Any questions on today’s homework (Sections 1.61.7

							   Any questions on
  today’s homework?
   (Sections 1.6/1.7)

 Reminder: You should be doing this
homework without using a calculator,
because calculators can’t be used for
Quizzes 1 or 2, Test 1 or the Gateway
                 Quiz.
   Next Class Session:
1. HW 1.2/1.8 is due at the start of the
   next class session.
2. Gateway HW #3 also due next class
   session (paper worksheet required!)
3. Review lecture for Gateway Quiz 1 (this
  online quiz will be given at the next class
  session after this review lecture)
4. Take a Preview Gateway Quiz during
  last half of class session (worth 5
  points, for extra practice).
            NOW
       CLOSE
YOUR LAPTOPS
(You may reopen them when I finish the
lecture, at which time you can start this
        homework assignment.)
                   Section 1.2
Sets of numbers:
  • Natural numbers: {1, 2, 3, 4, 5, 6 . . .}
  (These are also called the “Counting Numbers” –
    think about how you count out loud: 1, 2, 3,…
  • Whole numbers: {0, 1, 2, 3, 4 . . .}
  (Just add the number zero to the natural numbers)
  • Integers: {. . . –3, -2, -1, 0, 1, 2, 3 . . .}
  (All positive and negative counting numbers and
    zero)
More sets of numbers:
  • Rational numbers – the set of all numbers that can be
    expressed as a ratio (or quotient) of integers, with
    denominator  0. (In other words, any number that can
    be written as a fraction or a ratio.)
  Comments on the set of rational numbers:
  • All integers are also rational numbers, because to make them
    into a fraction, you just write 1 on the bottom (the bottom
    number is called the denominator, top number is called the
    numerator.)
  • If you divide the numerator by the denominator on your
    calculator, you will get a decimal number that either ENDS or
    REPEATS.
  Examples:
  1 = 0.25        8 = 1.6      -2 = -0.666         2 = 0.181818…
  4              5             3                  11
More sets of numbers:
  • Irrational numbers – the set of all numbers that
    can NOT be expressed as a ratio of integers
    (“irrational” literally mean “NOT rational”
                    _
  Examples: Π , √2

  These numbers CAN’T be expressed as a fraction,
   and their decimal form never ends and never
   repeats. (Try entering these numbers on a scientific
   calculator and you’ll see this behavior.)
And one last set of numbers (the big one…):
  • Real numbers – the set of all rational and
    irrational numbers combined
  Comments about the real numbers:
   • When you draw a number line, every point on the line is
     associated with a real number – the number tells how far
     the point is from zero, the middle of the number line.
   • A negative number means the point is to the LEFT of zero
     on a number line; positive numbers are to the right of zero.

               –5 –4 –3 –2 –1    0   1   2   3   4   5


                   Negative           Positive
                   numbers            numbers
   • The actual DISTANCE of a point from zero is called the
     ABSOLUTE VALUE of the number, and it’s always
     positive (except of course the number zero, whose absolute
     value is just zero.)
Check on page 11 of your textbook (either the online version or the paper copy) to
  see this diagram that shows the relationship between all of the sets of numbers:
(It would be worth copying this into your class notebook and studying it as you prepare for Quiz 1.)
Accessing the online textbook, power point lecture slides, and the
two homework assignments due at the next class session:
Answer: a
Answer: e
•   A number line used to represent ordered real numbers
    has negative numbers to the left of 0 and positive
    numbers to the right of 0.
•   Order Property for Real Numbers indicates how to
    use inequality signs (< , which means “less than”, and
    >, meaning “greater than”).
    If a and b are real numbers,
        • a < b means a is to the left of b on a number line.
        • a > b means a is to the right of b on a number line.
Examples:

•    Fill in the blank with either < or > :
     3 ___ 10
    -2 ___ 5
    -2 ___ -5 (be careful on this one!)
    -5 ___ -2

When in doubt, draw the two numbers on a number line.
   If the first number is farther LEFT, put in the < sign.
If the first number is to the RIGHT of the second one,
   put in the > sign.
Answer: True
The absolute value of a number is the distance of that
  number away from 0.
   a 0, since distances are non-negative.
   Note:  means “greater than or equal to”, so “ 0” really just
     means “not negative”, i.e. “either positive or zero”
Sample problems: find the absolute value:
|3| = 3   |-3| = 3 |0| = 0      |- ½ | = ½      -|-3| = -3

•     Fill in the blank with either < or > :
     |3| ___ |10|
    |-2| ___ | 5 |
    |-2| ___ |-5|
    |-5| ___ |-2| (be careful on this one!)
    -|-5| ___ -|-2| (and this one!)

When in doubt, draw the two numbers on a number line.
   If the first number is farther LEFT, put in the < sign.
If the first number is to the RIGHT of the second one,
   put in the > sign.
Answer: <
Answer: 16 > 6
                     Section 1.8
Properties of real numbers
    • Commutative property
       • of addition: a + b = b + a
       • of multiplication: a · b = b · a
Examples:
Complete each statement using the commutative property:
     x + 16 = __________
                                     Answer: 16 + x
     xy = _______
                                     Answer: yx
More properties of real numbers:
  • Associative property
      • of addition: (a + b) + c = a + (b + c)
      • of multiplication: (a · b) · c = a · (b · c)

 Examples:
  Complete each statement using the associative property:
       (x + 16) + 2y = __________
                                          Answer: x + (16 + 2y)
       4·(xy) = _______
                                          Answer: (4x)·y
More examples:
Complete each statement using the associative property:
     (x + 16) + 2y = __________
                                       Answer: x + (16 + 2y)
     4·(xy) = _______
                                       Answer: (4x)·y

Use the commutative and associative properties to simplify:
   8 + (9 + b)
                 Solution: 8 + (9 + b) = (8 + 9) + b = 17 + b
   2(42x)
                 Solution: 2(42x) = (2·42)x = 84 x
   13 + (a + 13)
              Solution: 13 + (a + 13) = 13 + (13 + a) =
                                            (13 + 13) + a = 26 + a
Answer: s
    • Distributive property of multiplication over addition
       • a(b + c) = ab + ac
Examples:
Use the distributive property to write each statement without
   parentheses, and then simplify the result where possible:
 8(x + 2)
               Solution: 8(x + 2) = 8·x + 8·2 = 8x + 16
 2(3x – 4y + 7)
              Solution: 2(3x – 4y + 7) = 2·3x + 2·(-4y) + 2·7
                                            = 6x – 8y + 14
 1 (6x - 2)
 4
   Solution: 1 (6x - 2) = 1·6x - 1·(-2) = 1 · 6·x + 1 · (-2) = 3 x - 1
            4             4      4        4 1      4 1         2     2
Answer: -63x - 13
Answer: a
More examples:
Use the distributive property to write each sum as a product.


   4x + 4y
               Solution: 4x + 4y = 4(x + y)


   (-1)·5 + (-1)·x
               Solution: (-1)· 5 + (-1) · x = (-1)(5 + x) or -(5 + x)
Answer: 2(x +y)
          REMINDERS:
HW 1.2/1.8 AND Gateway HW # 3 (with
 worksheet!) are due at the start of the
 next class session.

You should also take Practice Gateway
 Quiz # 1 at least once before the next
 class session.
    You may now
       OPEN
   your LAPTOPS
and begin working on the
 homework assignment.