REAL NUMBERS as opposed to fake numbers Why
![REAL NUMBERS (as opposed to fake numbers? ) REAL NUMBERS (as opposed to fake numbers? )](https://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-1.jpg)
REAL NUMBERS (as opposed to fake numbers? )
![Why do we have to review numbers Miss Price? Be a beast in Math Why do we have to review numbers Miss Price? Be a beast in Math](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-2.jpg)
Why do we have to review numbers Miss Price? Be a beast in Math when you know the basics!
![Objective SWBAT… • Identify and classify the parts of the Real Number System By… Objective SWBAT… • Identify and classify the parts of the Real Number System By…](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-3.jpg)
Objective SWBAT… • Identify and classify the parts of the Real Number System By… • Visualizing the number line
![Key Concepts • • • Real Number Rational Number Integer Whole Number Natural Number Key Concepts • • • Real Number Rational Number Integer Whole Number Natural Number](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-4.jpg)
Key Concepts • • • Real Number Rational Number Integer Whole Number Natural Number Irrational Number
![Key Concept Real Numbers • Real Numbers are every number. • Therefore, any number Key Concept Real Numbers • Real Numbers are every number. • Therefore, any number](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-5.jpg)
Key Concept Real Numbers • Real Numbers are every number. • Therefore, any number that you can find on the number line.
![What does it Mean? • The number line goes on forever. • Every point What does it Mean? • The number line goes on forever. • Every point](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-6.jpg)
What does it Mean? • The number line goes on forever. • Every point on the line is a REAL number. • There are no gaps on the number line. • Between the whole numbers and the fractions there are numbers that are decimals but they don’t terminate and are not recurring decimals. They go on forever.
![Real Numbers REAL NUMBERS 154, 769, 852, 354 1. 333 -5, 632. 1010101256849765… -8 Real Numbers REAL NUMBERS 154, 769, 852, 354 1. 333 -5, 632. 1010101256849765… -8](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-7.jpg)
Real Numbers REAL NUMBERS 154, 769, 852, 354 1. 333 -5, 632. 1010101256849765… -8 π 549. 23789 61 49%
![Two Kinds of Real Numbers • Rational Numbers • Irrational Numbers Two Kinds of Real Numbers • Rational Numbers • Irrational Numbers](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-8.jpg)
Two Kinds of Real Numbers • Rational Numbers • Irrational Numbers
![Key Concept Rational Numbers • A rational number is a real number that can Key Concept Rational Numbers • A rational number is a real number that can](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-9.jpg)
Key Concept Rational Numbers • A rational number is a real number that can be written as a fraction. • A rational number written in decimal form is terminating or repeating.
![Examples of Rational Numbers • 16 • 1/2 • 3. 56 • -8 • Examples of Rational Numbers • 16 • 1/2 • 3. 56 • -8 •](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-10.jpg)
Examples of Rational Numbers • 16 • 1/2 • 3. 56 • -8 • 1. 3333… • - 3/4
![Integers One of the subsets of rational numbers Integers One of the subsets of rational numbers](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-11.jpg)
Integers One of the subsets of rational numbers
![Key Concept What are integers? • Integers are the whole numbers and their opposites. Key Concept What are integers? • Integers are the whole numbers and their opposites.](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-12.jpg)
Key Concept What are integers? • Integers are the whole numbers and their opposites. • Examples of integers are 6 -12 0 186 -934
![How can you write an integer as a rational number by definition? • Integers How can you write an integer as a rational number by definition? • Integers](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-13.jpg)
How can you write an integer as a rational number by definition? • Integers are rational numbers because they can be written as fraction with 1 as the denominator.
![Key Concept Types of Integers • Natural Numbers(N): Natural Numbers are counting numbers from Key Concept Types of Integers • Natural Numbers(N): Natural Numbers are counting numbers from](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-14.jpg)
Key Concept Types of Integers • Natural Numbers(N): Natural Numbers are counting numbers from 1, 2, 3, 4, 5, . . . . N = {1, 2, 3, 4, 5, . . . . } • Whole Numbers (W): Whole numbers are natural numbers including zero. They are 0, 1, 2, 3, 4, 5, . . . . W = {0, 1, 2, 3, 4, 5, . . . } W = 0 + N
![Key Concept Irrational Numbers • An irrational number is a number that cannot be Key Concept Irrational Numbers • An irrational number is a number that cannot be](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-15.jpg)
Key Concept Irrational Numbers • An irrational number is a number that cannot be written as a fraction of two integers. • Irrational numbers written as decimals are non-terminating and non-repeating.
![Irrational numbers : If a whole number is not a perfect square, then its Irrational numbers : If a whole number is not a perfect square, then its](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-16.jpg)
Irrational numbers : If a whole number is not a perfect square, then its square root is an irrational number. Caution! A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
![Examples of Irrational Numbers • • Pi Examples of Irrational Numbers • • Pi](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-17.jpg)
Examples of Irrational Numbers • • Pi
![The Real Number System Rational numbers Integers Whole numbers Natural numbers Irrational numbers The Real Number System Rational numbers Integers Whole numbers Natural numbers Irrational numbers](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-18.jpg)
The Real Number System Rational numbers Integers Whole numbers Natural numbers Irrational numbers
![The Real Number System Irrational Rational numbers • Can be represented as a fraction The Real Number System Irrational Rational numbers • Can be represented as a fraction](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-19.jpg)
The Real Number System Irrational Rational numbers • Can be represented as a fraction of 2 integers numbers Integers. … -3, -2, -1, 0, 1, 2, 3, …. Whole numbers 0, 1, 2, 3, …. Natural numbers 1, 2, 3, …. • Cannot be represented as a fraction of 2 integers
![The Human Number Line Determine all of the classifications that fit for the number… The Human Number Line Determine all of the classifications that fit for the number…](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-20.jpg)
The Human Number Line Determine all of the classifications that fit for the number…
![A fraction with a denominator of 0 is undefined because you cannot divide by A fraction with a denominator of 0 is undefined because you cannot divide by](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-21.jpg)
A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.
![Determining the Classification of All Numbers State if each number is rational, irrational, or Determining the Classification of All Numbers State if each number is rational, irrational, or](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-22.jpg)
Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. A. 21 irrational B. 0 3 0 =0 3 rational
![Determining the Classification of All Numbers State if each number is rational, irrational, or Determining the Classification of All Numbers State if each number is rational, irrational, or](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-23.jpg)
Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. C. 4 0 not a real number
![Objective SWBAT… • compare rational and irrational numbers By… • Ordering numbers on a Objective SWBAT… • compare rational and irrational numbers By… • Ordering numbers on a](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-24.jpg)
Objective SWBAT… • compare rational and irrational numbers By… • Ordering numbers on a number line
![Comparing Rational and Irrational Numbers • When comparing different forms of rational and irrational Comparing Rational and Irrational Numbers • When comparing different forms of rational and irrational](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-25.jpg)
Comparing Rational and Irrational Numbers • When comparing different forms of rational and irrational numbers, convert the numbers to the same form. Compare -3 (convert -3 3 7 and -3. 571 to -3. 428571… > -3. 571
![Practice • Practice •](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-26.jpg)
Practice •
![Ordering Rational and Irrational Numbers • To order rational and irrational numbers, convert all Ordering Rational and Irrational Numbers • To order rational and irrational numbers, convert all](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-27.jpg)
Ordering Rational and Irrational Numbers • To order rational and irrational numbers, convert all of the numbers to the same form. • You can also find the approximate locations of rational and irrational numbers on a number line.
![Example Order these numbers from least to greatest. ¹/₄, 75%, . 04, 10%, ⁹/₇ Example Order these numbers from least to greatest. ¹/₄, 75%, . 04, 10%, ⁹/₇](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-28.jpg)
Example Order these numbers from least to greatest. ¹/₄, 75%, . 04, 10%, ⁹/₇ ¹/₄ becomes 0. 25 75% becomes 0. 75 0. 04 stays 0. 04 10% becomes 0. 10 ⁹/₇ becomes 1. 2857142… Answer: 0. 04, 10%, ¹/₄, 75%, ⁹/₇
![Practice Order these from least to greatest: Practice Order these from least to greatest:](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-29.jpg)
Practice Order these from least to greatest:
![Objectives SWBAT… • compute with integers By… • Using the basic operation rules for Objectives SWBAT… • compute with integers By… • Using the basic operation rules for](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-30.jpg)
Objectives SWBAT… • compute with integers By… • Using the basic operation rules for integers
![Examples: Use the number line if necessary. 1) (-4) + 8 = 4 2) Examples: Use the number line if necessary. 1) (-4) + 8 = 4 2)](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-31.jpg)
Examples: Use the number line if necessary. 1) (-4) + 8 = 4 2) (-1) + (-3) = -4 3) 5 + (-7) = -2
![Addition Rule 1) When the signs are the same, ADD and keep the sign. Addition Rule 1) When the signs are the same, ADD and keep the sign.](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-32.jpg)
Addition Rule 1) When the signs are the same, ADD and keep the sign. (-2) + (-4) = -6 2) When the signs are different, SUBTRACT and use the sign of the larger number. (-2) + 4 = 2 2 + (-4) = -2
![-1 + 3 = ? 1. 2. 3. 4. -4 -2 2 4 Answer -1 + 3 = ? 1. 2. 3. 4. -4 -2 2 4 Answer](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-33.jpg)
-1 + 3 = ? 1. 2. 3. 4. -4 -2 2 4 Answer Now
![-6 + (-3) = ? 1. 2. 3. 4. -9 -3 3 9 Answer -6 + (-3) = ? 1. 2. 3. 4. -9 -3 3 9 Answer](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-34.jpg)
-6 + (-3) = ? 1. 2. 3. 4. -9 -3 3 9 Answer Now
![The additive inverses (or opposites) of two numbers add to equal zero. Example: The The additive inverses (or opposites) of two numbers add to equal zero. Example: The](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-35.jpg)
The additive inverses (or opposites) of two numbers add to equal zero. Example: The additive inverse of 3 is -3 Proof: 3 + (-3) = 0 We will use the additive inverses for subtraction problems.
![What’s the difference between 7 - 3 and 7 + (-3) ? 7 - What’s the difference between 7 - 3 and 7 + (-3) ? 7 -](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-36.jpg)
What’s the difference between 7 - 3 and 7 + (-3) ? 7 - 3 = 4 and 7 + (-3) = 4 The only difference is that 7 - 3 is a subtraction problem and 7 + (-3) is an addition problem. “SUBTRACTING IS THE SAME AS ADDING THE OPPOSITE. ” (Keep-change)
![When subtracting, change the subtraction to adding the opposite (keep-change) and then follow your When subtracting, change the subtraction to adding the opposite (keep-change) and then follow your](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-37.jpg)
When subtracting, change the subtraction to adding the opposite (keep-change) and then follow your addition rule. Example #1: - 4 - (-7) - 4 + (+7) Diff. Signs --> Subtract and use larger sign. 3 Example #2: -3 -7 - 3 + (-7) Same Signs --> Add and keep the sign. -10
![Which is equivalent to -12 – (-3)? 1. 2. 3. 4. Answer Now 12 Which is equivalent to -12 – (-3)? 1. 2. 3. 4. Answer Now 12](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-38.jpg)
Which is equivalent to -12 – (-3)? 1. 2. 3. 4. Answer Now 12 + 3 -12 - 3
![7 – (-2) = ? 1. 2. 3. 4. Answer Now -9 -5 5 7 – (-2) = ? 1. 2. 3. 4. Answer Now -9 -5 5](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-39.jpg)
7 – (-2) = ? 1. 2. 3. 4. Answer Now -9 -5 5 9
![Review 1) If the problem is addition, follow your addition rule. 2) If the Review 1) If the problem is addition, follow your addition rule. 2) If the](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-40.jpg)
Review 1) If the problem is addition, follow your addition rule. 2) If the problem is subtraction, change subtraction to adding the opposite (keep-change) and then follow the addition rule.
![State the rule for multiplying and dividing integers…. If the signs are the same, State the rule for multiplying and dividing integers…. If the signs are the same,](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-41.jpg)
State the rule for multiplying and dividing integers…. If the signs are the same, the answer will be positive. If the signs are different, the answer will be negative.
![Independent Practice Copy down the problems in your notes. The problems you don’t finish Independent Practice Copy down the problems in your notes. The problems you don’t finish](http://slidetodoc.com/presentation_image_h/0e5441c36790a71bb13bf33403f54fe5/image-42.jpg)
Independent Practice Copy down the problems in your notes. The problems you don’t finish will be homework. We will review a proper homework assignment format tomorrow. 1. -8 - -7 = 2. (-3)(-4) = 3. (2)(-2) + 1 = 4. (28 – 8) ÷ (-9 - -5) = 5. If x = -10, then -2 – x = 6. Compare: ⅗ and ⅝ 7. Identify on a number line: ½, ⅜, √ 12, -0. 8 8. Prove 0. 6666666…. . is a rational number.
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