#1 ­ 3.1 What is a Rational Number.notebook
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TITLE:
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THE NUMBER SYSTEM
N ­ NATURAL NUMBERS
All positive non­zero numbers; in other words, all positive numbers. This does not include zero. These are the numbers we use to count. Ex: 1, 2, 3, 4, 5, …
W ­ WHOLE NUMBERS
All positive numbers as well as zero. The whole number set expands upon the natural number set to include zero. Ex: 0, 1, 2, 3, 4, 5, …
I ­ INTEGERS
All positive and negative numbers as well as zero. Integers expand upon the whole number set to include negative numbers. Ex: … , ­3, ­2, ­1, 0, 1, 2, 3, …
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#1 ­ 3.1 What is a Rational Number.notebook
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Q ­ RATIONAL NUMBERS
A number that can be expressed as the quotient of two integers; in other words, a rational number is any number that can be expressed as a fraction. (The denominator cannot be 0.) Ex: 0.2 , ­0.2 , 0. 3 , 4 , ­4 , 0 , ½ , ­½ , 4 , 9 … Q ­ IRRATIONAL NUMBERS
A number that cannot be expressed as a quotient of integers; in other words, an irrational number is any number that cannot be expressed as a fraction. This includes all non­terminating and non­
repeating decimals.
Ex: π (3.141592…) , 1.23456738… , √15 , ­ π , …
R ­ REAL NUMBERS
All rational and irrational numbers.
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"The quotient of two integers":
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#1 ­ 3.1 What is a Rational Number.notebook
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WARM UP: PAGE 103, #26 AND #27
26. Use the definition of a rational number to
show that each of the following numbers is rational:
a) 3
b) ­2
c)
­0.5
d) ­7.45
27. Which of the following numbers do you think are rational numbers? Explain why.
a) 4.21
b) ­3.121 121 112 111 12...
c)
d) ­2.122 222 22...
2.78
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Which number groups do the following
numbers belong to? (NOTE: Every
number belongs to AT LEAST 2 number
groups.)
1. 2
2. -3
3.
4.
5.
6.
7.
8.
15
-0.9
25
0
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#1 ­ 3.1 What is a Rational Number.notebook
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TRUE or FALSE:
1. ALL integers are rational numbers.
2. ALL natural numbers are whole
numbers.
3. ALL rational numbers are natural
numbers.
4. ALL integers are irrational numbers.
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NUMBER SYSTEM QUIZ
(5 ­ 10 min.)
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#1 ­ 3.1 What is a Rational Number.notebook
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Have you ever heard the "Mathematical Pi" song?
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RATIONAL NUMBERS
DEFINITION: A rational number is any number that can be expressed in the form of a/b where a and b are integers, and b 0. This includes all terminating and repeating decimal numbers.
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#1 ­ 3.1 What is a Rational Number.notebook
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DOES A RATIONAL NUMBER ALWAYS HAVE TO LOOK LIKE A FRACTION?
EXAMPLES OF RATIONAL NUMBERS:
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NEGATIVE FRACTIONS:
It doesn't matter where the negative symbol (­) is in a fraction. As long as there is one, the fraction is considered to be negative; however, we will put the negative sign with the numerator (on top) when doing calculations.
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#1 ­ 3.1 What is a Rational Number.notebook
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EQUIVALENT FRACTIONS:
Example:
To form equivalent fractions, you multiply or divide the numerator and denominator of the original fraction by the same number.
Example:
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REDUCING FRACTIONS TO LOWEST TERMS:
Example:
To reduce fractions to lowest terms, we find the GCF (greatest common factor) of the numerator and denominator, then divide them both by this GCF. You know that your fraction is in lowest terms when the only GCF you can find is 1.
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#1 ­ 3.1 What is a Rational Number.notebook
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EXPRESSING DECIMAL NUMBERS AS FRACTIONS:
TERMINATING DECIMAL NUMBERS:
Examples: 0.4 =
0.27 =
0. 375 =
3.9 =
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EXPRESSING DECIMAL NUMBERS AS FRACTIONS:
REPEATING DECIMAL NUMBERS:
Examples: 0.4 =
0.27 =
0. 375 =
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#1 ­ 3.1 What is a Rational Number.notebook
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IMPROPER FRACTIONS AND MIXED NUMBERS:
An improper fraction has a numerator that is greater than its denominator. It can be changed to a mixed number.
Example:
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IMPROPER FRACTIONS AND MIXED NUMBERS:
A mixed number has a larger number to the left of a fraction. It can be changed to an improper fraction. This is what you must do if you are asked to express mixed numbers in the form . Examples:
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#1 ­ 3.1 What is a Rational Number.notebook
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CONCEPT REINFORCEMENT
MATH MAKES SENSE 9 (MMS9):
PAGE 101: #5, #6 and #7
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PLEASE TURN TO PAGE 94 IN MMS9.
o
C
­17
­18
­19
­20
­21
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­24
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PLEASE TURN TO PAGE 95 IN MMS9.
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PLEASE TURN TO PAGE 96 IN MMS9.
Rational numbers can be written in many ways, including fractions, terminating decimal numbers, and repeating decimal numbers.
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COMPARING RATIONAL NUMBERS:
Rational numbers are similar to integers when you are deciding which one is larger than the other. Think of a number line... the one the furthest to the left is the smaller one, and the one the furthest to the right is the larger one. Also, with fractions, we sometimes use common denominators in order to compare. We use the denominators' LCM (lowest common multiple) as the common denominator.
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Example:
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Example:
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HOW CAN I COMPARE A FRACTION AND A DECIMAL?
0.3
Example:
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PLEASE TURN TO PAGE 97 IN MMS9.
EXAMPLE 1: Writing a Rational Number Between Two Given Numbers
a) ­3.26 and 1.25 (I prefer to think of the numbers in the order in which they appear on a number line) Nov 6­12:26 PM
b) ­0.26 and ­0.25 (I prefer to think of the numbers in the order in which they appear on a number line) Nov 6­12:30 PM
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c) ­1/2 and 1/4 Nov 6­12:30 PM
d) ­1/2 and ­1/4 Nov 6­12:31 PM
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Nov 6­12:31 PM
PLEASE TURN TO PAGE 98 IN MMS9.
EXAMPLE 2: Ordering Rational Numbers in Decimal or Fraction Form
a) Use a number line. Order these numbers from least to greatest:
0.35 ; 2.5 ; ­0.6 ; 1.7 ; ­3.2 ; ­0.6
I first determine which number is the smallest and which is the largest to decide which integers I need on my number line.
We also need to determine which is larger:
­0.6 OR ­0.6 .
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For least to greatest, read the numbers from left to right: ­3.2 ; ­0.6 ; ­0.6 ; 0.35 ; 1.7 ; 2.5
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b) Use a number line. Order these numbers from greatest to least: ­3/8 ; 5/9 ; ­10/4 ; ­1 1/4 ; 7/10 ; 8/3
METHOD 1:
Consider the positive numbers:
5/9 ; 7/10 ; 8/3
Which one is the largest? Which positive integer is closest to that?
Consider the negative numbers:
­3/8 ; ­10/4 ; ­ 1 1/4
Which one is the smallest? Which negative integer is closest to that?
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For the positive numbers, we've already determined that 8/3 is the largest. How will we compare 5/9 and 7/10?
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For the negative numbers, we've already determined that ­10/4 is the smallest. How will we compare ­3/8 and ­ 1 1/4?
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b) Use a number line. Order these numbers from greatest to least: ­3/8 ; 5/9 ; ­10/4 ; ­1 1/4 ; 7/10 ; 8/3
METHOD 2:
Write each number as a decimal:
­3/8 ; 5/9 ; ­10/4 ; ­1 1/4 ; 7/10 ; 8/3
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#1 ­ 3.1 What is a Rational Number.notebook
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Using the decimals to guide you, mark each fraction on a number line:
From greatest to least, the numbers are:
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PLEASE TURN TO PAGE 100 IN MMS9.
EXAMPLE 3: Ordering Rational Numbers in Fraction and Decimal Form
Use a number line. Order these rational numbers from least to greatest:
1.13 ; ­10/3 ; ­3.4 ; 2.7 ; 3/7 ; ­2 2/5
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Write the fractions and mixed numbers as decimals:
1.13 ; ­10/3 ; ­3.4 ; 2.7 ; 3/7 ; ­2 2/5
Using the smallest and the largest number, determine which integers are needed on the number line:
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From least to greatest, the numbers are:
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CONCEPT REINFORCEMENT
MMS9:
PAGE 101: #8, #9, #10, #11 & #12abgh
PAGE 102: #13, #14abef, #15, #16, #17, #18 & #21
PAGE 103: #22, #23b, #24b & #25a
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