6-1 The Set of Rational Numbers - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/6-1_the_set_of_rational_numb… · When you hear ‘rational number’, what do you

§ 6-1 The Set of Rational Numbers

What Are Rational Numbers

When you hear ‘rational number’, what do you think of?

Where do we see rational numbers?

DefinitionA number is said to be rational if the number can be expressed in theform a

b where a and b are integers with b 6= 0. A rational number issaid to have numerator a and denominator b.

NotationThe set of rational numbers is denoted by Q.

A number that is not rational is called irrational.












b where a and b are integers with b 6= 0.

A rational number issaid to have numerator a and denominator b.























A number that is not rational is called

irrational.








Notation

The following represent the same rational number:

−ab

−aba−b

Why We Need Rational Numbers

How many of you have cited a Common Core standard involvingfractions and the use of them with arithmetic operations?

The last problem of the week involved the multiplication and divisionof fractions.

Why We Need Rational Numbers

How many of you have cited a Common Core standard involvingfractions and the use of them with arithmetic operations?

The last problem of the week involved the multiplication and divisionof fractions.

Types of Fractions

What do we call a fraction that looks like 94 ?

DefinitionA rational number is said to be improper if the denominator is smallerthan the numerator.

DefinitionA rational number is said to be proper if the denominator is largerthan the numerator.

Types of Fractions




Types of Fractions




Converting Improper Fractions

Whereas it is easier for many calculations to work with improperfractions, it is sometimes desirable to express as fraction as amixed number.

Are there any times you can think of where an improper fractionwould make no sense but a mixed number would make more sense?


Whereas it is easier for many calculations to work with improperfractions, it is sometimes desirable to express as fraction as amixed number.

Are there any times you can think of where an improper fractionwould make no sense but a mixed number would make more sense?


Example

Convert 94 to a mixed number.

To find this, we equivalently find 9÷ 4 using the Division Algorithm.

9 = 4(2) + 1

This means 94 = 2 1

4 .


Example



9 = 4(2) + 1

This means 94 = 2 1

4 .


Example



9 = 4(2) + 1

This means 94 = 2 1

4 .


Example



9 = 4(2) + 1

This means 94 = 2 1

4 .

Converting Mixed Numbers

Example

Convert 4 25 to an improper fraction.

What basic arithmetic operation is being implied by this expression?

425= 4 +

25

= 4 · 55+

25

=205

+25

=225

Is there an easier way?


Example



425= 4 +

25

= 4 · 55+

25

=205

+25

=225



Example



425=

4 +25

= 4 · 55+

25

=205

+25

=225



Example



425= 4 +

25

= 4 · 55+

25

=205

+25

=225



Example



425= 4 +

25

= 4 · 55+

25

=205

+25

=225



Example



425= 4 +

25

= 4 · 55+

25

=205

+25

=225



Example



425= 4 +

25

= 4 · 55+

25

=205

+25

=225



Example



425= 4 +

25

= 4 · 55+

25

=205

+25

=225


Visually Representing Rational Numbers

When using the number line, there is little difference, except formaking good choices with the delineations.

Example

Represent −35 and 2

5 on a number line.

What delineations should we use here?

−1 − 45 −

35 −

25 −

15 0 1

525

35

45 1



Example


5 on a number line.


−1 − 45 −

35 −

25 −

15 0 1

525

35

45 1



Example


5 on a number line.


−1 − 45 −

35 −

25 −

15 0 1

525

35

45 1



Example


5 on a number line.


−1 − 45 −

35 −

25 −

15 0 1

525

35

45 1



Example


5 on a number line.


−1 − 45 −

35 −

25 −

15 0 1

525

35

45 1


It will be of more use to us to represent rational numbers either withcircles or with boxes. This is because it will be useful to have theserepresentations when we get to the numeric operations.

Example

Represent 13 using colored chips.



Example




Example




Example



Example


Example

Represent − 34 using colored chips.


Example


Example



Example


Example



Example


Example



Example


Example



Example


Example



Example

The given represents 13 of the whole. What does the whole look like?

What relationship can we say we illustrated with these two pictures?


Example




Example




We can also use arrays to visualize rational numbers.

Example

The given represents 14 of the whole. Give a visual representation of

the whole.



Example


the whole.



Example


the whole.

Equivalent Fractions

We have seen a couple visuals of equivalent fractions. Can anyonedefine equivalent fractions?

The Fundamental Law of FractionsIf a

b ,cd ∈ Q then a

b is equivalent to cd , denoted a

b = cd , if there exists

n ∈ Q∗ such that cd = na

nb .

In other words, ab and c

d are equivalent when we can reduce bothfractions until the are equal.

We can use prime factorization and our divisibility rules to reducefractions.




b ,cd ∈ Q then a




nb .







b ,cd ∈ Q then a




nb .







b ,cd ∈ Q then a




nb .




Reducing Fractions

Example

Put 3645 in simplest form.

What does it mean for a fraction to be in simplest form?

DefinitionIf a

b ∈ Q, then we say ab is in simplest form if gcf (a, b) = 1.

Reducing Fractions

Example



DefinitionIf a


Reducing Fractions

Example



DefinitionIf a


Reducing Fractions

So let’s find the factorizations.

36 = 22 · 32

45 = 32 · 5

So what do we do with these?

3645

=22 · 32

32 · 5

=22· 6 32

6 32 · 5

=45

What does 32 = 9 represent here?

Reducing Fractions


36 =

22 · 32

45 = 32 · 5


3645

=22 · 32

32 · 5

=22· 6 32

6 32 · 5

=45


Reducing Fractions


36 = 22 · 32

45 = 32 · 5


3645

=22 · 32

32 · 5

=22· 6 32

6 32 · 5

=45


Reducing Fractions


36 = 22 · 32

45 =

32 · 5


3645

=22 · 32

32 · 5

=22· 6 32

6 32 · 5

=45


Reducing Fractions


36 = 22 · 32

45 = 32 · 5


3645

=22 · 32

32 · 5

=22· 6 32

6 32 · 5

=45


Reducing Fractions


36 = 22 · 32

45 = 32 · 5


3645

=22 · 32

32 · 5

=22· 6 32

6 32 · 5

=45


Reducing Fractions


36 = 22 · 32

45 = 32 · 5


3645

=

22 · 32

32 · 5

=22· 6 32

6 32 · 5

=45


Reducing Fractions


36 = 22 · 32

45 = 32 · 5


3645

=22 · 32

32 · 5

=22· 6 32

6 32 · 5

=45


Reducing Fractions


36 = 22 · 32

45 = 32 · 5


3645

=22 · 32

32 · 5

=22· 6 32

6 32 · 5

=45


Reducing Fractions


36 = 22 · 32

45 = 32 · 5


3645

=22 · 32

32 · 5

=22· 6 32

6 32 · 5

=45


Reducing Fractions


36 = 22 · 32

45 = 32 · 5


3645

=22 · 32

32 · 5

=22· 6 32

6 32 · 5

=45


Reducing Fractions

ExampleReduce the following expressions:

12x2y4xy2

= 3xy

x3y2z2x5yz3 = y

2x2z2

Reducing Fractions


12x2y4xy2 = 3x

y

x3y2z2x5yz3 = y

2x2z2

Reducing Fractions


12x2y4xy2 = 3x

y

x3y2z2x5yz3

= y2x2z2

Reducing Fractions


12x2y4xy2 = 3x

y

x3y2z2x5yz3 = y

2x2z2


So how can we determine if we have equivalent fractions?

Example

Is 3042 equivalent to 100

140 ?

We can determine this is 3 different ways:

Prime factorization

Common denominator

Cross multiplication


So how can we determine if we have equivalent fractions?

Example

Is 3042 equivalent to 100

140 ?

We can determine this is 3 different ways:

Prime factorization

Common denominator

Cross multiplication

Equivalent Fractions: Prime Factorization

3042

=

2 · 3 · 52 · 3 · 7

=57

100140

=22 · 52

22 · 5 · 7

=57


3042

=2 · 3 · 52 · 3 · 7

=57

100140

=22 · 52

22 · 5 · 7

=57


3042

=2 · 3 · 52 · 3 · 7

=57

100140

=22 · 52

22 · 5 · 7

=57


3042

=2 · 3 · 52 · 3 · 7

=57

100140

=

22 · 52

22 · 5 · 7

=57


3042

=2 · 3 · 52 · 3 · 7

=57

100140

=22 · 52

22 · 5 · 7

=57


3042

=2 · 3 · 52 · 3 · 7

=57

100140

=22 · 52

22 · 5 · 7

=57

Equivalent Fractions: Common Denominator

We need to determine the common denominator between 42 and 140.

Using the prime factorization, we get the common denominator to be...

22 · 3 · 5 · 7 = 420

3042· 2 · 5

2 · 5=

300420

100140· 3

3=

300420

Therefore, these fractions are equivalent.




22 · 3 · 5 · 7 = 420

3042· 2 · 5

2 · 5=

300420

100140· 3

3=

300420





22 · 3 · 5 · 7 = 420

3042

· 2 · 52 · 5

=300420

100140· 3

3=

300420





22 · 3 · 5 · 7 = 420

3042· 2 · 5

2 · 5

=300420

100140· 3

3=

300420





22 · 3 · 5 · 7 = 420

3042· 2 · 5

2 · 5=

300420

100140· 3

3=

300420





22 · 3 · 5 · 7 = 420

3042· 2 · 5

2 · 5=

300420

100140

· 33=

300420





22 · 3 · 5 · 7 = 420

3042· 2 · 5

2 · 5=

300420

100140· 3

3=

300420





22 · 3 · 5 · 7 = 420

3042· 2 · 5

2 · 5=

300420

100140· 3

3=

300420





22 · 3 · 5 · 7 = 420

3042· 2 · 5

2 · 5=

300420

100140· 3

3=

300420


Equivalent Fractions: Cross Multiplication

Cross Multiplication Test

For ab ,

cd ∈ Q, a

b = cd when ad = bc.

So in our example, what do we get?

3042

?=

100140

30(140) ?= 42(100)

4200 = 4200

Therefore, the fractions are equivalent.



For ab ,

cd ∈ Q, a



3042

?=

100140

30(140) ?= 42(100)

4200 = 4200




For ab ,

cd ∈ Q, a



3042

?=

100140

30(140) ?= 42(100)

4200 = 4200




For ab ,

cd ∈ Q, a



3042

?=

100140

30(140) ?= 42(100)

4200 = 4200




For ab ,

cd ∈ Q, a



3042

?=

100140

30(140) ?= 42(100)

4200 = 4200


Ordering Rational Numbers

We can use this same idea to order rational numbers.

Orderings

ad > bc only when ab > c

d

Example

Fill in the blank: 2032

4570

20(70) = 1400

32(45) = 1440

Conclusion?

2032

<4570



Orderings


d

Example


4570

20(70) = 1400

32(45) = 1440

Conclusion?

2032

<4570



Orderings


d

Example


4570

20(70) = 1400

32(45) = 1440

Conclusion?

2032

<4570



Orderings


d

Example


4570

20(70) = 1400

32(45) = 1440

Conclusion?

2032

<4570



Orderings


d

Example


4570

20(70) = 1400

32(45) = 1440

Conclusion?

2032

<4570



Orderings


d

Example


4570

20(70) = 1400

32(45) = 1440

Conclusion?

2032

<4570



Orderings


d

Example


4570

20(70) = 1400

32(45) = 1440

Conclusion?

2032

<4570


We can also use common denominators.

Example


4560

What is the common denominator?

50 = 2 · 52

60 = 22 · 3 · 5lcm(50, 60) = 22 · 3 · 52 = 300



Example


4560


50 = 2 · 52

60 = 22 · 3 · 5lcm(50, 60) = 22 · 3 · 52 = 300



Example


4560


50 = 2 · 52

60 = 22 · 3 · 5lcm(50, 60) = 22 · 3 · 52 = 300



Example


4560


50 = 2 · 52

60 = 22 · 3 · 5lcm(50, 60) = 22 · 3 · 52 = 300



Example


4560


50 = 2 · 52

60 = 22 · 3 · 5

lcm(50, 60) = 22 · 3 · 52 = 300



Example


4560


50 = 2 · 52

60 = 22 · 3 · 5lcm(50, 60) = 22 · 3 · 52 = 300


4250

4560

4250· 6

64560· 5

5252300

225300

Conclusion?

4250

>4560


4250

4560

4250· 6

64560· 5

5

252300

225300

Conclusion?

4250

>4560


4250

4560

4250· 6

64560· 5

5252300

225300

Conclusion?

4250

>4560


4250

4560

4250· 6

64560· 5

5252300

225300

Conclusion?

4250

>4560


4250

4560

4250· 6

64560· 5

5252300

225300

Conclusion?

4250

>4560

What Lies Between?

Example

Find a rational number that lies between 15 and 3

5 .

Example


5 .

There is always a rational number between any two rational numbers.

The Density of Rational NumbersGiven any two rational numbers, there exists a rational numberbetween them.

What Lies Between?

Example


5 .

Example


5 .



What Lies Between?

Example


5 .

Example


5 .



What Lies Between?

Example


5 .

Example


5 .



What Lies Between

Example


4 .

We can always find a common denominator and see what we aredealing with.

34· 5

5=

1520

35· 4

4=

1220

So, the easiest fractions that work are 1320 and 14

20 .

What Lies Between

Example


4 .


34· 5

5=

1520

35· 4

4=

1220


20 .

What Lies Between

Example


4 .


34· 5

5=

1520

35· 4

4=

1220


20 .

What Lies Between

Example


4 .


34· 5

5=

1520

35· 4

4=

1220


20 .

What Lies Between

Example


4 .


34· 5

5=

1520

35· 4

4=

1220


20 .

What Lies Between

Example

Find a rational number between 34 and 2

3 .

What would happen if we used common denominators?

34· 3

3=

912

23· 4

4=

812

So is there none?

What Lies Between

Example


3 .


34· 3

3=

912

23· 4

4=

812

So is there none?

What Lies Between

Example


3 .


34· 3

3=

912

23· 4

4=

812

So is there none?

What Lies Between

Example


3 .


34· 3

3=

912

23· 4

4=

812

So is there none?

What Lies Between

Another approach would be the following:

RuleFor the rational numbers a

b and cd , a rational number that lies between

them is a+cb+d .

So in our example, one rational number between the given rationalnumbers is

3 + 24 + 4

=58

What Lies Between

Another approach would be the following:

RuleFor the rational numbers a

b and cd , a rational number that lies between

them is a+cb+d .

So in our example, one rational number between the given rationalnumbers is

3 + 24 + 4

=58

Documents

6-1 The Set of Rational Numbers - Dr. Travers Page of Mathbtravers.weebly.com/.../7/2/9/6729909/6-1_the_set_of_rational_numb… · When you hear ‘rational number’, what do you