Section 1'2 Real Numbers

1
Section 1.2Real Numbers

• Number line
• Absolute Values and Opposites
• Reciprocals
• Geometry Facts

2
The Real Number Line
Positive direction
Negative direction
0
1
3
-1
-2
2
-3
Origin
Positive Numbers
Negative Numbers
3
Definition Coordinate

• Point A is located at 2.
• Point B is located at ½.
• Point C is located at -1.
• Definition The number associated with a point on
  the real number line is called the coordinate of
  that point.

4
Definition Real Numbers

• Definition The numbers that can be represented
  with points on the real number line are called
  real numbers.

5
Real Numbers Riddle
30
40
50
60
20
10
0
70
80
90
100

• Are there more real numbers between 0 and 100 or
  0 and 10?

10
0
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Definition Fraction

• Definition If a and b are real numbers then the
  expression is called a fraction.
• a is the numerator.
• b is the denominator.
• The restriction b ? 0 prevents us from writing an
  undefined expression.

7
Are Fractions Real Numbers?

• Definition The numbers that can be represented
  with points on the real number line are called
  real numbers.

8
Using the Number Line to Visualize Fractions
1
0
0/3
2/3
3/3
1/3

• The denominator indicates the number of equal
  parts in the interval from 0 to 1.
• The numerator indicates how many of those parts
  we have.

9
Using the Number Line to Visualize Fractions
1
2/6
4/6
5/6
0
0/3
2/3
3/3
1/3

• Which is bigger? 2/3 or 5/6
• Here we can see that 1/3 2/6
• And 2/3 4/6
• So 5/6 gt 4/6 2/3

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Definition Equivalent Fractions

• Fractions that represent the same number are said
  to be equivalent.
• Equivalent fractions may look different, but they
  must have the same value.

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Property 1 page 16

• Multiplying the numerator and the denominator of
  a fraction by the same nonzero number never
  changes the value on the fraction.

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Property 2 page 16

• Dividing the numerator and the denominator of a
  fraction by the same nonzero number never changes
  the value on the fraction.

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Definition of Absolute Value

• The absolute value of a real number is its
  distance from zero on the number line.
• 3 3
• -2 2
• 0 0

0
1
3
-1
-2
2
-3
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Absolute Value Examples

• 5 7 3 8
• -2 -5
• 2 5
• -3

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Definition of Opposites

• Numbers that are the same distance from zero but
  in opposite directions are called opposites.
• What is the opposite of -7?
• What is the opposite of 4.5?

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Fun Facts About Opposites

• Each negative number is the opposite of some
  positive number.
• Each positive number is the opposite of some
  negative number.
• -(-a) a
• When you add any two opposites the result is
  always zero.
• a (-a) 0

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Fraction Multiplication

• The product of the numerators is 8.
• The product of the denominators is 15.

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Fraction Multiplication

• Write the understood denominator of 1.
• The product of the numerators is 16.
• The product of the denominators is 3.
• Improper fraction.
• Mixed number.

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Fraction Multiplication

• The product of the numerators is 40.
• The product of the denominators is 40.
• Reduce the fraction.
• The factors were reciprocals.

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Definition Reciprocals

• Two numbers whose product is 1 are called
  reciprocals.

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Find the next number in the sequence (example 21).
Reciprocals of Fibonacci
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Facts from GeometryArea of a Rectangle
What is the Area?
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Area of a Rectangle
3
5
3 ? 5 15
l ? w A
24
Perimeter of a Rectangle
5
2l 2w P
3
3
5
3 5 3 5 16
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Perimeter Application

• A Farmer has 800 meters of fencing material to
  enclose a rectangular field. The width of the
  field is 175 meters. Find the length of the
  field.

l
175
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Facts from Geometry

• Square
• Perimeter
• P 4s
• Area
• A s2

s
s
s
s
Let s 5
A s2 (5)2
(5)(5)
25
P 4s 4(5)
20
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Area of a Triangle
But we wanted the area of a triangle.
Area of the Rectangle
4 ? 4 16
4
½(4 ? 4) 8
4
Area of the Triangle
½(b ? h) A
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Perimeter of a Triangle
4
6
5

• To find the perimeter of a triangle add up all
  three sides.
• 4 5 6 15

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Getting Ready for Class

• What is a real number?
• Explain Multiplication with fractions.
• How do you find the opposite of a number?
• Explain how you find the perimeter and area of a
  rectangle.

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Homework

• 1.2
• p.19
• 1-91 odd

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1 8 Draw a number line. Label the points with
the following coordinates.

• 1. 5
• 2. -2
• 3. -4
• 4. -3

• 5. 1.5
• 6. -1.5
• 7. 9/4
• 8. 8/3

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9. Write each of the following fractions as an
equivalent fraction with a denominator of 24.
6
18
6
33
16. Write each of the following fractions as an
equivalent fraction with a denominator of 60.
5
25
5
34
20 For the number 8 find the

• Opposite
• Reciprocal
• Absolute value

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30 For the number a find the

• Opposite
• Reciprocal
• Absolute value