Transcript
An Ordering Activity
Microsoft Office Word 97 - 2003 Document
Fraction Equality: dc
ba if and only if ad = bc.
Figure 9.2
Figure 9.5
9.1 THE RATIONAL NUMBERS Definition: The set of rational numbers is the set
0,integers areand bbaba
Examples of Rational Numbers: 2
3,
9
4
,
34
7, 3, .7
Explanation: 3 31
47 7
, 3 6
3 or1 2
, 7
.710
9.1 THE RATIONAL NUMBERS Definition: The set of rational numbers is the set
0,integers areand bbaba
Examples of Rational Numbers: 2
3,
9
4
,
34
7, 3, .7
Explanation: 3 31
47 7
, 3 6
3 or1 2
, 7
.710
Equality of Rational Numbers: dc
ba if and only if ad = bc.
To Do: Show that 12 4
9 3
Solution: (–12)×(–3) = 36 = 9×4
Equality of Rational Numbers: dc
ba if and only if ad = bc.
To Do: Show that 12 4
9 3
Solution: (–12)×(–3) = 36 = 9×4
Equality of Rational Numbers: dc
ba if and only if ad = bc.
To Do: Show that 12 4
9 3
Solution: (–12)×(–3) = 36 = 9×4
Equality of Rational Numbers: dc
ba if and only if ad = bc.
To Do: Show that 12 4
9 3
Solution: (–12)×(–3) = 36 = 9×4
Example: 3 6
4 8
Definition: a / b is in simplest form if a and b have no common prime factors and b is positive.
Example: The following are not in simplest form:
6 7 3, ,
8 4 18
Why not?
Example: 3 6
4 8
Definition: a / b is in simplest form if a and b have no common prime factors and b is positive.
Example: The following are not in simplest form:
6 7 3, ,
8 4 18
Why not?
Example: 3 6
4 8
Definition: a / b is in simplest form if a and b have no common prime factors and b is positive.
Example: The following are not in simplest form:
6 7 3, ,
8 4 18
Why not?
Example: 3 6
4 8
Definition: a / b is in simplest form if a and b have no common prime factors and b is positive.
Example: The following are not in simplest form:
6 7 3, ,
8 4 18
Why not?
Example: 2 3 2 8 7 3 37
7 8 7 8 56
Notice that ( )a c ab bc b a c a c
b b b b b b b
Example: 2 3 2 8 7 3 37
7 8 7 8 56
Notice that ( )a c ab bc b a c a c
b b b b b b b
Example: 2 3 2 8 7 3 37
7 8 7 8 56
Notice that ( )a c ab bc b a c a c
b b b b b b b
To Do: Add 5 3
12 20
Answer: 5 3 5 20 3 12
12 20 12 20 12 20
100 36 136 17 8 17
240 240 240 30 8 30
To Do: Add 5 3
12 20
Answer: 5 3 5 20 3 12
12 20 12 20 12 20
100 36 136 17 8 17
240 240 240 30 8 30
Notice that 3 3
4 4
since ( 3) ( 4) 4 3 .
Also, notice that
3 3 3 ( 3) 00
4 4 4 4
.
Therefore, 3
4
is the additive inverse of
3
4.
That is, 3 3
4 4
.
So, 3 3 3
4 4 4
Notice that 3 3
4 4
since ( 3) ( 4) 4 3 .
Also, notice that
3 3 3 ( 3) 00
4 4 4 4
.
Therefore, 3
4
is the additive inverse of
3
4.
That is, 3 3
4 4
.
So, 3 3 3
4 4 4
Notice that 3 3
4 4
since ( 3) ( 4) 4 3 .
Also, notice that
3 3 3 ( 3) 00
4 4 4 4
.
Therefore, 3
4
is the additive inverse of
3
4.
That is, 3 3
4 4
.
So, 3 3 3
4 4 4
Rational numbers on the number line:
Rational numbers on the number line:
To Do: Subtract 7 3
9 5
Solution:
7 3 7 3 7 3 35 ( 27) 35 27 8
9 5 9 5 9 5 45 45 45
To Do: Subtract 7 3
9 5
Solution:
7 3 7 3 7 3 35 ( 27) 35 27 8
9 5 9 5 9 5 45 45 45
To Do: Subtract 7 3
9 5
Solution:
7 3 7 3 7 3 35 ( 27) 35 27 8
9 5 9 5 9 5 45 45 45
To Do: Multiply and simplify 3 25
10 27
Answer: 5
18
To Do: Multiply and simplify 3 25
10 27
Answer: 5
18
To Do: Multiply and simplify 3 25
10 27
Answer: 5
18
Why does 3 2 3 7 21
4 7 4 2 8 ?
Well, why does 6 2 = 3?
Because 2 3 = 6.
Let’s check 3 2 21
4 7 8 :
2 21 42 14 3 3
7 8 56 14 4 4
. Yay!
To Do: Divide and simplify 21 3
25 5
Answer: 7
5
Why does 3 2 3 7 21
4 7 4 2 8 ?
Well, why does 6 2 = 3?
Because 2 3 = 6.
Let’s check 3 2 21
4 7 8 :
2 21 42 14 3 3
7 8 56 14 4 4
. Yay!
To Do: Divide and simplify 21 3
25 5
Answer: 7
5
Why does 3 2 3 7 21
4 7 4 2 8 ?
Well, why does 6 2 = 3?
Because 2 3 = 6.
Let’s check 3 2 21
4 7 8 :
2 21 42 14 3 3
7 8 56 14 4 4
. Yay!
To Do: Divide and simplify 21 3
25 5
Answer: 7
5
Why does 3 2 3 7 21
4 7 4 2 8 ?
Well, why does 6 2 = 3?
Because 2 3 = 6.
Let’s check 3 2 21
4 7 8 :
2 21 42 14 3 3
7 8 56 14 4 4
. Yay!
To Do: Divide and simplify 21 3
25 5
Answer: 7
5
Why does 3 2 3 7 21
4 7 4 2 8 ?
Well, why does 6 2 = 3?
Because 2 3 = 6.
Let’s check 3 2 21
4 7 8 :
2 21 42 14 3 3
7 8 56 14 4 4
. Yay!
To Do: Divide and simplify 21 3
25 5
Answer: 7
5
Why does 3 2 3 7 21
4 7 4 2 8 ?
Well, why does 6 2 = 3?
Because 2 3 = 6.
Let’s check 3 2 21
4 7 8 :
2 21 42 14 3 3
7 8 56 14 4 4
. Yay!
To Do: Divide and simplify 21 3
25 5
Answer: 7
5
Why does 3 2 3 7 21
4 7 4 2 8 ?
Well, why does 6 2 = 3?
Because 2 3 = 6.
Let’s check 3 2 21
4 7 8 :
2 21 42 14 3 3
7 8 56 14 4 4
. Yay!
To Do: Divide and simplify 21 3
25 5
Answer: 7
5
Example: 14 2
15 5
14 5 70 7(1)
15 2 30 3
14 2 14 6 14 7(2)
15 5 15 15 6 3
Example: 14 2
15 5
14 5 70 7(1)
15 2 30 3
14 2 14 6 14 7(2)
15 5 15 15 6 3
14 2 14 2 7(3)
15 5 15 5 3
To Do: Divide 10 5
9 4
Solution: 10 5 10 4 ( 10)(4) (2)(4) 8
9 4 9 5 (9)( 5) 9 9
To Do: Divide 10 5
9 4
Solution: 10 5 10 4 ( 10)(4) (2)(4) 8
9 4 9 5 (9)( 5) 9 9
Error in book:
This should read a c
b b if and only if a < c and b > 0.
Example: Compare 4
9
and
3
7
.
Solution: (–4)(7) = -28 and (9)(–3) = -27.
So, (–4)(7) < (9)(–3) 4 3
9 7
To Do: Compare 10
9
and
9
8
.
Solution: (–10)(8) ??? (9)(–9)
–90 ?? –91
–90 > –91
So, 10 9
9 8
Example: Compare 4
9
and
3
7
.
Solution: (–4)(7) = -28 and (9)(–3) = -27.
So, (–4)(7) < (9)(–3) 4 3
9 7
To Do: Compare 10
9
and
9
8
.
Solution: (–10)(8) ??? (9)(–9)
–90 ?? –91
–90 > –91
So, 10 9
9 8
Example: Compare 4
9
and
3
7
.
Solution: (–4)(7) = -28 and (9)(–3) = -27.
So, (–4)(7) < (9)(–3) 4 3
9 7
To Do: Compare 10
9
and
9
8
.
Solution: (–10)(8) ??? (9)(–9)
–90 ?? –91
–90 > –91
So, 10 9
9 8
Example: Compare 4
9
and
3
7
.
Solution: (–4)(7) = -28 and (9)(–3) = -27.
So, (–4)(7) < (9)(–3) 4 3
9 7
To Do: Compare 10
9
and
9
8
.
Solution: (–10)(8) ??? (9)(–9)
–90 ?? –91
–90 > –91
So, 10 9
9 8
Example: Compare 4
9
and
3
7
.
Solution: (–4)(7) = -28 and (9)(–3) = -27.
So, (–4)(7) < (9)(–3) 4 3
9 7
To Do: Compare 10
9
and
9
8
.
Solution: (–10)(8) ??? (9)(–9)
–90 ?? –91
–90 > –91
So, 10 9
9 8
Assignment
9.1 A: 2-14 (& e-mail me something interesting about yourself plus a picture)
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