303B Section 09.1
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Transcript of 303B Section 09.1
![Page 1: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/1.jpg)
![Page 2: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/2.jpg)
An Ordering Activity
Microsoft Office Word 97 - 2003 Document
![Page 3: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/3.jpg)
Fraction Equality: dc
ba if and only if ad = bc.
![Page 4: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/4.jpg)
Figure 9.2
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Figure 9.5
![Page 6: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/6.jpg)
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
![Page 7: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/7.jpg)
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
![Page 8: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/8.jpg)
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
![Page 9: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/9.jpg)
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
![Page 10: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/10.jpg)
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
![Page 11: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/11.jpg)
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
![Page 12: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/12.jpg)
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?
![Page 13: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/13.jpg)
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?
![Page 14: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/14.jpg)
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?
![Page 15: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/15.jpg)
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?
![Page 16: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/16.jpg)
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
![Page 17: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/17.jpg)
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
![Page 18: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/18.jpg)
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
![Page 19: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/19.jpg)
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
![Page 20: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/20.jpg)
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
![Page 21: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/21.jpg)
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
![Page 22: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/22.jpg)
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
![Page 23: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/23.jpg)
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
![Page 24: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/24.jpg)
Rational numbers on the number line:
![Page 25: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/25.jpg)
Rational numbers on the number line:
![Page 26: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/26.jpg)
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
![Page 27: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/27.jpg)
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
![Page 28: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/28.jpg)
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
![Page 29: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/29.jpg)
To Do: Multiply and simplify 3 25
10 27
Answer: 5
18
![Page 30: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/30.jpg)
To Do: Multiply and simplify 3 25
10 27
Answer: 5
18
![Page 31: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/31.jpg)
To Do: Multiply and simplify 3 25
10 27
Answer: 5
18
![Page 32: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/32.jpg)
![Page 33: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/33.jpg)
![Page 34: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/34.jpg)
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
![Page 35: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/35.jpg)
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
![Page 36: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/36.jpg)
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
![Page 37: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/37.jpg)
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
![Page 38: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/38.jpg)
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
![Page 39: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/39.jpg)
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
![Page 40: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/40.jpg)
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
![Page 41: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/41.jpg)
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
![Page 42: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/42.jpg)
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
![Page 43: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/43.jpg)
14 2 14 2 7(3)
15 5 15 5 3
![Page 44: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/44.jpg)
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
![Page 45: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/45.jpg)
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
![Page 46: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/46.jpg)
Error in book:
This should read a c
b b if and only if a < c and b > 0.
![Page 47: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/47.jpg)
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
![Page 48: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/48.jpg)
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
![Page 49: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/49.jpg)
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
![Page 50: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/50.jpg)
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
![Page 51: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/51.jpg)
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
![Page 52: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/52.jpg)
Assignment
9.1 A: 2-14 (& e-mail me something interesting about yourself plus a picture)
![Page 53: 303B Section 09.1](https://reader031.fdocuments.us/reader031/viewer/2022020803/548eb04ab4795959398b49cb/html5/thumbnails/53.jpg)