REAL NUMBERS

1. REAL NUMBERS (as opposed to fake numbers?)

2. Why do we have to review numbers Miss Price? Be a beast in Math when you know the basics!

3. Objective SWBAT… • Identify and classify the parts of the Real Number System By… • Visualizing the number line

4. Key Concepts • Real Number • Rational Number • Integer • Whole Number • Natural Number • Irrational Number

5. Real Numbers Real Numbers are every number. Therefore, any number that you can find on the number line. Key Concept

6. 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.

7. Real Numbers REAL NUMBERS 154,769,852,354 1.333 -5,632.1010101256849765… -8 61 π 49% 549.23789

8. Two Kinds of Real Numbers • Rational Numbers • Irrational Numbers

9. 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.

10. 16 1/2 3.56 -8 1.3333… - 3/4 Examples of Rational Numbers

11. Integers One of the subsets of rational numbers

12. Key Concept What are integers? • Integers are the whole numbers and their opposites. • Examples of integers are 6 -12 0 186 -934

13. 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.

14. 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

15. 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.

16. 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.

17. Pi Examples of Irrational Numbers

18. The Real Number System Irrational numbers Rational numbers Integers Whole numbers Natural numbers

19. 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, ….

20. The Human Number Line Determine all of the classifications that fit for the number…

21. A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

22. 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

23. 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

24. Objective SWBAT… • compare rational and irrational numbers By… • Ordering numbers on a number line

25. 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

26. Practice

27. 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.

28. 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

29. Practice Order these from least to greatest:

30. Objectives SWBAT… • compute with integers By… • Using the basic operation rules for integers

31. 1) (-4) + 8 = Examples: Use the number line if necessary. 4 2) (-1) + (-3) = -4 3) 5 + (-7) = -2

32. 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

33. Answer Now -1 + 3 = ? • -4 • -2 • 2 • 4

34. Answer Now -6 + (-3) = ? • -9 • -3 • 3 • 9

35. 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

36. 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)

37. 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

38. Answer Now Which is equivalent to-12 – (-3)? • 12 + 3 • -12 + 3 • -12 - 3 • 12 - 3

39. Answer Now 7 – (-2) = ? • -9 • -5 • 5 • 9

40. 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.

41. 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.

42. 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.