CHAPTER 8

Integers

Problem. Prove that the square of any whole number n is a multiple of 4or 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 of4.

8.1. Addition and Subtraction

Whole numbers and fractions are insu�cient for expressing and solving manycommon problems.

Example.

(1) At 8:00 am the temperature was 15� below zero, but had risen 20� by 4:00pm. What was the temperature at 4:00 pm.

(2) A submarine is 200 ft below sea level. If it first dives 300 ft, then comesback up 150 ft, what is its current depth?

21

22 8. INTEGERS

(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 (thetext uses black chips for positive and red chips for negative integers - justlike 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.

8.1. ADDITION AND SUBTRACTION 23

Each integer a has an opposite , written as �a or (�a), as follows:

(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.

24 8. INTEGERS

Addition of Integers

8.1. ADDITION AND SUBTRACTION 25

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 wholenumbers.

3. (Adding two negatives) If a and b are positive, so that �a and �b arenegative, then (�a) + (�b) = �(a + b), where a + b is the whole numbersum of a and b.

4. (Adding a positive and a negative)

a. If a and b are positive and a � b, then a+(�b) = a�b, the whole numberdi↵erence of a and b.

b. If a and b are positive and a < b, then a + (�b) = �(b� a), where b� ais the whole number di↵erence of a and b.

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.

26 8. INTEGERS

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 �(�a) = a.

Proof. a + (�a) = 0 and �(�a) + (�a) = 0 =)a + (�a) = �(�a) + (�a) =) (cancellation)

a = �(�a). ⇤

Example.

5 + (�11) =

5 +⇥(�5) + (�6)

⇤=⇥

5 + (�5)⇤

+ (�6) =

0 + (�6) =

�6

8.1. ADDITION AND SUBTRACTION 27

Subtraction of Integers

1) Viewed as a Pattern.

5� 2 = 3

5� 1 = 4

5� 0 = 5

We see a pattern developing and just keep it going.

5� (�1) = 6

5� (�2) = 7

5� (�3) = 8 = 5

2)Viewed as Take-away.

28 8. INTEGERS

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).

8.2. MULTIPLICATION, DIVISION, AND ORDER 29

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� (�4).

7� (�4) = c if and only if 7 = �4 + c.

But 7 = �4 + 11, so 7� (�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 3days.

3⇥ (�4) = (�4) + (�4) + (�4) = �12.

30 8. INTEGERS

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⇥ (�1) = �2

2⇥ (�2) = �4

2⇥ (�3) = �6

2⇥ (�4) = �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⇥ (�1) = 2

�2⇥ (�2) = 4

�2⇥ (�3) = 6

�2⇥ (�4) = 8

8.2. MULTIPLICATION, DIVISION, AND ORDER 31

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) = �8

Example. (�2)⇥ 4

Take away (the minus sign) two groups of 4 �.

(�2)⇥ 4 = �8

32 8. INTEGERS

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

8.2. MULTIPLICATION, DIVISION, AND ORDER 33

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 · cTheorem. 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.

Thena + a(�1) = a + (�a)

soa(�1) = �a

by additive cancellation. ⇤

“Multiplying an integer by -1 reflects it about the origin.”

34 8. INTEGERS

Theorem. Let a and b be any integers. Then

(�a)b = �(ab).

Proof.

(�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 = 0 or b = 0| {z }or a=b=0

.

8.2. MULTIPLICATION, DIVISION, AND ORDER 35

Division of Integers – viewed as an extension of whole number division usingthe missing factor approach.

Definition (Division of Integers).

Let a and b be any integers where b 6= 0. Then

a ÷ b = c if and only if a = bc

for a unique integer c.Example.

(1) 12 ÷ 4 = 3 since 12 = 4 · 3.

(2) 10 ÷ (�2) = �5 since 10 = (�2)(�5).

(3) �20 ÷ 5 = �4 since �20 = 5(�4).

(4) �48 ÷ (�6) = 8 since �48 = (�6)8.

Negative Exponents and Scientific Notation

a3 = a · a · a+ ÷a

a2 = a · a+ ÷a

a1 = a

+ ÷a

a0 = 1

+ ÷a

a�1 = 1a

+ ÷a

a�2 = 1a2

+ ÷a

a�3 = 1a3

36 8. INTEGERS

Definition (Negative Integer Exponent).

Let a be any nonzero number and n be a positive integer. Then

a�n =1

an.

Example.

7�3 =1

73

5�2 =1

52

1

2�3=

1

1/23= 23

Thus n in the above definition can be any integer.

Note. As a base with exponent moves from numerator to denominator orvice-versa, the base remains the same, but the exponent sign changes.

Example.

8�4 =1

84

73 =1

7�3

1

8�5= 85

1

64= 6�4

8.2. MULTIPLICATION, DIVISION, AND ORDER 37

Theorem (Exponential Properties). For any nonzero numbers a and band 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

an= am�n 75

73= 75�3 = 72

Scientific Notation

.characteristic

a⇥ 10n

mantissa%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

1.42⇥ 103=

6.524

1.42⇥ 107

103⇡ 4.594⇥ 104 = 45, 940.

(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.

38 8. INTEGERS

Ordering Integers – less than and greater than are defined as extensions of or-dering of the whole numbers. Number Line Approach: the integer a is less thanthe integer b, written a < b (or b > a) if a is to the left of b on the integernumber 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, ifand only if there is a positive integer p such that

a + p = b.Example.

�5 < �3 since �5 + 2 = �3.

�3 < 0 since �3 + 3 = 0.

�2 < 4 since �2 + 6 = 4.

0 < 5 since 0 + 5 = 5.

2 < 7 since 2 + 5 = 7.

8.2. MULTIPLICATION, DIVISION, AND ORDER 39

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.”