2.1 integers & rational numbers

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2.1 integers & rational numbers

  1. 1. Chapter 2Properties of Real Numbers<br />
  2. 2. In this chapter, you will learn to work with the REALS – a set of numbers that include both positive and negative numbers, decimals, fractions, and more. <br />Learn to identify SETS of Numbers<br />We’ll look at all four operations and learn the number properties for each.<br />Find square roots of given numbers<br />
  3. 3. Using Integers andRational Numbers<br />Section 2.1<br />P. 64 - 70<br />
  4. 4. Natural or Counting Numbers<br /> { 1, 2, 3, 4, 5, . . .}<br />Whole numbers {0, 1, 2, 3, 4, 5, . . .}<br />Integers { . . . -3, -2, -1, 0, 1, 2, 3, . . .}<br />Rationals: a number a/b, where a & b are integers and b is not zero. Includes all terminating and repeating decimals.<br />
  5. 5. learn classify rational numbers into different sets; alsoTSW be able to compare rational numbers (including absolute value<br /> Natural<br />
  6. 6. Two points that are the same distance from the origin but on opposite sides (of the origin) are opposites.<br />Name some opposites on this #-line<br />-4 -3 -2 -1 0 1 2 3 4<br />
  7. 7. The expression “ -3” can be stated as “negative three” or “the opposite of three”<br />How should you read “-a” ? Why?<br />Does zero have an opposite?<br /> - (-4) = _____ - [ -(-5)] = _____<br />
  8. 8. Tell whether each of the following numbers is a whole<br />number, an integer, or a rational number:5, 0.6,<br />–2 and – 24.<br />Rational number?<br />Integer?<br />Whole number?<br />Number<br />Rational number?<br />Integer?<br />Whole number?<br />Number<br />2<br />2<br />2<br />Yes<br />Yes<br />Yes<br />5<br />Yes<br />Yes<br />Yes<br />5<br />3<br />3<br />3<br />Yes<br />No<br />No<br />0.6<br />Yes<br />No<br />No<br />0.6<br />Yes<br />No<br />No<br />Yes<br />No<br />No<br />–2<br />–2<br />Yes<br />Yes<br />No<br />–24<br />Yes<br />Yes<br />No<br />–24<br />EXAMPLE 2<br />Classify numbers<br />
  9. 9. – 2.1, – ,0.5 ,– 2.1.(Order the numbers from least to greatest).<br />5. 4.5, – , – 2.1, 0.5 <br />Rational number?<br />Integer?<br />Whole number?<br />Number<br />Rational number?<br />Integer?<br />Whole number?<br />Number<br />3<br />3<br />3<br />3<br />4<br />4<br />4<br />4<br />Yes<br />No<br />No<br />4.5<br />Yes<br />No<br />No<br />4.5<br />Yes<br />No<br />No<br />Yes<br />No<br />No<br />–<br />–<br />Yes<br />No<br />No<br /> –2 .1<br />Yes<br />No<br />No<br /> –2 .1<br />Yes<br />No<br />No<br />0.5<br />Yes<br />No<br />No<br />0.5<br />for Examples 2 and 3<br />GUIDED PRACTICE<br />ANSWER<br />
  10. 10. for Examples 2 and 3<br />GUIDED PRACTICE<br />Tell whether each numbers in the list is a whole number, an integer, or a rational number.Then order the numbers from least list to greatest.<br />4. 3, –1.2, –2,0<br />
  11. 11. ANSWER<br />–2, –1.2, 0, 3. (Ordered the numbers from least to greatest).<br />for Examples 2 and 3<br />GUIDED PRACTICE<br />
  12. 12. ANSWER<br />On the number line,– 3is to the right of– 4.So, –3 > – 4.<br />EXAMPLE 1<br />Graph and compare integers<br />Graph– 3and– 4on a number line. Then tell which number is greater.<br />learn classify rational numbers into different sets; <br />Also TSW be able to compare rational numbers (including absolute value<br />
  13. 13. 0<br />4<br /> – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6<br />ANSWER<br />On the number line,4is to the right of0.So, 4 > 0.<br />for Example 1<br />GUIDED PRACTICE<br />Graphthe numbers on a number line. Then tell which number is greater.<br />1.4and0<br />
  14. 14. –5<br />2<br /> – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6<br />ANSWER<br />On the number line,2is to the right of–4.So, 2 > –5.<br />for Example 1<br />GUIDED PRACTICE<br />2.2and–5<br />learn classify rational numbers into different sets; <br />alsoTSW be able to compare rational numbers (including absolute value<br />
  15. 15. –1<br />–6<br /> – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6<br />ANSWER<br />On the number line,–1 is to the right of–6.So, –1 > –6.<br />for Example 1<br />GUIDED PRACTICE<br />3.–6and–1<br />learn classify rational numbers into different sets; <br />alsoTSW be able to compare rational numbers (including absolute value<br />
  16. 16. EXAMPLE 3<br />Order rational numbers<br />ASTRONOMY<br />A star’s color index is a measure of the temperature of the star. The greater the color index, the cooler the star. Order the stars in the table from hottest to coolest. <br />SOLUTION<br />Begin by graphing the numbers on a number line.<br />
  17. 17. ANSWER<br />From hottest to coolest, the stars are Shaula, Rigel, <br />Denebola, and Arneb.<br />learn classify rational numbers into different sets; <br />alsoTSW be able to compare rational numbers (including absolute value<br />EXAMPLE 3<br />Read the numbers from left to right:– 0.22, – 0.03, 0.09, 0.21.<br />
  18. 18. Absolute Value<br />The absolute value of a real number is the distance between the origin and the point representing the number. The symbol| a | represents the absolute value of a.<br />The absolute value of a number is never negative.<br />
  19. 19. If a is a positive number, then | a| = a<br />If a is zero, then |a | = 0<br />If a is a negative #, then | a | = -a<br />Examples:<br /> | 6 | = _______<br /> | 0 | = _______<br /> | -5 | = _______<br />learn classify rational numbers into different sets; <br />alsoTSW be able to compare rational numbers (including absolute value<br />
  20. 20. Simplify: - | -8 | = _____<br /> - | 5 | = ______<br /> - ( -5) = ______<br /> - ( 0 ) = _____<br />learn classify rational numbers into different sets; <br /> TSW be able to compare rational numbers (including absolute value<br />
  21. 21. b.Ifa = ,then – a = – .<br />3<br />3<br />4<br />4<br />EXAMPLE 4 <br />Find opposites of numbers<br />a. Ifa=– 2.5, then –a=–(–2.5) =<br />learn classify rational numbers into different sets; <br />alsoTSW be able to compare rational numbers (including absolute value<br />
  22. 22. a.Ifa = – , then|a|= || = – ()=<br />2<br />2<br />2<br />2<br />3<br />3<br />3<br />3<br />EXAMPLE 5<br />Find absolute values of numbers<br />b.Ifa= 3.2,then|a|=|3.2|= 3.2.<br />learn classify rational numbers into different sets; <br />TSW be able to compare rational numbers (including absolute value<br />
  23. 23. for Example 4, 5 and 6<br />GUIDED PRACTICE<br />For the given value of a, find –a and |a|.<br />8. a = 5.3<br />SOLUTION<br />If a = 5.3, then –a = – (5.3) =<br /> |a| = |5.3| = <br />
  24. 24. ( – )<br />4<br />4<br />4<br />4<br />4<br />4<br />4<br />| – |<br />9<br />9<br />9<br />9<br />9<br />9<br />9<br />–<br />–<br />10. a = <br />If a = , then –a = – ( ) = <br />–<br />–<br />|a|<br />=<br />=<br />=<br />for Example 4, 5 and 6<br />GUIDED PRACTICE<br />9. a = – 7<br />SOLUTION<br />If a = – 7, then –a = – (– 7) =<br /> |a| = | – 7| =<br />SOLUTION<br />
  25. 25. A conditional statement has a hypothesis and a conclusion. An if-then statement is a form of a conditional statement.<br />The “if” is the hypothesis, the “then” is the conclusion. <br />A counterexample– proving it is false with just one example.<br />
  26. 26. EXAMPLE 6<br />Analyze a conditional statement<br />Identify the hypothesis and the conclusion of the statement “If a number is a rational number, then the number is an integer.” Tell whether the statement is true or false. If it is false, give a counterexample.<br />SOLUTION<br />Hypothesis: a number is a rational number<br />Conclusion: the number is an integer<br />The statement is false. The number 0.5 is a counterexample, because 0.5 is a 0 rational number but not an integer.<br />
  27. 27. for Example 4, 5 and 6<br />GUIDED PRACTICE<br />Identify the hypothesis and the conclusion of the statement. Tell whether the statement is true or false. If it is the false, give a counterexample.<br />11. If a number is a rational number, then the number is positive <br />SOLUTION<br />Hypothesis: a number is a rational number<br />Conclusion: the number is positive which is false<br />Counterexample: The number –1 is rational, but not positive.<br />
  28. 28. 12.<br />If a absolute value of a number is a positive, then the number is positive <br />for Example 4, 5 and 6<br />GUIDED PRACTICE<br />SOLUTION<br />Hypothesis: the absolute value of a number is positive<br />Conclusion: the number is positive which is false false<br />Counter example: the absolute value of –2 is 2 but –2 negative..<br />
  29. 29. learn classify rational numbers into different sets; alsoTSW be able to compare rational numbers (including absolute value<br />Be ready to discuss / define these words:<br />Real Numbers *<br />Rational Numbers*<br />Integers<br />Irrational Numbers*<br />Whole Numbers<br />Absolute Value*<br /> * critical vocabulary<br />
  30. 30. Assignment: : <br /> P. 67 (#1 - 3,10,11,13 -number lines,16, 20, 23-25, 42-44)<br />

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