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## REAL NUMBERS

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**REAL NUMBERS**(as opposed to fake numbers?)**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… • Visualizing the number line**Key Concepts**• Real Number • Rational Number • Integer • Whole Number • Natural Number • Irrational Number**Real Numbers**Real Numbers are every number. Therefore, any number that you can find on the number line. Key Concept**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 61 π 49% 549.23789**Two Kinds of Real Numbers**• Rational Numbers • Irrational Numbers**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.**16**1/2 3.56 -8 1.3333… - 3/4 Examples of Rational Numbers**Integers**One of the subsets of rational numbers**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 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 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 written as a fraction of two integers. • Irrational numbers written as decimals are non-terminating and non-repeating.**Caution!**A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Irrational numbers :If a whole number is not a perfect square, then its square root is an irrational number.**Pi**Examples of Irrational Numbers**The Real Number System**Irrational numbers Rational numbers Integers Whole numbers Natural numbers**The Real Number System**• Irrational • numbers • Cannot be • represented • as a fraction • of 2 integers • Rational numbers • Can be represented as a fraction of 2 integers Integers .… -3, -2, -1, 0, 1, 2, 3, …. Whole numbers 0, 1, 2, 3, …. Natural numbers 1, 2, 3, ….**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 zero. So it is not a number at all.**0 3**= 0 Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. A. 21 irrational 0 3 B. rational**Determining the Classification of All Numbers**State if each number is rational, irrational, or not a real number. 4 0 C. not a real number**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 numbers, convert the numbers to the same form. Compare -3 and -3.571 (convert -3 to -3.428571… -3.428571… > -3.571 3 7 3 7**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 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:**Objectives**SWBAT… • compute with integers By… • Using the basic operation rules for integers**1) (-4) + 8 =**Examples: Use the number line if necessary. 4 2) (-1) + (-3) = -4 3) 5 + (-7) = -2**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**Answer Now**-1 + 3 = ? • -4 • -2 • 2 • 4**Answer Now**-6 + (-3) = ? • -9 • -3 • 3 • 9**The additive inverses(or opposites) of two numbers add to**equal zero. -3 Proof: 3 + (-3) = 0 We will use the additive inverses for subtraction problems. Example: The additive inverse of 3 is**What’s the difference between7 - 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-change)**When subtracting, change the subtraction to adding the**opposite (keep-change-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**Answer Now**Which is equivalent to-12 – (-3)? • 12 + 3 • -12 + 3 • -12 - 3 • 12 - 3**Answer Now**7 – (-2) = ? • -9 • -5 • 5 • 9**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-change) and then follow the addition rule.**State the rule for multiplying and dividing integers….**If the signs are the same, If the signs are different, the answer will be negative. the answer will be positive.**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. -8 - -7 = (-3)(-4) = (2)(-2) + 1 = (28 – 8) ÷ (-9 - -5) = If x = -10, then -2 – x = Compare: ⅗ and ⅝ Identify on a number line: ½, ⅜, √12, -0.8 Prove 0.6666666….. is a rational number.