Download Problem. Prove that the square of any whole number n is a multiple

CHAPTER 8
Integers
Problem. Prove that the square of any whole number n is a multiple of 4
or one more than a multiple of 4.
Strategy 13 – Use cases.
This strategy may be appropriate when
• A problem can be separated into several distinct cases.
• A problem involves distinct collections of numbers such as odds and evens,
primes and composites, and positives and negatives..
• Investigations in specific cases can be generalized.
Case 1 – n is even.
Then n = 2x =) n2 = 4x2, which is a multiple of 4.
Case 2 – n is odd.
Then n = 2x + 1 =) n2 = 4x2 + 4x + 1, which is one more than a multiple of
4.
8.1. Addition and Subtraction
Whole numbers and fractions are insufficient for expressing and solving many
common problems.
Example.
(1) At 8:00 am the temperature was 15 below zero, but had risen 20 by 4:00
pm. What was the temperature at 4:00 pm.
(2) A submarine is 200 ft below sea level. If it first dives 300 ft, then comes
back up 150 ft, what is its current depth?
142
8.1. ADDITION AND SUBTRACTION
143
(3) We would like an equation such as x + 5 = 2 to have a solution.
For all of the above, we need negative numbers.
Definition. The set of integers is the set
{. . . , 3, 2, 1, 0, 1, 2, 3, . . . }.
The numbers 1, 2, 3, . . . are the positive integers.
The numbers
1, 2, 3, . . . are the negative integers.
Zero is neither a positive nor negative integer.
Representations:
(1) Set model –we use for positive integers and for negative integers (the
text uses black chips for positive and red chips for negative integers - just
like accounting).
represents +1 and
represents 1. Thus each
cancels out an and vice versa.
Example. Set representations for
4.
Integer number line.
Note the symmetric arrangement to the right and left of 0.
144
8. INTEGERS
Each integer a has an opposite , written as
(1) Set model.
+5 and
5 are opposites of each other.
(2) Number line:
Note.
(1) If a is positive,
a is negative.
(2) If a is negative,
a is positive.
a or ( a), as follows:
8.1. ADDITION AND SUBTRACTION
Addition of Integers
145
146
8. INTEGERS
Definition (Addition of Integers). Let a and b be any integers.
1. (Adding 0) a + 0 = 0 + a = a.
2. (Adding two positives) If a and b are positive, they are added as whole
numbers.
3. (Adding two negatives) If a and b are positive, so that a and b are
negative, then ( a) + ( b) = (a + b), where a + b is the whole number
sum of a and b.
4. (Adding a positive and a negative)
a. If a and b are positive and a
di↵erence of a and b.
b, then a+( b) = a b, the whole number
b. If a and b are positive and a < b, then a + ( b) =
is the whole number di↵erence of a and b.
(b
a), where b
a
Example.
0 + ( 5) = 5
( 3) + ( 6) = (3 + 6 = 9
11 + ( 4) = 11 4 = 7
5 12 = (12 5) = 7
Properties of Integer Addition
Let a, b, and c be any integers.
(Closure) a + b is an integer.
(Commutative) a + b = b + a
(Associative) (a+b)+c=a+(b+c)
(Identity) 0 is the unique integer such that a + 0 = 0 + a = a for all a
(Additive inverse) For each integer a, there is a unique integer, written as a,
such that a + ( a) = 0 The integer a is called the additive inverse of a.
8.1. ADDITION AND SUBTRACTION
147
Note.
1) If a is positive,
a is negative.
2) If a is negative,
a is positive.
3) If a = 0,
a = 0 also.
Theorem (Additive Cancellation for Integers).
Let a, b, and c be any integers.
If a + c = b + c, then a = b.
Proof. a + c = b + c =)(addition)
a + c + ( c) = b + c + ( c) =) (associative)
⇥
⇤
⇥
⇤
a + c + ( c) = b + c + ( c) =) (additive inverse)
a + 0 = b + 0 =) (additive identity)
⇤
a = b.
Theorem. Let a be any integer. Then
Proof. a + ( a) = 0 and
a + ( a) =
a=
( a) = a.
( a) + ( a) = 0 =)
( a) + ( a) =) (cancellation)
⇤
( a).
Example.
5 + ( 11) =
⇥
⇤
5 + ( 5) + ( 6) =
⇥
⇤
5 + ( 5) + ( 6) =
0 + ( 6) =
6
148
8. INTEGERS
Subtraction of Integers
1) Viewed as a Pattern.
5
5
5
2=3
1=4
0=5
We see a pattern developing and just keep it going.
5
5
5
2)Viewed as Take-away.
( 1) = 6
( 2) = 7
( 3) = 8 = 5
8.1. ADDITION AND SUBTRACTION
3) Viewed as Adding the Opposite.
Definition (Subtraction of Integers: Adding the Opposite).
Let a and b be any integers. Then
a
b = a + ( b).
149
150
8. INTEGERS
Example.
3 ( 5) =
3 + 5 = 2.
4 6=
4 + ( 6) = 10.
4) Viewed as Missing Addend.
Definition (Subtraction of Integers: Adding the Opposite).
Let a, b, and c be any integers. Then
a
b = c if and only if a = b + c.
Example. Find 7
7
( 4).
( 4) = c if and only if 7 =
But 7 =
4 + 11, so 7
4 + c.
( 4) = 11.
Note. We have 3 di↵erent meanings for “ ”.
1) “negative”:
8 means negative 8.
2) “opposite or additive inverse of”: -6 is the opposite or additive inverse of 6.
3) “minus”: 7
3.
8.2. Multiplication, Division, and Order
Multiplication viewed as an extension of whole number multiplication:
1) As repeated addition:
Example. John has borrowed $4.00 from his sister Terri each of the last 3
days.
3 ⇥ ( 4) = ( 4) + ( 4) + ( 4) = 12.
8.2. MULTIPLICATION, DIVISION, AND ORDER
2) As an extension of patterns:
2⇥4=8
2⇥3=6
2⇥2=4
2⇥1=2
2⇥0=0
We see each step results in 2 less. So we continue the pattern:
2⇥(
2⇥(
2⇥(
2⇥(
1) =
2) =
3) =
4) =
2
4
6
8
Now using the results from above plus commutivity, which we want:
2⇥3= 6
2⇥2= 4
2⇥1= 2
2⇥0=0
Noticing that each step results in 2 more, we continue the pattern
2⇥(
2⇥(
2⇥(
2⇥(
1) = 2
2) = 4
3) = 6
4) = 8
151
152
8. INTEGERS
3) Chips model:
Example. 4 ⇥ ( 2)
4 ⇥ ( 2) = 8
The sign of the second number determines the kind of chips used.
Example. ( 2) ⇥ 4
Use the above model with commutivity.
( 2) ⇥ 4 = 4 ⇥ ( 2) =
Example. ( 2) ⇥ 4
Take away (the minus sign) two groups of 4
.
( 2) ⇥ 4 =
8
8
8.2. MULTIPLICATION, DIVISION, AND ORDER
Example. ( 2) ⇥ ( 4)
As above, but take away two groups of 4
.
( 2) ⇥ ( 4) = 8
Definition (Multiplication of Integers).
Let a and b be any integers.
1. a · 0 = 0 = 0 · a.
2. If a and b are positive, they are multiplied as whole numbers.
3. If a and b are positive (thus
b is negative),
a( b) =
(ab).
4. If a and b are positive, then
( a) · ( b) = ab.
Example.
(1) 5 · 0 = 0
(2) 5 · 7 = 35
(3) 3·( 4) =
(3 · 4) =
12
(4) ( 4) · ( 8) = 4 · 8 = 32
153
154
8. INTEGERS
Properties of Integer Multiplication
Let a, b, and c be any integers.
(Closure) a · b is an integer.
(Commutative) a · b = b · a.
(Associative) (a · b) · c = a · (b · c).
(Identity) 1 is the unique integer such that a · 1 = a = 1 · a.
(Distributive of Multiplication over Addition) a · (b + c) = a · b + a · c
Theorem. Let a be any integer. Then
a( 1) =
a.
Proof. We know a · 0 = 0 and a + ( a) = 0. But
⇥
⇤
a · 0 = a · 1 + ( 1)
= a · 1 + a · ( 1) = a + a( 1) = 0.
Then
a + a( 1) = a + ( a)
so
a( 1) =
a
by additive cancellation.
“Multiplying an integer by -1 reflects it about the origin.”
⇤
8.2. MULTIPLICATION, DIVISION, AND ORDER
155
Theorem. Let a and b be any integers. Then
( a)b =
Proof.
(ab).
⇥
⇤
( a)b = ( 1)a b
= ( 1)(ab)
= (ab)
⇤
Theorem. Let a and b be any integers. Then
( a)( b) = ab.
Proof.
⇥
⇤⇥
⇤
( a)( b) = ( 1)a ( 1)b
⇥
⇤
= ( 1)( 1) (ab)
= 1(ab) = ab
Theorem (Multiplicative Cancellation Property).
⇤
Let a, b,and c be any integers with c 6= 0.
If ac = bc, then a = b.
Why must we say c 6= 0?
5 · 0 = 8 · 0, but 5 6= 8.
Theorem (Zero Divisors Property). Let a and b be any integers. Then
ab = 0 if and only if a
or b = 0} .
| = 0 {z
or a=b=0
156
8. INTEGERS
Division of Integers – viewed as an extension of whole number division using
the missing factor approach.
Definition (Division of Integers).
Let a and b be any integers where b 6= 0. Then
for a unique integer c.
Example.
a ÷ b = c if and only if a = bc
(1) 12 ÷ 4 = 3 since 12 = 4 · 3.
(2) 10 ÷ ( 2) =
(3)
(4)
20 ÷ 5 =
5 since 10 = ( 2)( 5).
4 since
48 ÷ ( 6) = 8 since
20 = 5( 4).
48 = ( 6)8.
Negative Exponents and Scientific Notation
a3 = a · a · a
a2 = a · a
a1 = a
a
a
1
2
3
=
=
=
+ ÷a
+ ÷a
a0 = 1
a
+ ÷a
1
a
1
a2
1
a3
+ ÷a
+ ÷a
+ ÷a
8.2. MULTIPLICATION, DIVISION, AND ORDER
157
Definition (Negative Integer Exponent).
Let a be any nonzero number and n be a positive integer. Then
1
a n = n.
a
Example.
1
73
1
5 2= 2
5
1
1
=
= 23
3
3
2
1/2
7
3
=
Thus n in the above definition can be any integer.
Note. As a base with exponent moves from numerator to denominator or
vice-versa, the base remains the same, but the exponent sign changes.
Example.
1
84
1
73 = 3
7
1
5
=
8
8 5
1
4
=
6
64
8
4
=
158
8. INTEGERS
Theorem (Exponential Properties). For any nonzero numbers a and b
and integers m and n,
am · an = am+n
32 · 34 = 32+4 = 36
am · bm = (ab)m
42 · 52 = (4 · 5)2
(am)n = amn
(62)3 = 62·3 = 66
am
75
m n
5 3
2
=
a
=
7
=
7
an
73
Scientific Notation
mantissa%
a ⇥ 10n
.characteristic
1  a < 10
n is any integer
32, 500, 000 – move decimal 7 places to the left to get – 3.25 ⇥ 107.
3.25 ⇥ 107 – move decimal 7 places to the right to get –32, 500, 000.
0.000187 – move decimal 4 places to the right to get – 1.87 ⇥ 10 4.
1.87 ⇥ 10
4
– move decimal 4 places to the left to get – 0.000187
Example.
6.524 ⇥ 107 6.524 107
=
⇥ 3 ⇡ 4.594 ⇥ 104 = 45, 940.
3
1.42 ⇥ 10
1.42
10
(2.17 ⇥ 104)(5.2 ⇥ 10 8) = (2.17 ⇥ 5.2)(104 ⇥ 10 8)
11.284 ⇥ 10
4
= 1.1284 ⇥ 101 ⇥ 10
4
= 1.1284 ⇥ 10
3
= .0011284.
8.2. MULTIPLICATION, DIVISION, AND ORDER
159
Ordering Integers – less than and greater than are defined as extensions of ordering of the whole numbers. Number Line Approach: the integer a is less than
the integer b, written a < b (or b > a) if a is to the left of b on the integer
number line.
3<2
We also have
5<
3,
3 < 0,
2 < 4,
0 < 5,
2 < 7.
Addition Approach: the integer a is less than the integer b, written a < b, if
and only if there is a positive integer p such that
a + p = b.
Example.
5<
3 since
5+ 2 =
3 < 0 since
3 + 3 = 0.
2 < 4 since
2 + 6 = 4.
0 < 5 since 0 + 5 = 5.
2 < 7 since 2 + 5 = 7.
3.
160
8. INTEGERS
Properties of Ordering Integers Let a, b, c be any integers, p a positive integer,
and n a negative integer.
(Transitive for Less Than) If a < b and b < c, then a < c.
(Less Than and Addition) If a < b, then a + c < b + c.
(Less Than and Multiplication by a Positive) If a < b, then ap < bp.
(Less Than and Multiplication by a Negative) If a < b, then an > bn.
“Multiplying by a negative changes the direction.”