Chapter 3 Whole Numbers: Operations and Properties

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Chapter 3Whole NumbersOperations and Properties
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Addition of whole numbers
Set Model To find 2 3, we find two disjoint
sets, one with 2 objects and the other with 3
objects, form their union, and count the total.
(click to see animation) It is very important to
use disjoint sets.
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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.
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Alternative number line 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.
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4
5
0
1
2
6
7
Click whenever you are ready.
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Example 2 4

• Our rabbit starts at 0 facing right.
• It then hops forward 2 units because it sees 2 in
  the beginning.(Click to see animation.)

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4
5
0
1
2
6
7
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Example 2 4

• Our rabbit starts at 0 facing right.
• It then hops forward 2 units because it sees 2 in
  the beginning.

3
4
5
0
1
2
6
7
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Example 2 4

• Our rabbit starts at 0 facing right.
• It then hops forward 2 units because it sees 2 in
  the beginning.

3
4
5
0
1
2
6
7
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Example 2 4

• Our rabbit starts at 0 facing right.
• It then hops forward 2 units because it sees 2 in
  the beginning.

• After this the bunny has to jump 4 more steps
  forward because it sees 4. (click when
  you are ready)

3
4
5
0
1
2
6
7
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Example 2 4

• Our rabbit starts at 0 facing right.
• It then hops forward 2 units because it sees 2 in
  the beginning.

• After this the bunny has to jump 4 more steps
  forward because it sees 4.

3
4
5
0
1
2
6
7
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Example 2 4

• Our rabbit starts at 0 facing right.
• It then hops forward 2 units because it sees 2 in
  the beginning.

• 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.
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4
5
0
1
2
6
7
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Properties of Addition

• Closure property The sum of any two whole
  numbers is still a whole number.
• 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.)

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Properties of Addition

• Associative propertyFor any whole numbers a, b,
  and c,
  (a b) c a (b c)

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4. Identity property For any whole number a,
a 0 a 0 a
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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) Example 5
2 can be modelled by
The number a b is called the difference and
is read a minus b, where a is called the
minuend and b the subtrahend.
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Number line model for Subtraction

• There is a difference between addition and
  subtraction.
• To perform subtraction, the rabbit has to turn
  around (180 deg) first.

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4
5
0
1
2
6
7
Click whenever you are ready.
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Example 4 7 3

• Our rabbit still starts at 0 facing right.
• It then hops forward 7 units because it sees 7
  first.(Click to see animation.)

3
4
5
0
1
2
6
7
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Example 4 7 3

• Our rabbit still starts at 0 facing right.
• It then hops forward 7 units because it sees 7
  first.

3
4
5
0
1
2
6
7
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Example 4 7 3

• Our rabbit still starts at 0 facing right.
• It then hops forward 7 units because it sees 7
  first.

• Now the rabbit has to turn around because it sees
  the subtraction sign.
• (click to see animation)

3
4
5
0
1
2
6
7
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Example 4 7 3

• Our rabbit still starts at 0 facing right.
• It then hops forward 7 units because it sees 7
  first.

• Now the rabbit has to turn around because it sees
  the subtraction sign.

3
4
5
0
1
2
6
7
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Example 4 7 3

• Our rabbit still starts at 0 facing right.
• It then hops forward 7 units because it sees 7
  first.

• Now the rabbit has to turn around because it sees
  the subtraction sign.

• Finally the rabbit has to jump forward 3 steps
  because it sees the number 3.

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4
5
0
1
2
6
7
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Example 4 7 3

• Our rabbit still starts at 0 facing right.
• It then hops forward 7 units because it sees 7
  first.

• Now the rabbit has to turn around because it sees
  the subtraction sign.

• Finally the rabbit has to jump forward 3 steps
  because it sees the number 3.

3
4
5
0
1
2
6
7
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Example 4 7 3

• Our rabbit still starts at 0 facing right.
• It then hops forward 7 units because it sees 7
  first.

• Now the rabbit has to turn around because it sees
  the subtraction sign.

• Finally the rabbit has to jump forward 3 steps
  because it sees the number 3.

3
4
5
0
1
2
6
7
We now know that 7 3 is 4.
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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.
Remark This approach is helpful to students who
can do addition well but cannot perform the
take-away process mentally yet.
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Comparison approach Let a and b be whole numbers.
Construct a set A with a elements, another set B
with b elements. Match the elements in B with
elements in A. The number of elements left
unmatched in A is the value of a b .
Example 7 4
set B
set A
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Remarks

• 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.

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Multiplication of whole numbers
Repeated-addition approach Let a and b
be whole numbers where a ? 0, then
ab b b b
a addends
If a 0, then ab 0
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Rectangular array approach Let a and b be whole
numbers, the product ab is defined to be the
number of elements in a rectangular array having
a rows and b columns.
Example 53 is equal to the number of stars in
the following
array.
5 rows
3 columns
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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.
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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 ab is the number of
elements in the set AB, 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 can we make? Answer
(ham, white) (turkey, white) (roast beef,
white) (ham, wheat) (turkey, wheat) (roast
beef, wheat)
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Properties of Multiplication

• Closure propertyThe product of any two whole
  numbers is still a whole number.
• Commutative propertyFor any two whole numbers a
  and b, a b b a

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3. Associative property For any three whole
numbers a, b, and c, a (b
c) (a b) c4. Identity property
For any whole number a, we have
a 1 a 1 a
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5. Distributive Property of Multiplication over
addition For any whole numbers a, b,
and c, a (b c)
ab ac
Example we know that 3 (2 5) 3 7
21, but 32 35 6 15 which is also
equal to 21, hence 3 (2 5)
3 2 3 5
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6. Distributive property of Multiplication over
subtraction Let a, b, and c be whole
numbers, then a (b
- c) ab - ac
Example In order to calculate 314 39 , We
can use the property that 314 39
3(14 9) 35 15
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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 ab 0, then either a
0 or b 0.
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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 on each plate, then
you need 9 plates. Hence 54 6 9. (go to next
slide for an animation)
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Example If you have baked 54 cookies and you
want to put exactly 6 cookies on each plate, then
you need 9 plates. Hence 54 6 9. (click to
see animation)
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Partition 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)
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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 (click to see animation)
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Remarks
The difference between the above two approaches
is very subtle when we deal with whole numbers,
but it will become more apparent when we divide
decimals or fractions.
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Missing factor approach If a and b (b0) are
whole numbers, then a b c if
and only if a bc
Example 91 13 7 because 91 13 7
Remark This approach is most useful when we
divide numbers with exponents.
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Division properties of Zero
1. If a ? 0, then 0 a 02. If a ? 0, then a
0 is undefined3. 0 0 is also undefined
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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 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)
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Remark The easiest way to find the remainder is
by the repeated-subtraction
approach
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Ordering and Exponents
Definition A whole number a is said to be less
that another whole number b (written a there is a positive whole number n such that
a n b.
Example We say that 3 4 7 and 4 is a whole number.
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Transitive property of less than For all whole
numbers a, b, and c, if a

Properties of less than

If a c.

If a
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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.
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• Examples
• 25 22222 32
• 53 555 125
• 91 9
• 370 1 ( by definition)
• 04 0000 0

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Properties of Exponents

• For any whole numbers a, m(0), and n(0),
• a ma n a mn

Example 3435 (3333)(33333) 345 39
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Properties of Exponents

• For any whole numbers a, b, and m(0),
• a mb m (ab) m

Example 3474 (3333)(7777)
(37)(37)(37)(37) (37)4
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Properties of Exponents

• For any whole numbers a, n(0), and m(0),
• (a m) n a mn

Example (78)4 (78)(78)(78)(78)
78888 732
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Caution
(2 3)5 ? 25 35 and there is no property
involving raising powers of sums or differences.
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Properties of Exponents

• For any whole numbers a(0), n, and m, with
  mn0
• a m a n a m-n

Example 67 64 ? We need to use missing factor
approach. If 67 64 x, then 67 64 x, and
we can use guess and check to deduce that x 63.
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Exercises

• Rewrite the following with only one exponent
• 23 16
• 37 95 272
• 92 153 54
• 123 23

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Less than properties for exponents

• For any whole numbers a, b, and n0
  if a b then an bn
• For any whole numbers a(1), m and n
  if m n then am an
• For any whole numbers a, b, and m, n
  if a b and m n, then am bn

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Exercise Order the following exponents from
smallest to largest without calculators.
322, 414, 910, 810