Positive Number Notes
(Subset of The Number System)
Name _______________
Date _____
Exline Period __________
1
Standards
7.NS.1
Apply and extend previous understanding of addition and
subtraction to add and subtract rational numbers; represent
addition and subtraction on a horizontal or vertical number line
diagram
7.NS.2
Apply and extend previous understanding of multiplication and
division and of fractions to multiply and divide rational numbers
7.NS.3
Solve real-world and mathematical problems involving the four
operations with rational numbers
2
Topics
Number Sets:
• Natural numbers
• Whole numbers
• Rational numbers
• Irrational numbers
• Real
Terminating decimals
Repeating decimals
Exponents
Base
Order of operations
P E M A
D S
Operations with fractions
Multiplicative inverse
Reciprocal
Complex fractions
Operations with mixed numbers
Converting fractions and mixed numbers to decimals
Operations with decimals
Properties of Operations
• Associative property of addition
• Commutative property of addition
• Additive identity property of 0
• Associate property of multiplication
• Commutative property of multiplication
• Multiplicative identity property of 1
3
Vocabulary
Associative property
Commutative property
Divisor
Dividend
Identity property
Improper fractions
Mixed Number
Multiplicative inverse
Product
Quotient
Repeating decimals
Simplify
Sum
Terminating decimals
4
Sets of Numbers
Def. 1. Decimals in which a digit or group of digits repeats are repeating
decimals. Decimals in which only the 0 repeats are terminating decimals. Some
decimals are neither terminating nor repeating.
Def. 2: Numbers can be classified in different ways.
The natural numbers are 1, 2, 3, …
The whole numbers are 0, 1, 2, ....
Rational number are numbers which can be expressed as quotients.
These are numbers that can be written as fractions.
Irrational numbers are numbers that cannot be written as fractions.
They are decimals that do not repeat or terminate.
The set of rational and irrational numbers is the set of real numbers.
Note 1. All terminating and repeating decimals are rational numbers. That is,
they can be written as fractions. Non-terminating and non-repeating decimals
are irrational numbers.
Source: www.formysschoolstuff.com
Ex. 1. Identify the set(s) of numbers that each of the following belong to.
a. π
b. 1.6162636465...
c. 7.93
d. 3
5
8
5
e. 4.639 1
f. 39.62134...
g. 0
h. 5.3
Ex. 2. Determine whether the given numbers are rational or irrational.
a. 0.373373337…
b. 24.15971597…
c. 5.71571571571…
d. 0.5795579555…
e. 8.3121121112…
f. 0.555…
g. 0.9148
h. 0.21211211121111…
Ex. 3. Name some irrational numbers.
a.
b.
c.
Ex. 4. Name an irrational number between the given numbers.
a. 6.7 and 6.8
b. 17.3 and 17.4
6
Ex. 5. Name the set(s) of numbers to which each number belongs (natural,
whole, rational, irrational and/or real numbers).
1
a. 9
b. 1
c. 0.121131114…
2
d. 6.78
e.
24
3
f. 6.01001…
7
Powers and Exponents
Ex. 1. Simplify:
a. 3 × 3
c. 4(4)4
b. 2 • 2 • 2 • 2 • 2 • 2
d. 7 × 7
Note 1. In order to save time, mathematicians use exponents.
Def. 1. A product in which the same expression is being multiplied together can
be written with exponents. The exponent tells us how many times the expression
is being multiplied.
Ex. 2. How can we write 3 •3 •3 •3 •3 another way?
Ex. 3. Rewrite using exponents.
a. 3 × 3
b. 2 • 2 • 2 • 2
c. 4(4)4
d. 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7
e. x ⋅ x ⋅ x
f. 3
g. 2.3 (2.3)
h. eight to the fifth power
Ex. 4. Find the value.
1
a. 0.39
b. 2 5
8
Def. 2. Exponents are written as small numerals. The number that is being
multiplied is the base. Exponents are also called powers.
Ex. 5. Circle the exponent and underline the base.
10
a. 35
c. 214 2
b. 7
d. 1300
3
Note 2. The second power of a number is called the square of the number. The
third power of a number is called the cube of the number.
Ex. 6. Write each power as the product of the same factor.
a. 7 3
b. m5
Ex. 7. Complete the following pattern.
4
2 =
3
2 =
2
2 =
21 =
20 =
Def. 2. For every number a ≠ 0, we define a = 1 .
0
Note 3. We usually do not write an exponent of 1. i.e., 3 = 3
1
Ex. 8. Simplify
3
a. 7
c. 0
b. 20
6
d. 124 0
e. 3 × 5
4
2
2
f. 2 • 7
3
2
9
Order of Operations
Def. 1. A mathematical expression is any combination of numbers and
operations such as addition, subtraction, multiplication, and division.
Ex. 1. Which is a true statement 2 × 3 + 4 ÷ 2 = 5 or 2 × 3 + 4 ÷ 2 = 8 ?
Note 1. There is an order in which we combine expressions, PEMDAS
P
E
M
A.
D
S
P =Parentheses
E = Exponents
M = Multiplication
D = Division
A = Addition
S = Subtraction
Please Excuse My Dear Aunt Sally
Note 2. Multiplication and division have the same weighting. So we do
multiplication and division in the order in which they occur from left to right.
Note 3. Addition and subtraction have the same weighting. So we do addition
and subtraction in the order in which they occur from left to right.
Note 4. Use the underlining method to help you keep track of which two
numbers you are combining. You must rewrite the entire expression after you
combine two numbers!!
Ex. 2. Find the value of each expression.
a. 3(8 − 2)
c. 20 − 6 + 4 • 3
b. 8 × 3 + 9
d. 12 ÷ 4 + 5 × 2 − 24 ÷ 6
10
Recall 1. Multiplication can be represented by a raised dot, parentheses, or the “
× ” symbol.
Ex. 3. Rewrite 8 × 5 using other symbols for multiplication.
Recall 2. Division can be represented by “ ÷ ” or the fraction bar. When you
simplify a problem that is written in fraction form, first rewrite the expression
using parentheses. Then evaluate it using the order of operations.
Ex. 4. Rewrite each of the following problems using the fraction bar notation.
a. 8 ÷ 2
b. 21 ÷ 7
c. 4 ÷ 25
5
2
37 ÷ 9
Ex. 5. Simplify
6 + 24
a.
3
b.
7(8 − 2)
3 −1
Note 5. We use parenthese ( ) and brackets [ ] as grouping symbols.
Ex. 6. Evaluate each of the following.
a. 3(6 + 8) − 9 ⋅ 2
b. 2[(10 − 2) + 2(9)]
11
c.
84 − 12
6+2
d. 10[8(15 − 7) − 4 ⋅ 3]
e. 5 + 4⋅ 3
f. (5 + 4)3
g. 21 − 10 ÷ 2 + 3
h.
7 + 2 ⋅3 − 1 + 20
3(8 ÷ 4)
Ex. 7. VideoCity rents DVDs and DVD players. A newly released DVD rents for
$4.75 for 2 days, other DVDs rent for $2.50 for 2 days, and a DVD player rental
is $12 per day.
a. In one month, the Harnapp family rented 2 newly released DVDs for 3
days, 4 regular DVDs for 4 days, and a DVD player for a week. Write an
expression for the price of the rentals.
b. Find the cost of the rentals.
12
Ex. 8. Insert parentheses where necessary to make the equation true.
a. 3 ∙ 8 − 5 = 9
b. 20 + 12 ÷ 4 + 4 = 4
c. 15 − 3 ÷ 12 + 1 = 2
Ex. 9. In each exercise below, you are given some numbers. Insert operation
symbols ( +,−,⋅,÷ ) and parentheses so that a true mathematical sentence is
formed.
a. 5 4 3 2 1 = 3
b. 5 4 3 2 1 = 0
c. 5 4 3 2 1 = 1
d. 5 4 3 2 1 = 50
Ex. 10. Find the value of each expression.
a. 9 + 6 • 4
c. 8 + 12 ÷ 3
b. 7(2 + 5)
d. 5 + 4
3
13
e. 32(48 − 6)
f. 8⋅ 2(10 − 5)
g. 27 − 9 + 21
h. 3 + 2 • 4
i. 4 ÷ 8 − 2
j. 9 − 8 ÷ 2 ⋅3
2
2
k. (6 + 4) ⋅(11 − 5)
2
1 1
3
l. 5(3 − 2 ) ÷
4
2 2
m. (34 − 5⋅ 6 − 1) ÷ 3
14
Operations with Fractions and Mixed Numbers: Adding and Subtracting
Recall 1. To estimate, you can round. The symbol
≈ means " is approximately equal to".
Note 1. When working with fractions and mixed numbers, round in the following
manner:
3
1
1
• If < the fraction < , round the fraction to .
4
4
2
1
• If the fraction is less than , round the fraction to 0.
4
3
• If the fraction is greater than , round the fraction to 1.
4
1
Ex. 1. Round each fraction to 0, , or 1.
2
a.
1
6
b.
7
8
c.
3
5
Ex. 2. Estimate the sum or difference by rounding to the nearest half.
a.
b.
1
7
1
2
8
7
15
+
12
16
c.
d.
1
6
8
2
+ 1
5
1
9
9
-5
10
6
15
Recall 2: To add or subtract fractions, you need to have a common denominator.
For all numbers a, b, and c ≠ 0,
a b a+b
+ =
c
c c
a b a− b
− =
c
c c
Source: www.eduplace.com
Ex. 3. Estimate the sum or difference. Then find the sum or difference.
a.
1
5
+
2
6
b.
7
1
+
8
4
16
c.
7
2
−
3
8
d.
1
2
−
6
3
e.
1
1
2
+
+
4
6
3
f.
1
5
3
+
+
2
6
8
Ex. 4. Solve each equation. Write the solution in simplest form.
a. w =
8
3
+
11 11
b.
4 3
+ =t
5 4
c. p =
2 5
−
3 9
d.
5 1
− =u
9 6
Note 1. Steps for adding mixed numbers.
1. Rewrite the fractions using the LCD.
2. Add the whole numbers and the fractions separately.
3. Rewrite any improper fractions and reduce.
17
Ex. 1. Find the sum.
a.
b.
1
4
2
3
+ 3
4
1
8
3
+ 6
4
c.
d. 13
9
1
2
+ 9
3
5
7
8
2
+ 12
3
11
Note 2. Steps for subtracting mixed numbers
1. Rewrite the fractions using the LCD.
2. If necessary, borrow from the whole number.
3. Subtract the whole numbers and the fractions separately. Then
simplify and
reduce.
Ex. 2. Find the difference.
a.
1
8
2
2
− 5
3
b.
6
− 4
3
8
18
c.
d. 9
7
10
1
− 15
2
1
2
− 3
4
5
20
19
Operations with Fractions and Mixed Numbers: Multiplying and Dividing
Note 1. We can use rounding or compatible numbers to estimate multiplication
and division.
Ex. 1. Use rounding or compatible numbers to estimate each product or
quotient.
1
3
a. 4 • 6
b. 375 ÷ 19
2
5
c. 156 ÷ 12
14
15
d. (8.53)(4.86)
3
cups of flour. For the party, he is making 4
4
batches. Estimate how much flour he will need for all 4 batches.
Ex. 2. Jacob’s recipe calls for 1
Ex. 3. Estimate the quotient using compatible numbers.
2
7
a. 25 ÷ 8
9
3
1
1
b. 23 ÷ 6
3
5
2
4
c. 16 ÷ 4
5
5
1
4
d. 59 ÷ 2
7
7
Recall 1. To multiply fractions, multiply numerators and denominators. Then
simplify. You should simplify before multiplying by cross canceling.
20
Ex. 4. Estimate the product. Then find the product. Write the answer in simplest
form.
3
2
1
1
a.
b.
×
•
3
4
8
3
c.
5
6
•
12
7
5  7
d.    
8 10
e.
5  2 3
2  3  10
f.
Recall 2.
1.
2.
3.
5
7
17
×
×
34
10
6
Steps for multiplying mixed numbers:
Rewrite each number as improper fractions.
Multiply these fractions.
Reduce and simplify your answer.
Ex. 5. Find the product.
a. 1
2
1
× 2
3
4
 1  5 
c.  3   
 2  6 
b. 4
2
× 5
5
d. 7
5
1
• 3
6
3
21
e.
7
4
5
•1 • 1
9
5
8
f. 5 × 2
1
1
× 2
7
10
Def. 1. Two fractions whose product is 1 are called reciprocals or multiplicative
inverses.
Ex. 6. Write the reciprocal of each number.
1
a.
b. 6
7
c. 2
1
8
Ex. 7. Name the multiplicative inverse for each rational number.
2
2
a. 3
b.
3
5
Recall 3. To divide by a fraction, multiply by the reciprocal of the second fraction
(i.e., invert the second fraction and multiply; copy, change, flip).
Ex. 8. Solve each equation.
a.
1 2
• =w
4 3
b. g =
1 3
•
3 8
22
c. y =
5 6
•
12 7
d.
17 5 7
=p
• •
6 34 10
Ex. 9. Find the quotient. Write the answer in simplest form.
1
2
3
2
a.
b.
÷
÷
3
8
9
4
c.
3
3
÷
4
8
d.
1
7
÷
2
8
Note 4. Steps for dividing mixed numbers:
1. Rewrite each number as a fraction.
2. Divide the fractions by multiplying by the reciprocal of the second
fraction
(copy, change, flip).
3. Simplify and reduce.
Ex. 10. Estimate the quotient. Then find the quotient. Write the answer in
simplest form.
1
3
3
a.
b. 1 ÷
÷ 5
5
3
5
23
c. 16
2
1
÷ 6
3
4
d. 9
3
1
÷ 2
4
8
Def. 2. A complex fraction is a fraction in which the numerator and/or
denominator contains a fraction and/or mixed number. A complex fraction is also
called a compound fraction.
Note 2. You can simplify a complex fraction by rewriting the complex fraction into
its equivalent division problem.
Ex. 11. Simplify each complex fraction.
1
4
3
8
a.
c.
1
2
2
5
3
3
1
3
b.
6
d.
3
9
1
6
3
4
4
Ex. 12. Simplify. Write the answer as a proper fraction in lowest terms or as a
mixed number in simplest form.
2
1
−1
3
3
3
1
a. 5 + 6
4
4
b. 7
1
3
c. 5 + 12
5
4
3
1
d. 6 − 4
2
8
24
e. 5
2
5
−3
3
7
f. 6 − 5
1
1
g. 5 • 2
3 10
13
15
1
3
h. 8 ÷ 2
4
4
Ex. 12. Solve each equation. Write the answer in simplest form.
a.
3 1
÷ =y
4 8
b. p =
2 2
÷
9 3
25
Converting Fractions and Mixed Numbers to Decimals with Long Division
Recall 1: To change a fraction to a decimal, divide the numerator by the
denominator.
Recall 2. Decimals in which a digit or group of digits repeats are repeating
decimals. Decimals in which only the 0 repeats are terminating decimals. Some
decimals are neither terminating nor repeating.
Ex. 1. Rewrite each fraction or mixed number as a decimal using long division.
Then identify them as either a terminating or repeating decimal.
1
9
a.
b.
2
20
c.
2
3
d. 5
e.
5
11
f. 2
3
8
5
12
26
Note 1. Our decimal system is based on powers of 10, such as 10, 100, and
1,000. If the denominator of a fraction is a factor of 10, 100, 1000, or any greater
power of ten, you can use mental math and place value to convert the fraction to
a decimal.
Ex. 2. Write each fraction or mixed number as a decimal by rewriting the
fraction’s denominator as a multiple of powers of ten.
74
7
a. 100
b. 20
7
c. 5 8
27
Operations with Decimals: Adding and Subtracting
Note 1. Remember to line up your decimal points when you add or subtract
decimals. Use 0’s for placeholders if necessary.
Ex. 1. Find the sum or difference.
a.
25.3
+ 37.9
c. 53.4 − 2.78
Ex. 2. Simplify each of the following.
a. 80.7 − 5.63
b. 13.21 + 12.9
d. 17.2 − 5.98
b. 23 − 5.75
c. 4.5 + 684 + 90.36
28
Ex. 3. Solve each equation.
a. x = 52.5 + 8.62
c. r = 8.4 − 6.08
b. 238 – 112.9 = y
d. y = 36.09 − 7.1
29
Operations with Decimals: Multiplying and Dividing
Note 1. When you multiply decimals, the number of decimal places in the
product is the sum of the decimal places in the factors.
Ex. 1. Place the decimal point in each product. Add zeros where necessary.
a. 4.35 • 2.1 =
9135
b. 0.74 • 0.08 =
592
c. 2.4 • 36.02 =
86448
Ex. 2. Find each product.
a. 9.56 × 7
b. 38.23 • 4.8
c. 8.14 • 12
Ex. 3. Solve each equation.
a. d = 2.3 •1.94
b. y = 0.26 • 0.03
30
Note 2. To divide decimals, multiply the divisor by a power of ten, such as 10 or
100, to make it a whole number. Then multiply the dividend by the same power
of 10 (i.e., move the decimal to the right the same numbers of places in both the
divisor and the dividend).
Ex. 4. Find each quotient.
a. 7418 ÷ 0.9
b. 176.2 ÷ 6.3
Ex. 5. Solve each equation.
a. f = 2.73 ÷ 1.3
Ex. 6. Find the product.
a. 0.02 • 8
b.
95.04
1.6
b. 0.25 • 0.03
31
c. 0.6 • 0.002
Ex. 7. Find the quotient.
a. 45.6 ÷ 3
c. 303 ÷ 0.5
d. 0.032 • 6.4
b. 3 ÷ 0.2
d. 0.816 ÷ 0.012
e. 1.575 ÷ 0.03
Ex. 8. Estimate. Then find the quotient of 11.8 ÷ 6 to the nearest
a. whole number.
b. hundredth
32
Note 1. When rounding quotients, you must divide to one more place value than
the place that you are rounding to.
Ex. 9. Several classes from Sterling Middle School are going to visit the art
museum. There are 148 students and 12 adults going. Each bus can hold 48
passengers. How many buses will be needed to transport the students and
adults to the museum?
33
Order of Operations with Rational Numbers
Recall 1. There is an order in which we combine expressions, PEMDAS
P
E
M
A.
D
S
P =Parentheses
E = Exponents
M = Multiplication
D = Division
A = Addition
S = Subtraction
Please Excuse My Dear Aunt Sally
Recall 2. Multiplication and division have the same weighting. So we do
multiplication and division in the order in which they occur from left to right.
Recall 3. Addition and subtraction have the same weighting. So we do addition
and subtraction in the order in which they occur from left to right.
Recall 4. Use the underlining method to help you keep track of which two
numbers you are combining. You must rewrite the entire expression after you
combine two numbers!!
Ex. 1. Simplify each expression using the order of operations. Show all your
work.
1
1 2 3
a. 5 4 − �2 ∙ 3�
2 1
2
c. 3 �2 − 5�
b.
d.
4.1
1 2
2 3
1 −
1 2
+
5 3
10
34
1
1
1
1
1
e. 3 �12 3 − 1 2 − 3 3�
f. �2 3 − 1.5� ∙ 3 3
g. 27.2 ∙ 1.8 − 5.4 ∙ 3.7
h. [4.06 ÷ (0.8 + 0.4)]2
i. (7.3 + 4) ÷ 4 + 0.3 ∙ 2
j. 8 − 6.2 ÷ 5 + 7(0.91)
k. 20 + 24 ÷ 2 − (8 + 5)
l. 4 6 − 3 �6 − 2� + 4
5
2
1
3
35
Ex. 2. A framed picture 24 inches wide is to be hung on a wall where the
1
distance from the floor to ceiling is 8 feet. The center of the picture must be 5 4
feet from the floor. Determine the distance from the ceiling to the top of the
picture frame.
36
Properties of Operations
Recall 1. Properties of operations can help you with evaluating expressions.
Def. 1. For any real numbers a, b, and c, the following properties exist.
Property
Associative property
of addition
Commutative
property of addition
Additive identity
property of 0
Associate property
of multiplication
Commutative
property of
multiplication
Multiplicative
identity property of 1
Algebra
Arithmetic
(𝑎 + 𝑏) + 𝑐 = 𝑎 + (𝑏 + 𝑐) (2.5 + 6) + 4 = 2.5 + (6 + 4)
Changing the grouping of the addends does not change
the sum.
𝑎+𝑏 =𝑏+𝑎
1.2 + 3.4 = 3.4 + 1.2
Changing the order of the addends does not change the
sum.
𝑎+0= 0+𝑎 = 𝑎
5.6 + 0 = 5.6
0 + 5.6 = 5.6
The sum of 0 and a is a.
(𝑎 ∙ 𝑏) ∙ 𝑐 = 𝑎 ∙ (𝑏 ∙ 𝑐)
(3 ∙ 2) ∙ 5 = 3 ∙ (2 ∙ 5)
Changing the grouping of factors does not change the
product.
𝑎∙𝑏 =𝑏∙𝑎
5∙2= 2∙5
Changing the order of factors does not change the
product.
𝑎∙1= 1∙𝑎 = 𝑎
5∙1=5
1∙5=5
The product of 1 and a is a.
Note 1. Addition and multiplication are both associative.
Note 2. Addition and multiplication are both commutative.
Ex. 1. Identify the property illustrated.
a. 9 ∙ 5 = 5 ∙ 9
b. 3 + (2 + 1) = (3 + 2) + 1
c. 7.1 + 0 = 7.1
d. 5 + (55 + 31) = (55 + 31) + 5
e. 6.2 + 30 = 30 + 6.2
f. 33 + (7 + 18) = (33 + 7) + 18
37
g. 11 ∙ (55 ∙ 75) = (11 ∙ 55) ∙ 75
h. 25 + 0 = 25
i. (2 ∙ 3) ∙ 34 = 34 ∙ (2 ∙ 3)
j. 16 ∙ 1 = 16
k. (7 + 8) + 0 = (7 + 8)
l. (39 ∙ 4) ∙ 25 = 39 ∙ (4 ∙ 25)
m. 225 + 375 = 375 + 225
n. 1 ∙ 9 = 9 ∙ 1
Ex. 2. Use the properties of operations to simplify each expression. State the
properties you used.
a. 0.7 + 12.5 + 1.3
b. 6.1 + 8.4 + 1.6
c. 27.4 + 0 + 12.1
d. 0.25 ∙ 3.58 ∙ 4
e. 1.09 ∙ 23.6 ∙ 0
f. 4 ∙ 12 ∙ 25
38
1
g. 7 ∙ 4 ∙ 4
Ex. 3. You buy a portable CD player for $48, rechargeable batteries with charger
for $25, and a CD case for $12. Find the total cost. Explain how you use the
properties.
Ex. 4. During the summer, you work 4 hours each day as a cashier and earn $7
each hour. Use properties of multiplication to find how much money you earn
during a 5 day work week. State the property used.
Ex. 5a. Does the associative property work with subtraction? Explain your
answer by giving examples or counterexamples.
b. Does the commutative property work with subtraction? Explain your
answer by giving examples or counterexamples.
39
Ex. 6a. Does the associative property work with division? Explain your answer
by giving examples or counterexamples.
b. Does the commutative property work with division? Explain your
answer by giving examples or counterexamples.
40
41