6-2 addition, subtraction and estimation with rational...

§ 6-2 Addition, Subtraction and Estimationwith Rational Numbers

Defining Addition

Thinking back to how we defined the addition of integers, how can wedefine the addition of rational numbers?

Definition

If ab ,

cb ∈ Q, then a

b + cb = a+c

What if we don’t have common denominators?

Definition

If ab ,

cd ∈ Q, then a

b + cd = ad+bc

Defining Addition

Definition

If ab ,

cb ∈ Q, then a

b + cb = a+c

Definition

If ab ,

cd ∈ Q, then a

b + cd = ad+bc

Defining Addition

Definition

If ab ,

cb ∈ Q, then a

b + cb = a+c

Definition

If ab ,

cd ∈ Q, then a

b + cd = ad+bc

Defining Addition

Definition

If ab ,

cb ∈ Q, then a

b + cb = a+c

Definition

If ab ,

cd ∈ Q, then a

b + cd = ad+bc

Properties of Addition

What is the statement that would be true if the rational numbers wereclosed under addition?

Closure of the Rational Numbers Under AdditionIf a

b ,cd ∈ Q, then a

b + cd ∈ Q.

Is this true statement? Yes ... why?

If we look at the formula ab + c

d = ad+bcbd , what are ad, bc and bd?

b ,cd ∈ Q, then a

b + cd ∈ Q.

b ,cd ∈ Q, then a

b + cd ∈ Q.

Is this true statement?

Yes ... why?

b ,cd ∈ Q, then a

b + cd ∈ Q.

b ,cd ∈ Q, then a

b + cd ∈ Q.

If the associative property held for the addition of rational numbers,what would the statement be?

Associativity of the Rational Numbers Under Addition

If ab ,

ef ∈ Q, then a

cd + e

( ab + c

Is this a true statement? Yes ... why?

What happens when we apply the sum formula on both expressions?

If ab ,

ef ∈ Q, then a

cd + e

( ab + c

If ab ,

ef ∈ Q, then a

cd + e

( ab + c

Is this a true statement?

Yes ... why?

If ab ,

ef ∈ Q, then a

cd + e

( ab + c

If ab ,

ef ∈ Q, then a

cd + e

( ab + c

If the commutative property held for the addition of rational numbers,what would the statement be?

Commutativity of the Rational Numbers Under Addition

If ab ,

cd ∈ Q, then a

b + cd = c

d + ab .

If ab ,

cd ∈ Q, then a

b + cd = c

d + ab .

If ab ,

cd ∈ Q, then a

b + cd = c

d + ab .

Yes ... why?

If ab ,

cd ∈ Q, then a

b + cd = c

d + ab .

If ab ,

cd ∈ Q, then a

b + cd = c

d + ab .

Is there some ‘magical number’ such that when we add it to a rationalnumber a

b , we get back ab ?

Additive Inverse for Rational Numbers

For any ab ∈ Q, there exists the unique rational number 0

b such that

What property did we apply in the last statement?

Note: There is only one additive identity element for rationalnumbers, but we can change the denominator to get an equivalentform to suit our needs.

b such that

Is there some magical rational number such that when we add it to ab ,

we get the identity?

Inverse Element for Rational NumbersFor each a

b ∈ Q, −ab is the unique rational number such that

0b= −a

Remember, we cannot have an inverse element without having anidentity element ...

ab+(−a

0b= −a

ab+(−a

0b= −a

Equal Additions

If ab ,

ef ∈ Q and if a

b = cd , then a

b + ef = c

d + ef .

We use this property more than we realize when solving equationswith variables.

Equal Additions

If ab ,

ef ∈ Q and if a

b = cd , then a

b + ef = c

d + ef .

We use this property more than we realize when solving equationswith variables.

Visual Representations

Example

Illustrate 14 + 2

Example

Illustrate 14 + 2

Example

Illustrate 14 + 2

Example

Illustrate 14 + 2

Example

Illustrate 14 + 2

Example

Illustrate 14 + 2

Example

Illustrate 14 + 2

Example

Illustrate 14 + 2

Example

Illustrate 14 + 2

Subtraction of Rational Numbers

How can we define subtraction of rational numbers in terms of whatwe know - addition of rational numbers?

DefinitionIf a

b ,cd ∈ Q, then a

b −cd is the unique e

f ∈ Q such that ab = c

d + ef .

We can also think of subtraction as ...

Subtraction of Rational NumbersFor a

b ,cd ∈ Q,

ab −

cd = ad−bc

bdab −

cd = a

b + −cd

DefinitionIf a

b ,cd ∈ Q, then a

d + ef .

b ,cd ∈ Q,

ab −

cd = ad−bc

bdab −

cd = a

b + −cd

DefinitionIf a

b ,cd ∈ Q, then a

d + ef .

b ,cd ∈ Q,

ab −

cd = ad−bc

ab −

cd = a

b + −cd

DefinitionIf a

b ,cd ∈ Q, then a

d + ef .

b ,cd ∈ Q,

ab −

cd = ad−bc

bdab −

cd = a

b + −cd

Properties of Subtraction

If the rational numbers were closed under subtraction, what would ourstatement be?

Closure of the Rational Numbers Under SubtractionIf a

b ,cd ∈ Q, then a

b −cd ∈ Q.

b ,cd ∈ Q, then a

b −cd ∈ Q.

b ,cd ∈ Q, then a

b −cd ∈ Q.

If the commutative property held for subtraction of rational numbers,what would the statement say?

Commutativity for the Subtraction of Rational Numbers

If ab ,

cd ∈ Q, then a

b −cd = c

d −ab .

Can you give a counterexample?

If ab ,

cd ∈ Q, then a

b −cd = c

d −ab .

If ab ,

cd ∈ Q, then a

b −cd = c

d −ab .

If ab ,

cd ∈ Q, then a

b −cd = c

d −ab .

If the associative property held for subtraction of rational numbers,what would the statement say?

Associativity for the Subtraction of Rational Numbers

If ab ,

ef ∈ Q, then a

b −(

cd −

( ab −

)− e

If ab ,

ef ∈ Q, then a

b −(

cd −

( ab −

)− e

If ab ,

ef ∈ Q, then a

b −(

cd −

( ab −

)− e

If ab ,

ef ∈ Q, then a

b −(

cd −

( ab −

)− e

Adding and Subtracting Mixed Numbers

There are two ways we can go about this. Ideas of what they are?

1 Combining whole number parts and rational parts separately,then combining as necessary

2 Converting to improper fractions, combining and then rewritingthe result as a mixed number

Example

Find 2 15 + 3 3

We first want to separate into whole number parts and rational parts.What property allows us to do this?

215+ 3

34= (2 + 3) +

)= 5 +

Example

Find 2 15 + 3 3

215+ 3

34= (2 + 3) +

)= 5 +

Example

Find 2 15 + 3 3

215+ 3

(2 + 3) +(

)= 5 +

Example

Find 2 15 + 3 3

215+ 3

34= (2 + 3) +

)= 5 +

Example

Find 2 15 + 3 3

215+ 3

34= (2 + 3) +

)= 5 +

Example

Find 2 15 + 3 3

215+ 3

34= (2 + 3) +

)= 5 +

= 51920

Example

Find 2 15 + 3 3

215+ 3

34= (2 + 3) +

)= 5 +

Adding and Subtractive Mixed Numbers

Example

Find 3 23 − 2 5

323− 2

56= (3− 2) +

(23− 5

)= 1 +

(23− 5

)= 1 +

(46− 5

)= 1 +

=66− 1

Example

Find 3 23 − 2 5

323− 2

(3− 2) +(

23− 5

)= 1 +

(23− 5

)= 1 +

(46− 5

)= 1 +

=66− 1

Example

Find 3 23 − 2 5

323− 2

56= (3− 2) +

(23− 5

)= 1 +

(46− 5

)= 1 +

=66− 1

Example

Find 3 23 − 2 5

323− 2

56= (3− 2) +

(23− 5

)= 1 +

(23− 5

(46− 5

)= 1 +

=66− 1

Example

Find 3 23 − 2 5

323− 2

56= (3− 2) +

(23− 5

)= 1 +

(23− 5

)= 1 +

(46− 5

= 1 +−16

=66− 1

Example

Find 3 23 − 2 5

323− 2

56= (3− 2) +

(23− 5

)= 1 +

(23− 5

)= 1 +

(46− 5

)= 1 +

=66− 1

Example

Find 3 23 − 2 5

323− 2

56= (3− 2) +

(23− 5

)= 1 +

(23− 5

)= 1 +

(46− 5

)= 1 +

=66− 1

Example

Find 3 23 − 2 5

323− 2

56= (3− 2) +

(23− 5

)= 1 +

(23− 5

)= 1 +

(46− 5

)= 1 +

=66− 1

Example

Find 4 27 + 1 3

What would our common denominator be here?

We first convert to improper fractions and then combine in the usualway.

427+ 1

=15035

=20635

= 53135

Example

Find 4 27 + 1 3

427+ 1

=15035

=20635

= 53135

Example

Find 4 27 + 1 3

427+ 1

=15035

=20635

= 53135

Example

Find 4 27 + 1 3

427+ 1

=15035

=20635

= 53135

Example

Find 4 27 + 1 3

427+ 1

=15035

=20635

= 53135

Example

Find 4 27 + 1 3

427+ 1

=15035

=20635

= 53135

Example

Find 4 27 + 1 3

427+ 1

=15035

=20635

= 53135

Example

Find 4 27 + 1 3

427+ 1

=15035

=20635

= 53135

Rational Numbers with Variables

Example

Find 2x −

31−x

There is no difference here as when we have numbers. What is ourcommon denominator?

2x− 3

1− x=

2x· 1− x

1− x− 3

1− x· x

=2− 2x

x(1− x)− 3x

x(1− x)

=2− 2x− 3x

x(1− x)

=2− 5xx− x2

Example

Find 2x −

31−x

2x− 3

1− x=

2x· 1− x

1− x− 3

1− x· x

=2− 2x

x(1− x)− 3x

x(1− x)

=2− 2x− 3x

x(1− x)

=2− 5xx− x2

Example

Find 2x −

31−x

2x− 3

1− x=

2x· 1− x

1− x− 3

1− x· x

=2− 2x

x(1− x)− 3x

x(1− x)

=2− 2x− 3x

x(1− x)

=2− 5xx− x2

Example

Find 2x −

31−x

2x− 3

1− x=

2x· 1− x

1− x− 3

1− x· x

=2− 2x

x(1− x)− 3x

x(1− x)

=2− 2x− 3x

x(1− x)

=2− 5xx− x2

Example

Find 2x −

31−x

2x− 3

1− x=

2x· 1− x

1− x− 3

1− x· x

=2− 2x

x(1− x)− 3x

x(1− x)

=2− 2x− 3x

x(1− x)

=2− 5xx− x2

Example

Find 2x −

31−x

2x− 3

1− x=

2x· 1− x

1− x− 3

1− x· x

=2− 2x

x(1− x)− 3x

x(1− x)

=2− 2x− 3x

x(1− x)

=2− 5xx− x2

Example

Find 2x −

31−x

2x− 3

1− x=

2x· 1− x

1− x− 3

1− x· x

=2− 2x

x(1− x)− 3x

x(1− x)

=2− 2x− 3x

x(1− x)

=2− 5xx− x2

Estimation

What is the value of estimation with rational numbers?

Reasonability of solution ...

Example

Estimate 2 13 + 1

10 − 1 78 .

Roughly speaking, we have ...

(using halves) 2 12 + 0− 2 ≈ 1

(using wholes) 2 + 0− 2 = 0

(using quarters) 2 14 + 0− 2 = 1

Estimation

Example

Estimate 2 13 + 1

10 − 1 78 .

(using halves) 2 12 + 0− 2 ≈ 1

Estimation

Example

Estimate 2 13 + 1

10 − 1 78 .

(using halves) 2 12 + 0− 2 ≈ 1

Estimation

Example

Estimate 2 13 + 1

10 − 1 78 .

(using halves)

2 12 + 0− 2 ≈ 1

Estimation

Example

Estimate 2 13 + 1

10 − 1 78 .

(using halves) 2 12 + 0− 2 ≈ 1

Estimation

Example

Estimate 2 13 + 1

10 − 1 78 .

(using halves) 2 12 + 0− 2 ≈ 1

(using wholes)

2 + 0− 2 = 0

Estimation

Example

Estimate 2 13 + 1

10 − 1 78 .

(using halves) 2 12 + 0− 2 ≈ 1

Estimation

Example

Estimate 2 13 + 1

10 − 1 78 .

(using halves) 2 12 + 0− 2 ≈ 1

(using quarters)

2 14 + 0− 2 = 1

Estimation

Example

Estimate 2 13 + 1

10 − 1 78 .

(using halves) 2 12 + 0− 2 ≈ 1

6-2 addition, subtraction and estimation with rational...

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