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2.1 integers & rational numbers

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  • 1. Chapter 2Properties of Real Numbers
  • 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.
    Learn to identify SETS of Numbers
    We’ll look at all four operations and learn the number properties for each.
    Find square roots of given numbers
  • 3. Using Integers andRational Numbers
    Section 2.1
    P. 64 - 70
  • 4. Natural or Counting Numbers
    { 1, 2, 3, 4, 5, . . .}
    Whole numbers {0, 1, 2, 3, 4, 5, . . .}
    Integers { . . . -3, -2, -1, 0, 1, 2, 3, . . .}
    Rationals: a number a/b, where a & b are integers and b is not zero. Includes all terminating and repeating decimals.
  • 5. learn classify rational numbers into different sets; alsoTSW be able to compare rational numbers (including absolute value
    Natural
  • 6. Two points that are the same distance from the origin but on opposite sides (of the origin) are opposites.
    Name some opposites on this #-line
    -4 -3 -2 -1 0 1 2 3 4
  • 7. The expression “ -3” can be stated as “negative three” or “the opposite of three”
    How should you read “-a” ? Why?
    Does zero have an opposite?
    - (-4) = _____ - [ -(-5)] = _____
  • 8. Tell whether each of the following numbers is a whole
    number, an integer, or a rational number:5, 0.6,
    –2 and – 24.
    Rational number?
    Integer?
    Whole number?
    Number
    Rational number?
    Integer?
    Whole number?
    Number
    2
    2
    2
    Yes
    Yes
    Yes
    5
    Yes
    Yes
    Yes
    5
    3
    3
    3
    Yes
    No
    No
    0.6
    Yes
    No
    No
    0.6
    Yes
    No
    No
    Yes
    No
    No
    –2
    –2
    Yes
    Yes
    No
    –24
    Yes
    Yes
    No
    –24
    EXAMPLE 2
    Classify numbers
  • 9. – 2.1, – ,0.5 ,– 2.1.(Order the numbers from least to greatest).
    5. 4.5, – , – 2.1, 0.5
    Rational number?
    Integer?
    Whole number?
    Number
    Rational number?
    Integer?
    Whole number?
    Number
    3
    3
    3
    3
    4
    4
    4
    4
    Yes
    No
    No
    4.5
    Yes
    No
    No
    4.5
    Yes
    No
    No
    Yes
    No
    No


    Yes
    No
    No
    –2 .1
    Yes
    No
    No
    –2 .1
    Yes
    No
    No
    0.5
    Yes
    No
    No
    0.5
    for Examples 2 and 3
    GUIDED PRACTICE
    ANSWER
  • 10. for Examples 2 and 3
    GUIDED PRACTICE
    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.
    4. 3, –1.2, –2,0
  • 11. ANSWER
    –2, –1.2, 0, 3. (Ordered the numbers from least to greatest).
    for Examples 2 and 3
    GUIDED PRACTICE
  • 12. ANSWER
    On the number line,– 3is to the right of– 4.So, –3 > – 4.
    EXAMPLE 1
    Graph and compare integers
    Graph– 3and– 4on a number line. Then tell which number is greater.
    learn classify rational numbers into different sets;
    Also TSW be able to compare rational numbers (including absolute value
  • 13. 0
    4
    – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6
    ANSWER
    On the number line,4is to the right of0.So, 4 > 0.
    for Example 1
    GUIDED PRACTICE
    Graphthe numbers on a number line. Then tell which number is greater.
    1.4and0
  • 14. –5
    2
    – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6
    ANSWER
    On the number line,2is to the right of–4.So, 2 > –5.
    for Example 1
    GUIDED PRACTICE
    2.2and–5
    learn classify rational numbers into different sets;
    alsoTSW be able to compare rational numbers (including absolute value
  • 15. –1
    –6
    – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6
    ANSWER
    On the number line,–1 is to the right of–6.So, –1 > –6.
    for Example 1
    GUIDED PRACTICE
    3.–6and–1
    learn classify rational numbers into different sets;
    alsoTSW be able to compare rational numbers (including absolute value
  • 16. EXAMPLE 3
    Order rational numbers
    ASTRONOMY
    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.
    SOLUTION
    Begin by graphing the numbers on a number line.
  • 17. ANSWER
    From hottest to coolest, the stars are Shaula, Rigel,
    Denebola, and Arneb.
    learn classify rational numbers into different sets;
    alsoTSW be able to compare rational numbers (including absolute value
    EXAMPLE 3
    Read the numbers from left to right:– 0.22, – 0.03, 0.09, 0.21.
  • 18. Absolute Value
    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.
    The absolute value of a number is never negative.
  • 19. If a is a positive number, then | a| = a
    If a is zero, then |a | = 0
    If a is a negative #, then | a | = -a
    Examples:
    | 6 | = _______
    | 0 | = _______
    | -5 | = _______
    learn classify rational numbers into different sets;
    alsoTSW be able to compare rational numbers (including absolute value
  • 20. Simplify: - | -8 | = _____
    - | 5 | = ______
    - ( -5) = ______
    - ( 0 ) = _____
    learn classify rational numbers into different sets;
    TSW be able to compare rational numbers (including absolute value
  • 21. b.Ifa = ,then – a = – .
    3
    3
    4
    4
    EXAMPLE 4
    Find opposites of numbers
    a. Ifa=– 2.5, then –a=–(–2.5) =
    learn classify rational numbers into different sets;
    alsoTSW be able to compare rational numbers (including absolute value
  • 22. a.Ifa = – , then|a|= || = – ()=
    2
    2
    2
    2
    3
    3
    3
    3
    EXAMPLE 5
    Find absolute values of numbers
    b.Ifa= 3.2,then|a|=|3.2|= 3.2.
    learn classify rational numbers into different sets;
    TSW be able to compare rational numbers (including absolute value
  • 23. for Example 4, 5 and 6
    GUIDED PRACTICE
    For the given value of a, find –a and |a|.
    8. a = 5.3
    SOLUTION
    If a = 5.3, then –a = – (5.3) =
    |a| = |5.3| =
  • 24. ( – )
    4
    4
    4
    4
    4
    4
    4
    | – |
    9
    9
    9
    9
    9
    9
    9


    10. a =
    If a = , then –a = – ( ) =


    |a|
    =
    =
    =
    for Example 4, 5 and 6
    GUIDED PRACTICE
    9. a = – 7
    SOLUTION
    If a = – 7, then –a = – (– 7) =
    |a| = | – 7| =
    SOLUTION
  • 25. A conditional statement has a hypothesis and a conclusion. An if-then statement is a form of a conditional statement.
    The “if” is the hypothesis, the “then” is the conclusion.
    A counterexample– proving it is false with just one example.
  • 26. EXAMPLE 6
    Analyze a conditional statement
    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.
    SOLUTION
    Hypothesis: a number is a rational number
    Conclusion: the number is an integer
    The statement is false. The number 0.5 is a counterexample, because 0.5 is a 0 rational number but not an integer.
  • 27. for Example 4, 5 and 6
    GUIDED PRACTICE
    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.
    11. If a number is a rational number, then the number is positive
    SOLUTION
    Hypothesis: a number is a rational number
    Conclusion: the number is positive which is false
    Counterexample: The number –1 is rational, but not positive.
  • 28. 12.
    If a absolute value of a number is a positive, then the number is positive
    for Example 4, 5 and 6
    GUIDED PRACTICE
    SOLUTION
    Hypothesis: the absolute value of a number is positive
    Conclusion: the number is positive which is false false
    Counter example: the absolute value of –2 is 2 but –2 negative..
  • 29. learn classify rational numbers into different sets; alsoTSW be able to compare rational numbers (including absolute value
    Be ready to discuss / define these words:
    Real Numbers *
    Rational Numbers*
    Integers
    Irrational Numbers*
    Whole Numbers
    Absolute Value*
    * critical vocabulary
  • 30. Assignment: :
    P. 67 (#1 - 3,10,11,13 -number lines,16, 20, 23-25, 42-44)