303B Section 09.1

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)