# Real numbers system

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### Real numbers system

• 1. The Real Number System W N Q Z IR Presented By: Prince
• 3. Real Number Real Number Real Number
• 4. What does it Mean?
• 5. Real Numbers REAL NUMBERS -8 -5,632.1010101256849765… 61 49% π 549.23789 154,769,852,354 1.333
• 6. The Real Number Line Any real number corresponds to a point on the real number line. Order Property for Real Numbers Given any two real numbers a and b, - if a is to the left of b on the number line, then a < b. - if a is to the right of b on the number line, then a > b.
• 7. Real Number System Tree Diagram Real Numbers Integers Terminating Decimals Repeating Decimals Whole Numbers Rational Numbers Irrational Numbers Negative #’s Natural #’s Zero Non-Terminating And Non-Repeating Decimals
• 8. Two Kinds of Real Numbers
• 10. Examples of Rational Numbers •16 •1/2 •3.56 •-8 •1.3333… •- 3/4
• 11. Integers One Of the subsets Of ratiOnal numbers
• 12. What are integers? • Integers are the whole numbers and their opposites. • Examples of integers are 6 -12 0 186 -934
• 13. What are integers?....... • Integers are rational numbers because they can be written as fraction with 1 as the denominator.
• 15. WHOLE Numbers REAL NUMBERS IRRATIONAL Numbers NATURAL Numbers INTEGERS RATIONAL Numbers
• 17. Irrational numbers can 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. Caution! A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
• 18. Examples of Irrational Numbers • Pi
• 19. Try this! • a) Irrational • b) Irrational • c) Rational • d) Rational a) 2 b) 12 c) 25 d) 11 5 e) 66 • e) Irrational
• 20. Additional Example 1: Classifying Real Numbers Write all classifications that apply to each number. 5 is a whole number that is not a perfect square. 5 irrational, real –12.75 –12.75 is a terminating decimal. rational, real 16 16 = 4 = 2 2 2 2 whole, integer, rational, real A. B. C.
• 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. Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. 21 irrational 0 3 rational 0 3 = 0 A. B.
• 23. Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. 4 C. 0 not a real number
• 25. Comparing Rational and Irrational Numbers • When comparing different forms of rational and irrational numbers, convert the numbers to the same form. 37 Compare -3 and -3.571 (convert -3 to -3.428571… -3.428571… > -3.571 37
• 27. Ordering Rational and Irrational Numbers
• 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 • TSW identify the rules associated computing with integers. • TSW compute with integers
• 31. Examples: Use the number line if necessary. -5 0 5 4 2) (-1) + (-3) = -4 3) 5 + (-7) = -2 1) (-4) + 8 =
• 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
• 34. -1 + 3 = ? 1. -4 2. -2 3. 2 4. 4 Answer Now
• 35. -6 + (-3) = ? 1. -9 2. -3 3. 3 4. 9 Answer Now
• 36. 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.
• 37. 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-change)
• 38. 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
• 39. Which is equivalent to -12 – (-3)? 1. 12 + 3 2. -12 + 3 3. -12 - 3 4. 12 - 3 Answer Now
• 40. 7 – (-2) = ? 1. -9 2. -5 3. 5 4. 9 Answer Now
• 41. 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.
• 42. State the rule for multiplying and dividing integers…. If the signs are the same, If the signs are different, + the answer will be positive. the answer will be negative.
• 43. 1. -8 * 3 What’s Different The Signs Negative Rule? Answer -24 2. -2 * -61 Same Signs Positive Answer 122 3. (-3)(6)(1) (-18)(1) -18 Just take Two at a time 4. 6 ÷ (-3) -2 Start inside ( ) first 5. - (20/-5) - (-4) 4 6. - - 68 408 6
• 44. 7. At midnight the temperature is 8°C. If the temperature rises 4°C per hour, what is the temperature at 6 am? How long How much Is it from does the Midnight temperature to 6 am? rise each hour? 6 hours +4 degrees (6 hours)(4 degrees per hour) = 24 degrees 8° + 24° = 32°C Add this to the original temp.
• 45. 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 (5 steps) (11 feet) mean? This (55 feet) 5 * 11 = 55
• 46. Summary • What did you learn in this lesson? • What are some important facts to remember about the real number system? • Is there something within the lesson that you need help on?
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