Lesson 2 Operations with Rational and Irrational Numbers

NCSCOS Obj.: 1.01; 1.02

ObjectiveTLW State the coordinate of a point on a number line

TLW Graph integers on a number line.

TLW Add and Subtract Rational numbers.

TLW Multiply and Divide Rational numbers

The Number Line

Integers = {…, -2, -1, 0, 1, 2, …}

Whole Numbers = {0, 1, 2, …}

Natural Numbers = {1, 2, 3, …}

-5 0 5

Addition Rule1) When the signs are the same,

ADD and keep the sign.(-2) + (-4) = -6

2) When the signs are different,

SUBTRACT and use the sign of the larger number.

(-2) + 4 = 2

2 + (-4) = -2

Karaoke Time!Addition Rule: Sung to the tune of

“Row, row, row, your boat”

Same signs add and keep,different signs subtract,

keep the sign of the higher number,then it will be exact!

-1 + 3 = ?1. -4

2. -2

3. 2

4. 4

Answer Now

-6 + (-3) = ?1. -9

2. -3

3. 3

4. 9

Answer Now

The additive inverses (or opposites) of two numbers add to equal zero.

-3

Proof: 3 + (-3) = 0

We will use the additive inverses for subtraction problems.

Example: The additive inverse of 3 is

What’s the difference between7 - 3 and 7 + (-3) ?7 - 3 = 4 and 7 + (-3) = 4

The only difference is that 7 - 3 is a subtraction problem and 7 + (-3) is an addition problem.

“SUBTRACTING IS THE SAME AS ADDING THE OPPOSITE.”

(Keep-change-change)

When subtracting, change the subtraction to adding the opposite (keep-change-change)

and then follow your addition rule.

Example #1: - 4 - (-7)- 4 + (+7)

Diff. Signs --> Subtract and use larger sign.3

Example #2: - 3 - 7- 3 + (-7)

Same Signs --> Add and keep the sign.-10

11b + (+2b)

Same Signs --> Add and keep the sign.

13b

Okay, here’s one with a variable!Example #3: 11b - (-2b)

Which is equivalent to-12 – (-3)?

Answer Now

1. 12 + 3

2. -12 + 3

3. -12 - 3

4. 12 - 3

7 – (-2) = ?

Answer Now

1. -9

2. -5

3. 5

4. 9

1) If the problem is addition, follow your addition rule.

2) If the problem is subtraction, change subtraction to adding the opposite (keep-change-change) and then follow the addition rule.

Review

Absolute Valueof a number is the distance from

zero.

Distance can NEVER be negative!

The symbol is |a|, where a is any number.

Examples

7 = 7

10

100

10 =

-100 =

5 - 8 = -3= 3

|7| – |-2| = ?1. -9

2. -5

3. 5

4. 9

Answer Now

|-4 – (-3)| = ?

Answer Now

1. -1

2. 1

3. 7

4. Purple

Find the sum.1) -2.304 + (-0.26)Line up the decimals and add (same signs).

-2.564

2)5

8

3

4

Get a common denominator and subtract.

5

8

6

8

1

8

Find the difference.

3)5 3

9 5

3

5

5

9

25

45

27

45

2

45

Change subtraction to adding the opposite.

Get a common denominator.

Subtract and keep sign of the larger number.

Find the difference.

4)1

2

6

7

61

2 7

7

14

12

14

5

14

Get a common denominator and subtract.

Change subtraction to adding the opposite.

Substitute for y: 6.32 - (-3.42)6.32 + 3.42

9.74

5) Solve 6.32 – y if y = -3.42

Find the solution6.5 – 9.3 = ?

Answer Now

1. -3.2

2. -2.8

3. 2.8

4. 3.2

Find the solution

Answer Now

2 1?

3 4

1

7

11

12

5

12

3

7

1. .

2.

3.

4.

A rational number is a number

that can be written as a fraction.How can these be written as a fraction?

3 = 3

1

23

4

11

4

Inequality Symbols

Symbol Meaning < is less than > is greater than ≠ is not equal to ≤ is less than or equal to ≥ is greater than or equal to = is equal to

Ordering Rational Numbers

2 ways to order from least to greatest

1. Get a common denominator

2. Change the fractions to decimals (numerator demoninator)

Which rational number is bigger?

1) Get a common denominator.32

88

33

88

4

11or

3

8

2) or convert the fraction to a decimal.0.363 < 0.375

4

11<

3

8

Which rational number is bigger?

1) Get a common denominator

2) or convert to a decimal1.75 < 1.83

7

4

11

6

7

4or

11

6

42

24< 44

24

Which symbol makes this true? 5

7__

3

41. <

2. >

3. =

Answer NowAnswer Now

Which symbol makes this true? -2

9__

-1

41. <

2. >

3. =

Answer NowAnswer Now

Multiplying Rules

1) If the numbers have the same signs then the product is positive.

(-7) • (-4) = 28

2) If the numbers have different signs then the product is negative.

(-7) • 4 = -28

Examples

1) (3x)(-8y)

20 4

5

2) Write both numbers as a fraction.

Cross-cancel if possible.

20

1

4

5

80

5= -16

Multiplying fractions:

top # • top #

Bottom # • bottom#

-24xy

When multiplying two negative numbers, the product is negative.

1. True

2. False

Answer NowAnswer Now

When multiplying a negative number and a positive number, use the sign of

the larger number.1. True

2. False

Answer NowAnswer Now

2

5

5

8

3)

10

40=

1

4=

Multiply: (-3)(4)(-2)(-3)

1. 72

2. -72

3. 36

4. -36

Answer NowAnswer Now

an easy way to determine the sign of the answer

3 negative signs -> Odd -> answer is negative-480

When you have an odd number of negatives, the answer is negative.

When you have an even number of negatives, the answer is positive.

4) (-2)(-8)(3)(-10)Do you have an even or odd number of

negative signs?

Last one!

5) 1

2

6 5 2

5

2

Positive or negative answer?Positive - even # of negative signs (4)

Write all numbers as fractions and multiply.

=12

1

2

6

1

5

1

2

5

2

1

12010

What is the sign of the product of (-3)(-4)(-5)(0)(-1)(-6)(-91)?

1. Positive

2. Negative

3. Zero

4. Huh?

Answer NowAnswer Now

Dividing Rules

1) If the numbers have the same signs then the quotient is positive.

-32 ÷ (-8 )= 4

2) If the numbers have different signs then the quotient is negative.

81 ÷ (-9) = -9

When dividing two negative numbers, the quotient is positive.

1. True

2. False

Answer NowAnswer Now

When dividing a negative number and a positive number, use the sign of the

larger number.1. True

2. False

Answer NowAnswer Now

The reciprocal of is

where a and b 0.

a

b

b

a

The reciprocal of a number is called its multiplicative inverse.

A number multiplied by its reciprocal/multiplicative inverse is

ALWAYS equal to 1.

Example #1

7

2.

2

7The reciprocal of is

2

7

7

2 1

11

Example #2

The reciprocal of -3 is

1

3.

1

11

3

1

1

3

Basically, you are flipping the fraction!

We will use the multiplicative inverses

for dividing fractions.

Which statement is false about reciprocals?

1. Reciprocals are also called additive inverses

2. A number and its reciprocal have same signs

3. If you flip a number, you get the reciprocal

4. The product of a number and its reciprocal is 1

Answer NowAnswer Now

When dividing fractions, change division to multiplying by the reciprocal.

Examples

1) 3

4

5

8

3

4

8

5

6

5

24

20

2) 1

5

3

10

1

5

10

3

10

15

2

3

What is the quotient of-21 ÷ -3?

1. 18

2. -18

3. 7

4. -7

Answer NowAnswer Now

1 10

?5 3

Answer NowAnswer Now

2

3

2

33

503

50

1. .

2. .

3. .

4. .