Unit 1 Notes

Name________________ Math 9R Unit 1: Number Systems, Operations and Properties Counting Numbers The counting numbers, which are also called ________________ ______________ are represented by 1, 2, 3, 4, 5, 6, 7, … The smallest counting number is ______ and there is no largest counting number. Whole Numbers If we combine 0 with all the counting numbers, then we form the set of _____________ _________________. The whole numbers are represented by the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, … What is the smallest whole number?

What is the largest whole number?

Integers We can now extend the set of whole numbers to include _______________ numbers. This new set of numbers is called the integers and consists of the positive whole numbers and their opposites (or ________________). We can write the integers as {………-3, -2. –1, 0, +1, +2, +3………..}

Integers Countin g Numbe rs

Absolute Value

Whol

The _____________ ______________ of a number is the ________________

he number

number is from zero. We use the symbol 10. Because -1 -10 is 10 units away from zero on the number line, the absolute value or −10 = 10.

1

− 10

to represent the absolute value of

Ex. Find the value of each expression:

a.

1 2

+

b.

− 3

12 −3

Symbols of Inequality Symbol Example > 9>2 < 2<9 ≥ 9 ≥2 ≤ 2 ≤9 ≠ 9≠ 2

Read 9 is greater than 2

Ex. 1 Tell whether each statement is true or false. a. -3 > -5

b. 0 < 4

d. (2)(7) ≤ 14

c. 12 – 5 > 2

Ex. 2 Use the symbol < to order the numbers -4, 2, and -7

Ex. 3 Write three statements to compare the numbers in the order they were given a. 8 and 2 b. 12 and 12 Rational Numbers The rational numbers are all numbers that can be expressed in the form where a and b are integers and b ≠ 0. (Why can’t b be zero?)

a b

Rational numbers are sometimes written as decimals. In order to be a rational number, a decimal must terminate (end) or repeat. We write a bar over numbers to indicate that they repeat, ex _____________. Some examples of rational numbers are:

2

Now lets write them in the form of

a b

Irrational Numbers A nonrepeating decimal that does not end is called an _________________ __________________. When we write an irrational number, we use three dots (…) after a series of digits to show that the number does not end. If a number is not a perfect square, it is an irrational number. What is a perfect square? Some examples of irrational numbers are: 3

0 .1

− 2

5

.312526…..

Look at what the calculator displays when you enter the above square roots. These answers are called _______________ ___________________________. The calculator only displays a certain number of places in its answer, but this does not mean the decimal terminates. BE CAREFUL OF THIS. Rational approximations are close to, but not equal to, the value of the irrational number. One popular example of an irrational number is π . EX 1: Find a rational approximation for each irrational number, to the nearest hundredth. a) 3 b) 0.1

EX 2: Which of the following four numbers is an irrational number? (a) 0.12 (b) 0.12121212… (c) 0.12111111… (d) 0.12112111211112…

EX 3: Determine if the following numbers are rational or irrational. 1)

1 4

3) π

2) 3.14 4) 5.52652

3

5)

6)

11

__ 7) 0.545454545…

4

8) 2.715489578157992482…

Real Numbers The set of real numbers is the set that consists of all ____________________ numbers and all___________________numbers. Real Numbers

Irration al Numbers

Ex.

Rationa l Numbe rs

Write the following numbers in order from smallest value to largest value: 2 3 3 , 1 , , 175 . ,1 3 2

Steps: 1. ____________________________________ 2.____________________________________ 3.____________________________(pretend it’s money)!!

Try this one: In which list are the numbers in order from least to greatest? 1 1 (1) 3.2, π , 3 , 3 (3) 3, π , 3.2, 3 3 3 1 1 (2) 3, 3.2, π , 3 (4) 3.2, 3 , 3, π 3 3

4

Properties of Operations PROPERTY

MEANING

EXAMPLES

Commutative property of addition or multiplication Associative property of addition or multiplication Distributive property

Closure property

Additive inverse

Additive identity

Multiplicative identity

Multiplicative inverse (or reciprocal)

Ex. 1 For −

2 find the following: 3

5

a.

Additive Inverse

b.

Multiplicative Identity

c.

Reciprocal

d.

Additive Identity

e. Multiplicative Inverse Ex. 2. x + 9 = 9 + x is an example of which property? ______________________

Ex. 3. 2(x + 3) = 2x + 6 is an example of which property? ___________________

Ex. 4 x + (y + 3) = x + (3 + y) is an example of which property? _________________

Ex. 5. (5y) • (1) = 5y is an example of which property? ____________________

Ex. 6 Name an operation that is not commutative and give an example of why.

Closure

Ex. 7. If you add two even numbers do you always get an even number?

Is the set of even numbers closed under addition?

Ex. 8. If you divide two even numbers do you always get an even number?

6

Is the set of even numbers closed under division?

Ex. 9. When you subtract two positive numbers, do you always get a positive number?

Is the set of positive numbers closed under subtraction? Ex. 10 Which set is closed under division? [A] integers

[B] whole numbers

[C] counting numbers

[D] {1}

Ex. 11 Ramón said that the set of integers is not closed for one of the basic operations (addition, subtraction, multiplication, or division). You want to show Ramón that his statement is correct. For the operation for which the set of integers is not closed, write an example using: - a positive even integer and a zero - a positive and a negative even integer - two negative even integers Be sure to explain why each of your examples illustrates that the set of integers is not closed

Types of Sets •

A ___________ is a collection of distinct numbers or objects. Each object or number in the set is called an ___________ of the set. The symbol ∈ is used to indicate that something is a element of a set. A set is usually indicated by using a pair of braces { }. The set of whole numbers can be written as {0, 1, 2, 3, 4, 5…….}

•

A ____________ set is a set whose elements can be counted. For example, the set of even integers between 1 and 10 is a finite set and can be written as {2, 4, 6, 8, 10}.

7

An _________________ set is a set whose elements cannot be counted because there is no ___________ to the set. Both the counting numbers and _________________ ___________________ are infinite sets. • The empty set or _________ ___________ is a set that has no elements written as { } or ∅. For example, the set of negative counting numbers is empty. (There is no such thing as a negative counting number!) Words Symbol Meaning A is a Subset of B •

The Intersection of Sets A and B The Union of Sets A and B

The complement of Subset A

EX 1: Using the proper notation write that ___________ are a subset of people in the classroom.

EX 2: Let U be a set of students who are members of a committee. Let B be a set of boys who are members of this committee. Set U = {harry, Marie, Susan, Ted, Bill} Set B = {Harry, Ted, Bill} Write using the correct subset notation.

EX 3: If A = {5, 6,7, 8, 9, 10, 11} and B = {9, 10, 11, 12, 13, 14}, then what is the intersection of set A and set B?

8

EX 4: Using the given sets in example 3, what is the union of set A and set B?

Ex5.

•

If A ⊂ B and A={2,3,5) and B={1,2,3,4,5,6,7} what is A’?

The following diagram shows that the rational numbers are a subset of the real numbers, and irrational numbers are also a subset of the real numbers. Notice, however, that the rationals and irrationals take up different spaces in the diagram because they have no numbers in common. Therefore, there is no __________________ of these two sets. The _______________ of these two sets are the _____________ numbers.

EX 6: Tell whether each of the following statements is true or false: 1) Every real number is a rational number. 2) Every rational number is a real number. 3) Every point on the real number line corresponds to an irrational number. 9

4) Some numbers are both rational and irrational

EX 7: Suppose A={1,3,5,7} and B{1,2,3,4,5}. 1) Write A ∪ B.

2)

Write A

∩

B.

3) Write a subset of set A.

⊂

4)

True or False? {4}

A

5)

True or False? {1,3}

6)

True or False? B

⊂

A

7)

True or False? A

⊂

B

⊂

B

8) One subset of set A is C{1,7} What is C`?

EX8: If A= {Natural Numbers} and B= {0}, 1) What is A ∪ B?

2)

What is A

∩

B?

10

3) Write a subset of set A that has 4 elements.

4)

If C= {positive even integers} and C

⊂

A, what is C`?

Example 9: If A = {#, $, %, &} and B = {*, %, ^} 1)

What is A

∪

B?

2)

What is A

∩

B?

Venn Diagrams A ________ __________ is a drawing, in which circular areas represent groups of items usually sharing common properties. The drawing consists of two or more circles, each representing a specific group or set. Each Venn diagram begins with a __________ representing the universal set. Then each set of values in the problem is represented by a ________. Any values that belong to more than one set will be placed in the sections where the circles ___________.

Values that belong to both set A and set B are located in the center region labeled _______where the circles overlap. This region is called the _____________ of the two sets. The notation __________represents the entire region covered by both sets A and B (and the section where they overlap). This region is called the _________ of the two sets. (Union, like marriage, brings all of both sets together.) Ex U (the universal set) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} (a subset of the positive integers) A = {2, 4, 6, 8} B = {1, 2, 3, 4, 5} 11

a) What are the elements of A ∪ B c) What is A’

b) What are the elements of A ∩ B d) What is B’

EX 1. The accompanying Venn diagram shows the number of students who take various courses. All students in circle A take mathematics. All in circle B take science. All in circle C take technology. What percentage of the students takes mathematics or technology?

Ex 2 In a class of 450 students, 300 are taking a mathematics course and 260 are taking a science course. If 140 of these students are taking both courses, how many students are not taking either of these courses? (1) 30 (3) 110 (2) 40 (4) 140

Ex 3 The senior class at South High School consists of 250 students. Of these students, 130 have brown hair, 160 have brown eyes, and 90 have both brown hair and brown eyes. How many members of the senior class have neither brown hair nor brown eyes?

12

Ex 4 In a telephone survey of 100 households, 32 households purchased Brand A cereal and 45 purchased Brand B cereal. If 10 households purchased both items, how many of the households surveyed did not purchase either Brand A or Brand B cereal?

Ex5. In a survey of 400 teenage shoppers at a large mall, 240 said they shopped at Abernathy's, 210 said they shopped at Bongo Republic, and 90 said they shopped at both stores. How many of the teenage shoppers surveyed did not shop at either store?

Ex6. A school district offers hockey and basketball. The result of a survey of 300 students showed: 120 students play hockey, only 90 students play basketball, only 30 students do not participate in either sport Of those surveyed, how many students play both hockey and basketball?

Ex7

There are 30 students on a school bus. Of these students, 24 either play in the school band or sing in the chorus. Six of the students play in the school band but do not sing in the chorus. Fourteen of the students sing in the chorus and also play in the school band. How many students on the school bus sing in the chorus but do not play in the band?

Ex8. A school newspaper took a survey of 100 students. The results of the survey showed that 43 students are fans of the Buffalo Bills, 27 students are fans of the New York Jets, 13

and 48 students do not like either team. How many of the students surveyed are fans of both the Buffalo Bills and the New York Jets? (1) 16 (3) 52 (2) 18 (4) 70 Graphing Points on the Coordinate Plane The coordinate plane is made up of 4 ______________. The middle point is called the _______________. Every point on the _______________ ____________ Can be described by two numbers, called the coordinates or ____________ _____________.. The first of the pair is called the _____ ___________ and the second is the ___ ______________. They are represented as (x, y). Name the points: Right Triangle: ( , ) Triangle: ( , ) Square: ( , ) Rectangle: ( , ) Star: ( , )

14

15