View
3
Download
0
Category
Preview:
Citation preview
§ 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.
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.
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.
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.
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.
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.
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.
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
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
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.
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?
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?
Converting Improper Fractions
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 .
Converting Improper Fractions
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 .
Converting Improper Fractions
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 .
Converting Improper Fractions
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 .
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?
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?
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?
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?
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?
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?
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?
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?
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
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
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
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
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
Visually Representing Rational Numbers
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.
Visually Representing Rational Numbers
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.
Visually Representing Rational Numbers
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.
Visually Representing Rational Numbers
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.
Visually Representing Rational Numbers
Example
Represent 26 using colored chips.
Example
Represent − 34 using colored chips.
Visually Representing Rational Numbers
Example
Represent 26 using colored chips.
Example
Represent − 34 using colored chips.
Visually Representing Rational Numbers
Example
Represent 26 using colored chips.
Example
Represent − 34 using colored chips.
Visually Representing Rational Numbers
Example
Represent 26 using colored chips.
Example
Represent − 34 using colored chips.
Visually Representing Rational Numbers
Example
Represent 26 using colored chips.
Example
Represent − 34 using colored chips.
Visually Representing Rational Numbers
Example
Represent 26 using colored chips.
Example
Represent − 34 using colored chips.
Visually Representing Rational Numbers
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?
Visually Representing Rational Numbers
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?
Visually Representing Rational Numbers
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?
Visually Representing Rational Numbers
We can also use arrays to visualize rational numbers.
Example
The given represents 14 of the whole. Give a visual representation of
the whole.
Visually Representing Rational Numbers
We can also use arrays to visualize rational numbers.
Example
The given represents 14 of the whole. Give a visual representation of
the whole.
Visually Representing Rational Numbers
We can also use arrays to visualize rational numbers.
Example
The given represents 14 of the whole. Give a visual representation of
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.
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.
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.
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
ExampleReduce the following expressions:
12x2y4xy2
= 3xy
x3y2z2x5yz3 = y
2x2z2
Reducing Fractions
ExampleReduce the following expressions:
12x2y4xy2 = 3x
y
x3y2z2x5yz3 = y
2x2z2
Reducing Fractions
ExampleReduce the following expressions:
12x2y4xy2 = 3x
y
x3y2z2x5yz3
= y2x2z2
Reducing Fractions
ExampleReduce the following expressions:
12x2y4xy2 = 3x
y
x3y2z2x5yz3 = y
2x2z2
Equivalent Fractions
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
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
Equivalent Fractions: Prime Factorization
3042
=2 · 3 · 52 · 3 · 7
=57
100140
=22 · 52
22 · 5 · 7
=57
Equivalent Fractions: Prime Factorization
3042
=2 · 3 · 52 · 3 · 7
=57
100140
=22 · 52
22 · 5 · 7
=57
Equivalent Fractions: Prime Factorization
3042
=2 · 3 · 52 · 3 · 7
=57
100140
=
22 · 52
22 · 5 · 7
=57
Equivalent Fractions: Prime Factorization
3042
=2 · 3 · 52 · 3 · 7
=57
100140
=22 · 52
22 · 5 · 7
=57
Equivalent Fractions: Prime Factorization
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.
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.
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 · 52 · 5
=300420
100140· 3
3=
300420
Therefore, these fractions are equivalent.
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.
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.
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
· 33=
300420
Therefore, these fractions are equivalent.
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.
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.
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.
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.
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.
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.
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.
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.
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
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
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
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
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
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
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
Ordering Rational Numbers
We can also use common denominators.
Example
Fill in the blank: 4250
4560
What is the common denominator?
50 = 2 · 52
60 = 22 · 3 · 5lcm(50, 60) = 22 · 3 · 52 = 300
Ordering Rational Numbers
We can also use common denominators.
Example
Fill in the blank: 4250
4560
What is the common denominator?
50 = 2 · 52
60 = 22 · 3 · 5lcm(50, 60) = 22 · 3 · 52 = 300
Ordering Rational Numbers
We can also use common denominators.
Example
Fill in the blank: 4250
4560
What is the common denominator?
50 = 2 · 52
60 = 22 · 3 · 5lcm(50, 60) = 22 · 3 · 52 = 300
Ordering Rational Numbers
We can also use common denominators.
Example
Fill in the blank: 4250
4560
What is the common denominator?
50 = 2 · 52
60 = 22 · 3 · 5lcm(50, 60) = 22 · 3 · 52 = 300
Ordering Rational Numbers
We can also use common denominators.
Example
Fill in the blank: 4250
4560
What is the common denominator?
50 = 2 · 52
60 = 22 · 3 · 5
lcm(50, 60) = 22 · 3 · 52 = 300
Ordering Rational Numbers
We can also use common denominators.
Example
Fill in the blank: 4250
4560
What is the common denominator?
50 = 2 · 52
60 = 22 · 3 · 5lcm(50, 60) = 22 · 3 · 52 = 300
Ordering Rational Numbers
4250
4560
4250· 6
64560· 5
5252300
225300
Conclusion?
4250
>4560
Ordering Rational Numbers
4250
4560
4250· 6
64560· 5
5
252300
225300
Conclusion?
4250
>4560
Ordering Rational Numbers
4250
4560
4250· 6
64560· 5
5252300
225300
Conclusion?
4250
>4560
Ordering Rational Numbers
4250
4560
4250· 6
64560· 5
5252300
225300
Conclusion?
4250
>4560
Ordering Rational Numbers
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
Find a rational number that lies between 14 and 1
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
Find a rational number that lies between 15 and 3
5 .
Example
Find a rational number that lies between 14 and 1
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
Find a rational number that lies between 15 and 3
5 .
Example
Find a rational number that lies between 14 and 1
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
Find a rational number that lies between 15 and 3
5 .
Example
Find a rational number that lies between 14 and 1
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
Find a rational number that lies between 35 and 3
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
Find a rational number that lies between 35 and 3
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
Find a rational number that lies between 35 and 3
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
Find a rational number that lies between 35 and 3
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
Find a rational number that lies between 35 and 3
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
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
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
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
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
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
Recommended