REAL NUMBERS

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

2. Real Numbers Real Numbers are every number. Therefore, any number that you can find on the number line. Real Numbers have two categories.

3. 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.

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

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

6. 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.

7. 16 1/2 3.56 -8 1.3333… - 3/4 Examples of Rational Numbers

8. Integers One of the subsets of rational numbers

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

10. Integers • Integers are rational numbers because they can be written as fraction with 1 as the denominator.

11. 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

12. Real Rational (fraction) Integer Irrational Number Systems Whole Natural Natural – counting numbers beginning with 1, 2, 3, … Whole – counting numbers beginning with 0, 1, 2, 3, … Integer– All the whole numbers and their opposites {…, –3, –2, –1, 0, 1, 2, 3, …} Rational – all numbers that can be written as a ratio (fraction) Irrational– a decimal that never ends and never repeats Real– all numbers that can be put on a number line

13. 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.

14. Caution! A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Irrational numberscan be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number.

15. Pi Examples of Irrational Numbers

16. Try this! a) Irrational b) Irrational c) Rational d) Rational e) Irrational

17. 16 2 4 2 = = 2 Additional Example 1: Classifying Real Numbers Write all classifications that apply to each number. A. 5 is a whole number that is not a perfect square. 5 irrational, real B. –12.75 –12.75 is a terminating decimal. rational, real 16 2 C. whole, integer, rational, real

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

19. 0 3 = 0 Additional Example 2: 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

20. Additional Example 2: 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

21. Exit Ticket • Draw the Real Number System. • Explain the difference between a natural number and whole number.

22. Objective • TSW compare rational and irrational numbers • TSW order rational and irrational numbers on a number line

23. 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

24. Practice

25. 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.

26. 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

27. Practice Order these from least to greatest:

28. Objectives • TSW identify the rules associated computing with integers. • TSW compute with integers

29. 1) (-4) + 8 = Examples: Use the number line if necessary. 4 2) (-1) + (-3) = -4 3) 5 + (-7) = -2

30. 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

31. Karaoke Time! Addition Rule: Sung to the tune of “Row, row, row, your boat” Same signs add and keep,different signs subtract,keep the sign of the higher number,then it will be exact! Can your class do different rounds?

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

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

34. 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

35. 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)

36. 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

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

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

39. 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.

40. 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.

41. Different Signs Negative Answer What’s The Rule? 1. -8 * 3 4. 6 ÷ (-3) -24 -2 Start inside ( ) first 2. -2 * -61 5. - (20/-5) 122 - (-4) Same Signs Positive Answer 4 3. (-3)(6)(1) (-18)(1) 6. -18 Just take Two at a time 68

42. 7. At midnight the temperature is 8°C. If the temperature rises 4°C per hour, what is the temperature at 6 am? How much does the temperature rise each hour? How long Is it from Midnight to 6 am? +4 degrees 6 hours (6 hours)(4 degrees per hour) Add this to the original temp. = 24 degrees 8° + 24° = 32°C

43. 8. A deep-sea diver must move up or down in the water in short steps in order to avoid getting a physical condition called the bends. Suppose a diver moves up to the surface in five steps of 11 feet. Represent her total movements as a product of integers, and find the product. Multiply What does This mean? (11 feet) (5 steps) (55 feet) 5 * 11 = 55

44. Exit Ticket • What is operation is called the additive inverse? • What are the rules for subtracting numbers?