Keepin

1
Keepin It Real

• The Real Number System
• Unit 2

2
Objectives

• I can classify real numbers
• I can plot real numbers on the number line and
  order real numbers.
• I can interpret change as positive or negative
  numbers.
• I can find the opposite and the absolute value of
  a real number.

3
With a partner, list examples of as many
different types of numbers as you can think of.
4
Real Numbers

• Integers Positive and Negative Natural Numbers
  (and 0)
• Rational Numbers Fractions of Integers, Decimals
  with repeating patterns.
• Irrational Numbers
• Have infinite decimal expansions with no
  patterns

5
Real Numbers

• Natural numbers are integers.
• Integers are rational numbers
• 6 6/1
• Rational numbers are NOT irrational.

6
Natural Number? Integer? Rational Number?
Irrational Number?

• List all that apply
• 51351390
• -456.14589
• 3½
• ¼
• -14/5
• v13
• v9

7
Create a real number system
8
The Real Number Line
The negative real numbers are the coordinates of
points to the left of the origin 0.
The real number zero is the coordinate of the
origin O.
The positive real numbers are the coordinates of
points to the right of the origin O.
9
Think of I-10

• Exit 194 is 0, the origin.
• The positive direction is to the right.
• The negative direction is to the left.

10
Think of I-10

• Imagine you are trying to go as far to the
  right as possible.
• What is closer to your goal 5 miles from the
  exit or -3 miles from the exit?

11
Think of I-10

• Imagine you are trying to go as far to the
  right as possible.
• What is closer to your goal -7 miles from the
  exit or -3 miles from the exit?

12
Ordering Real Numbers

• The symbols
• a lt b (a is less than b)
• a gt b (a is greater than b)
• a b (a is equal to b)

• The new rules
• If a is negative and b is postive a lt b
• If a and b are positive and a gt b, than a lt -
  b.
• Examples
• -3 lt 5
• -7 lt -3

13
Do you know HOW?

• On your number line, plot
• -7, 9, -3/2, 2.7, 5.9, and ¼
• Which is greater, -143 or 12?
• Which is greater, -41 or -1?
• Which is greater, 0 or 5?
• Which is greater, 0 or -5?

14
What do Positive and Negative Numbers MEAN?

• To which of the following words describing change
  would you associate with positive numbers? Which
  with negative numbers?
• Can you think of any more?

-
-
-
-

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15
Use an integer to describe the following

• Kalamazoo is 780 feet above sea level.
• I lost 5 betting at the track.
• The temperature decreased by 7 degrees.
• I dove 20 feet below sea level.
• I made 143 on that stock!
• The temperature warmed up by 3 degrees.
• Illegal formation 10 yard penalty!

16
Opposites

• To find the opposite of a (nonzero) real number,
  change its sign.
• The opposite is equally far from the origin, but
  in the opposite direction.

17
Opposites

• To find the opposite of a (nonzero) real number,
  change its sign.
• Find the opposite of
• 679
• -34
• -13
• ¼

18
Distance and Absolute Value

• A distance is never negative
• The absolute value of a number is its distance
  from the origin on the number line.

19
Number line

• How far is 3 from zero?
• How far is -3 from zero?

20
x the absolute value of x

• 3 asks how far from zero is 3?
• -5 asks how far from zero is -5?

21
Absolute Value

• Always gives a positive answer or zero.
• If there is arithmetic inside the absolute value
  symbol do that first, then take the absolute
  value of the answer.

22

• Real numbers include natural numbers, whole
  numbers, integers, rational numbers, and
  irrational numbers.
• Real numbers can be laid out along a number line.
• Positive numbers gt Negative Numbers
• Negative numbers are ordered in reverse
• Positive and negative numbers can describe
  change.
• Changing the sign of a real number gives its
  opposite.
• Absolute value is like distance, sign is like a
  direction.