Transcript 2.1 Rational Numbers

Chapter 02 – Section 01
Rational Numbers
To compare and order rational numbers, and to find a
number between two rational numbers.
• rational number – any number that can be written as a fraction
DEFINITION OF A RATIONAL NUMBER
a
,
b
A rational number is a number that can be expressed in the form
where a and b are integers and b is not equal to 0.
Rational numbers are any number that can be written as a fraction.
Examples of rational numbers:
Rational Numbers
Form
a
b
4
4
1
3
4
15

4
3
0.250
0
0.333
1
4
0
1
1
3
© William James Calhoun
Rational numbers can be graphed on a number line the same way as
integers.
Examples of the graphs of rational numbers.
-5
-4
Rational Numbers
4
a
Form
b
4
1
-3
-2
-1
0
3
3
4
15

4
1
2
0.250
0
0.333
1
4
0
1
1
3
3
4
5
6
7
8
© William James Calhoun
We use the inequality signs, < and > to compare rational numbers.
You might remember the shark (or PacMan or alligator) always eats the
larger number.
The set {-5, -2, 11/2, 3.5} is graphed below.
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
You can say the following about these numbers:
A. The graph of -2 is to the left of the graph of 3.5.
-2 < 3.5
B. The graph of 11/2 is to the right of the graph of -5.
11/2 > -5
© William James Calhoun
COMPARING NUMBERS ON THE NUMBER LINE
If a and b represent any numbers and the graph of a is to the left of the
graph of b, then a < b. If the graph of a is to the right of the graph of
b, then a > b.
The larger number is always to the right of the smaller number.
COMPARISON PROPERTY
For any two numbers a and b, exactly one of the following sentences
is true.
a<b
a=b
a>b
A number is either smaller than, greater than, or equal to any other
number.
© William James Calhoun
Here is a chart of symbols you need to know.
Symbol
Meaning
<
is less than
>
is greater than
=
is equal to
≠
is not equal to
≤
is less than or equal to
≥
is greater than or equal to
Notice these statements have verbs – mathematical sentences to follow.
© William James Calhoun
EX1β
EXAMPLE 1α:
Replace each _?_ with <, >, or = to make each sentence true.
a. -75 _?_ 13
Since any negative
number is always
less than any
positive number,
the true sentence
is:
-75 < 13.
b. -14 _?_ -22 + 9
Simplify the righthand side.
-14 _?_ -22 + 9
-14 _?_ -13
Since -14 is less
than -13, the true
sentence is:
-14 < -22 + 9.
c. 3/8 _?_ -7/8
Since 3 is greater
than -7, the true
sentence is:
3/ > -7/ .
8
8
© William James Calhoun
EXAMPLE 1β:
Replace each _?_ with <, >, or = to make each sentence true.
a. -11 _?_ -20 + 9
b. -15 + 6 _?_ 0
c. 5 _?_ 8 + (-12)
© William James Calhoun
Example 1C had fractions in it.
The problem was easy because the denominators were the same.
If you have a similar problem that has different denominators, you can
use cross products to compare the two fractions.
2.4.4 COMPARISON PROPERTY FOR RATIONAL NUMBERS
a
c
For any rational numbers
and , with b > 0 and d > 0:
b
d
a c
1. if  then ad < bc, and
b d
a c
2. if ad < bc, then
 .
b d
© William James Calhoun
EX2β
EXAMPLE 2α:
Replace each _?_ with <, >, or = to make each sentence true.
A.
7
4
_?_
13
15
B.
7
8
_?_
8
9
7(15) _?_ 13(4)
9(7) _?_8(8)
105 > 52
63 < 64
EXAMPLE 2β:
Replace each _?_ with <, >, or = to make each sentence true.
A.
6 _?_ 9
7
10
B. 7 _?_ 21
10
30
© William James Calhoun
Every rational number can be expressed as a terminating or repeating
decimal.
MORE HERE ON RE-WRITE
EXAMPLE 3α: Use a calculator to write the fractions
3
4
1
, , and
8
5
6
as decimals. Then write the fractions in order from least to greatest.
3
= 0.375
8
4
= 0.8
5
1
= 0.166666… or 0.16
6
This is a terminating decimal.
This is also a terminating decimal.
This is a repeating decimal.
Ordering the decimals, you get: 0.16 , 0.375, 0.8. So, the fractions in
1 3 4
order would be ,
, .
6 8 5
© William James Calhoun
3 4
5
,
,
and
EXAMPLE 3β: Use a calculator to write the fractions
11 25
16
as decimals. Then write the fractions in order from least to greatest.
© William James Calhoun
You can always find the midpoint of any distance.
Eventually you will get down to the atomic level and be working with
miniscule lengths.
However, you can always find a point mid-way between any two other
points - no matter how small the gap.
When you are asked to find a point in-between two numbers, always
calculate the midpoint – its easier for me to grade.
2.4.5 DENSITY PROPERTY FOR RATIONAL NUMBERS
Between every pair of distinct rational numbers, there are infinitely
many rational numbers.
© William James Calhoun
2
3
EXAMPLE 4α: Find a rational number between  and  .
5
8
Since we can choose which one, we will just find the midpoint. To do this, we take
the average of the two points - add them and divide by 2.
2  3
16  15 
31
31
        

or -0.3875
2 
5  8
40  40 
40
80
1
3
EXAMPLE 4β: Find a rational number between and .
9
5
© William James Calhoun