10.1 INTRODUCTION TO INTEGERS
Objective A - Identifying the Order Relation between Integers
 Up to this point, we have only dealt with positive numbers.
o Meaning the numbers greater than zero on the number line.
o However, there are also negative numbers, referring to all numbers less than zero!
o Where have you encountered negative numbers?
 Examples!
 We are ready to talk about more than just the whole numbers.
o Recall: whole numbers: 0, 1, 2, 3, 4, 5 …
 There is another set of numbers called integers.
o Integers are: …-5,-4,-3,-2,-1, 0, 1, 2, 3, 4, 5…
 A number line with arrows at both ends to helps picture the set of all integers.
 The same relationship regarding order, which was true for the whole numbers &
fractions, is also true for the integers.
 When comparing two integers:
o Smaller integers are always on the left, decreasing in order through zero, into negatives.
o Larger integers are on the right, increasing in order through zero, into positives.
Place the correct inequality symbol between the following numbers:
Ex.  4 7
Ex.  5  2
Ex.
6
8
Objective B - Evaluating Absolute Value Expressions
 Two numbers that are the same distance from zero on the number line but are on
opposite sides of zero are called opposites.
o Therefore, the opposite of 9 is (-9).
o In addition, the opposite of –7 is 7.
 This means that a negative sign (–) can be thought of as meaning “the opposite of”
“the opposite of 10”
“the opposite of –3”
“the opposite of 17”
“the opposite of –21”
– (10)
–(–3)
– (17)
–(–21)
 Notice on the graph that 4 and –4 are both the same distance from zero on the number line.
 This idea of distance on the number line has a name;
o It is the absolute value of a number.
 The symbol for absolute value is | 4| and is read as, “the absolute value of 4”.
 Definition:
o The absolute value of a number is the distance between zero and the number on the
number line. The absolute value of a number is always positive.
 Therefore, the absolute value of – 4 is the distance of – 4 from zero on the number line.
o |4|=
and,
|  4| =
 Note: The absolute value of a positive number is the number itself (positive). The absolute
value of a negative number is the opposite of that negative number (positive).
Evaluate the following:
Ex.  27 =
Ex.
Ex.
 45 =
Ex.
  81 =
Ex.
34 =
 45 =
Ex.
  112 =
Ex.
|0|=
10.2 ADDITION AND SUBTRACTION OF INTEGERS
Objective A - Adding Integers
 Remember that integers can be positive, negative, or zero.
o There are three different situations when adding:
 When both numbers have the same sign;
 OR, when both numbers have opposite signs;
 OR, if either number is zero just employ Addition Property of Zero!
o Recall: Addend + Addend = Sum
Situation #1: Adding Two Numbers with the Same Sign
Procedure:
 To add integers having the same sign, keep the same sign and add the absolute value
of each number.
1. Temporarily disregard the signs.
2. Add the absolute values of the numbers.
3. Place the common sign of the numbers (addends) in front of the answer (sum).
Ex.
78
Ex.
 7  8 
Situation #2: Adding Two Numbers with Opposite Signs
Procedure:
 To add integers with different signs, keep the sign of the number with the largest
absolute value and subtract the smallest absolute value from the largest value.
1. Temporarily disregard the signs.
2. Subtract the smaller absolute value from the larger absolute value.
3. Place the sign of the larger absolute value (addend) in front of the answer (sum).
Ex.
78
Ex.
7  8 
Objective B - Subtracting Integers
 The process is even easier with subtraction!
 Recall: Minuend – Subtrahend = Difference
Situation: Subtracting Two Numbers
Procedure:
1. Change ALL subtraction problems into addition problems, where the second
number is changed to the opposite sign. (switch the subtrahend’s sign)
2. Add the second number to the first number.
3. Put the sign of the larger absolute value in front of the answer (difference).
 If both signs are positive, the difference will be positive.
14 - (-6) = 14 + 6 = 20
 If both signs are negative, the difference will be negative.
-14 - (+6) = -14 - 6 = -20
 If the signs are different, subtract the smaller absolute value from the larger
absolute value. The sign will be the sign of the integer that produced the
larger absolute value.
14 - (+6) = 14 - 6 = 8
-14 - (-6) = -14 + 6 = -8
Ex.
 7  ( 8) 
Ex.
78
Notice the distinction between the subtraction & negative signs in front of the numbers
Ex.
78
Ex.
10.3 MULTIPLICATION AND DIVISION OF INTEGERS
Objective A - Multiplying Integers
DEFINE:
 Factors: the numbers (or expressions) being multiplied.
 Product: the result (or answer) to a multiplication problem.
7  ( 8) 
BACKGROUND:
 Recall: Factor • Factor = Product
 As discussed in previous chapters, the usual symbol for multiplication is not usually used
in algebra, most often you DO NOT encounter 4  6 .
 The “dot” signifies multiplication: 4  6
 Also, parentheses indicates multiplication: 4(6) or (4)(6)
 When you multiply integers:
o Numbers can have the same sign (either positive or negative);
o OR, can have opposite signs.
Situation #1: Multiplying Two Numbers with the Same Sign
Procedure:
1. Temporarily disregard the signs of the numbers.
2. Multiply the absolute value of the numbers.
3. The result is positive (even if both numbers were negative).
Ex.
46 
Ex.
(-4)(-6) =
Situation #2: Multiplying Two Numbers with Opposite Signs
Procedure:
1. Temporarily disregard the signs of the numbers.
2. Multiply the absolute value of the numbers.
3. The result is negative.
Ex.
–4(6) =
Ex.
4(-6) =
Notice that this problem could be written as: (–4)(6) or  4  6
Objective B - Dividing Integers
 Remember that fractions are really division problems.
12
 12  6 can also be written as
6
12  12 12
 Notice also that: 
→ the negative sign can be in any of those 3 locations.


6
6
6
 There are two different situations when dividing:
o Numbers can have the same sign (either positive or negative);
o OR, can have opposite signs.
Situation #1: Dividing Two Numbers with the Same Sign
Procedure:
1. Temporarily disregard the signs of the numbers.
2. Divide the absolute value of the numbers.
3. The result is positive (even if both numbers were negative).
Ex.
24  6 =
Ex.
 24  6 =
Situation #2: Dividing Two Numbers with Opposite Signs
Procedure:
1. Temporarily disregard the signs of the numbers.
2. Divide the absolute value of the numbers.
3. The result is negative.
Ex.
1.)
2.)
3.)
4.)
 24  6 =
Ex.
Properties of Zero and One in Division:
0
0
Any Number Other Than Zero
a
 1, (a  0) Any number divided by itself equals one
a
Any Number
 Any Number
1
AnyNumber
is undefined
0
24  6 
Example:
0
0
9
 17
1
 17
 23
 -23
1
7
 undefined
0
10.4 OPERATIONS WITH RATIONAL NUMBERS
Objective A - Adding and Subtracting Rational Numbers
 A rational number is the quotient of two integers.
 Definition: a rational number is a number that can be written in the form a , where a & b
b
are integers and b ≠ 0.
 Another way of saying this is → a rational number is an integer or mixed number.
 Examples of rational numbers:
o An integer can be written as the quotient of that integer and one, therefore every
integer is a rational number.
For example, 7 is a rational number because it can be written as 7
1
o A mixed number can be written as an improper fraction, which cab be written as the
quotient of two integer, therefore every mixed number is a rational number
For example, 5 1 is a rational number because it can be written as 16
3
3
 This means that every number we’ve used in this course is also a rational number!
o You have already encountered rational numbers with out knowing it.
 Every rational number written in decimal form will either terminate or repeat.
When you have a negative rational number, you need to know that:
1 1 1
 

3 3 3
The negative sign can be in front OR in the numerator OR in the denominator.
What happens if you try to put it in BOTH the numerator and the denominator?
Main Point: When you add or subtract rational numbers, you will add and subtract
fractions and remember → that means you will need to find the LCM.
Note: The new part here is that you can have negative results.
1 5
5 5
Ex.
Ex.  

9 27
6 9
Ex.

7  3
  
3  2
Ex.
3  7 7
 
4  12  8
Objective B - Multiplying or Dividing Rational Numbers
Main Points:
1. When you multiply or divide rational numbers, multiply or divide as you have in the
past with fractions.
2. Use the same sign rules you used when multiplying and dividing integers.
Ex.
 3  4 
    
 8  15 
Ex.
 5  42 
   
 12  65 
Ex.
1  1
 
3  2
Ex.

3  5
 
8  12 
Ex.

2
4
3
Ex.

5 15

12 32
10.5 SCIENTIFIC NOTATION & THE ORDER OF OPERATIONS AGREEMENT
Objective A - Writing a Number in Scientific Notation
 Scientific Notation enables us to write very large and very small numbers more easily.
 Form: a number  10exponent
o Where the number is between 1 and 10 (not including 10);
o And the exponent is an integer.
 Small numbers use an exponent that is negative, so let’s look at negative exponents first.
Negative Exponents 105 =
104 =
103 =
102 =
101 =
100 =
10-1 =
10-2 =
10-3 =
 So you can see that, when the exponent of 10 is negative, you end up a small number
that is a fraction.
Examples:
3.1  105
4.16  107
7.03  10-3
1.2  10-5
 But what do these numbers mean?
o We want to be able to change back and forth between decimal numbers and
scientific notation.
Change the following from scientific notation to decimal notation:
 Method: If the exponent of 10 is positive, move the decimal point that many places to
the right (to make the number larger). If the exponent of 10 is negative, move the
decimal point that many places to the left (to make the number smaller).
Ex.
3.1  105 =
Ex.
4.16  107 =
Ex.
7.03  10-3 =
Ex.
1.2  10-5 =
Change the following decimal numbers to scientific notation:
 Method: If the number is larger than 1, move the decimal point to the right of the first
digit, count how many places you moved it, and make that the exponent of 10 (a positive
exponent).
 If the number is less than 1, move the decimal point to the right of the first non-zero
digit, count how many places you moved it, and make that the exponent of 10 (a
negative exponent).
Ex.
3,149,000 =
Ex.
523, 000, 000, 000 =
Ex.
0.0000000625 =
Ex.
0.0001436 =
Ex.
Write the mass of the planet Earth, which is approximately
5,980,000,000,000,000,000,000,000 kg, in scientific notation.
Objective B - Using the Order of Operations Agreement
 Remember PEMDAS from Chapter #1:
 Do your work vertically – complete one step per line.
Ex.
3  12  2
Ex.
621
Ex.
4  ( 2  7)  5
Ex.
24  ( 2)  4
Ex.
4  2  7  32
Ex.
3  22  5(3  2)  17