12 sticks of gum. How many sticks of gum are in
10 packages?
4.
In a parking lot, there are 7 rows of cars with 15 cars in each row. How many
cars are there?
5.
A small restaurant makes 6 types of salads and 5 types of soup. Each lunch
consists of a soup and a salad. How many different lunches can be
ordered?
Properties of Multiplication
1. Closure property
The product of any two whole numbers is still a whole
number.
2. Commutative property
For any two whole numbers a and b, a × b = b × a
3. Associative property
For any three whole numbers a, b, and c,
a × (b × c) = (a × b) × c
4. Identity property
For any whole number a, we have
a×1=a=1×a
5. Distributive Property of Multiplication over addition
For any whole numbers a, b, and c,
a × (b + c) = a×b + a×c
Example:
we know that 3 × (2 + 5) = 3 × 7 = 21,
but
hence
3×2 + 3×5 = 6 + 15 which is also equal to 21,
3 × (2 + 5) = 3 × 2 + 3 × 5
2+5
3
Equals to
+ 3
3
2
5
6. Distributive property of Multiplication over subtraction
Let a, b, and c be whole numbers, then
a × (b − c) = a×b − a×c
Example:
In order to calculate 3×14 – 3×9 ,
We can use the property that
3×14 – 3×9 = 3×(14 – 9) = 3×5 = 15
7. Multiplication property of Zero.
For any whole number a, we have a × 0 = 0.
8. Zero Divisors Property
For any whole numbers a and b, if a×b = 0, then either
a = 0 or b = 0.
Division of whole numbers
Repeated-subtraction approach
(also called measurement approach)
For any whole numbers m and d, m ÷ d is the maximum
number of times that d objects can be successively
taken away from a set of m objects (possibly with a
remainder).
Example:
If you have baked 54 cookies and you want to put exactly 6 cookies
in each bag, how many bags will you need?
(go to next slide for an animation)
Example:
If you have baked 54 cookies and you want to put exactly 6
cookies in each bag, then you need 9 bags.
Hence 54 ÷ 6 = 9. (click to see animation)
Partition approach (or Sharing approach)
If m and d are whole numbers, then m ÷ d is the
number of objects in each group when m objects are
separated into d equal groups.
Example:
If we have 20 children and we want to separate them into 4
teams of equal size, then each group will have 5 children
because 20 ÷ 4 = 5 (go to next slide for animation)
Example:
If we have 20 children and we want to separate them into 4
teams of equal size, how many children will be in each
group? (click to see animation)
Team 1
Team 2
Team 3
Team 4
Remarks
The above two approaches are very similar and are
interchangeable when we are working with whole numbers.
However, when we need to divide fractions or decimals, only
the repeated subtraction approach has a realistic meaning.
Missing factor approach
If a and b (b>0) are whole numbers, then
a ÷ b = c if and only if c × b = a
Example:
91 ÷ 13 = ? Think ? × 13 = 91
Remark:
This approach is most useful when we divide numbers with
exponents.
Examples
Identify each of the following problems as an example of either
sharing or repeated-subtraction division.
a.Gabriel bought 15 pints of paint to redo all the doors in his
house. If each door requires 3 pints of paint, how many doors
can Gabriel paint?
b.Hideko cooked 12 tarts for her family of 4. If all of the family
members receive the same amount, how many tarts will each
person have?
c.Ms. Ivanovich need to give 3 straws to each student in her class
for their art activity. If she uses 51 straws, how many students
does she have in her class?
d. Mr. Roth wants to put 280 chairs in the school auditorium,
but he can only put 35 chairs in a row. How many rows of
chairs does he need to make?
e. There are 2.54cm in an inch, how many inches are in
120cm?
f. If a pack of lined paper is $1.80 (including tax), and you
have $12.80, how many packs can you buy?
g. If you can read 9 pages in 20 minutes, what is your
reading rate in pages/minute?
Division properties of Zero
1. If a ≠ 0, then 0 ÷ a = 0
2. If a ≠ 0, then a ÷ 0 is undefined
3. 0 ÷ 0 is also undefined
The Division Algorithm
If a and b are whole numbers with b ≠ 0, then there exist unique
whole numbers q and r such that
a = bq + r where 0 ≤ r < b
(r is called the remainder)
Example:
Let a = 57 and b = 9 then clearly b ≠ 0, and we can write
57 = 6 × 9 + 3
and this expression is unique if we require that the remainder is
between 0 and 9 (not including 9)
Remark: The easiest way to find the remainder is by the
repeated-subtraction approach
Ordering and Exponents
Definition
A whole number a is said to be less that another whole
number b (written a < b) if there is a positive whole
number n such that
a + n = b.
Example:
We say that 3 < 7 because we know that 3 + 4 = 7 and 4 is a
whole number.
Transitive property of “less than”
For all whole numbers a, b, and c,
if a < b and b < c, then a < c
Properties of “less than”
(1) If a < b then a + c < b + c for any whole number c.
(2) If a < b and c ≠ 0, then ac < bc.
Exponents
Repeated multiplication approach
For any whole numbers a and n with n ≠ 0,
a n = a ×a × ∙∙∙ × a
n copies.
For all whole numbers a ≠ 0, we define a 0 = 1, and
00 is left as undefined.
The number n above is called the exponent or power of a.
Examples
(1) 25 = 2×2×2×2×2 = 32
(2) 53 = 5×5×5 = 125
(3) 91 = 9
(4) 370 = 1 ( by definition)
(5) 04 = 0×0×0×0 = 0
Properties of Exponents
(1) For any whole numbers a, m(>0), and n(>0),
a m×a n = a m+n
Example
34×35 = (3×3×3×3)×(3×3×3×3×3) = 34+5 = 39
Properties of Exponents
(2) For any whole numbers a, b, and m(>0),
a m×b m = (ab) m
Example
34×74 = (3×3×3×3)×(7×7×7×7)
= (3×7)×(3×7)×(3×7)×(3×7)
= (3×7)4
Properties of Exponents
(3) For any whole numbers a, n(>0), and m(>0),
(a m) n = a m×n
Example
(78)4 = (78)×(78)×(78)×(78)
= 78+8+8+8
= 732
Caution
(2 + 3)5 ≠ 25 + 35 and there is no property involving
raising powers of sums or differences.
Properties of Exponents
(4) For any whole numbers a(>0), n, and m, with m>n>0
a m ÷ a n = a m−n
Example
67 ÷ 64 = ?
We need to use missing factor approach.
If 67 ÷ 64 = x, then x × 64 = 67 , and we can use guess
and check to deduce that x = 63.
Exercises
Rewrite the following with only one exponent
a) 23 × 16
b) 37 × 95 × 272
c) 92 × 153 × 54
d) 123 ÷ 26
“Less than” properties for exponents
(1) For any whole numbers a, b, and n>0
if a > b then an > bn
(2) For any whole numbers a(>1), m and n
if m > n then am > an
(3) For any whole numbers a, b, and m, n
if a > b and m > n, then am > bn
Exercise
Order the following exponents from smallest to largest without
calculators.
322, 414, 910, 810