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Subunit: 1.Positive and Negative Numbers 2.Addition and subtraction of Integers3.Multiplication and division of integers

14.Integers

Number line

• When points on a line represent numbers, then that line is called number line.

0 1 2 3 4 5 6-1-2-3-4-5-6

Definition

• Positive number – a number greater than zero.

0 1 2 3 4 5 6

Definition

• Negative number – a number less than zero.

0 1 2 3 4 5 6-1-2-3-4-5-6

Definition• Opposite Numbers – numbers

that are the same distance from zero in the opposite direction

0 1 2 3 4 5 6-1-2-3-4-5-6

Definition

• Integers – Integers are all the whole numbers and all of their opposites on the negative number line including zero.

7 opposite -7

Negative Numbers Are Used to Measure Temperature

Negative Numbers Are Used to Measure Under Sea Level

01020

30

-10-20-30

Hint• If you don’t see a negative or

positive sign in front of a number it is positive.

9+

Subunit: Addition and subtraction of Integers

14.Integers-2

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

-1 + 3 = ?1. -4

2. -2

3. 2

4. 4

Answer Now

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

2. -3

3. 3

4. 9

Answer Now

Solve the Problems• -3 + -5 =

• 4 + 7 =

• (+3) + (+4) =

• -6 + -7 =

• 5 + 9 =

• -9 + -9 =

-8

-18 14

-137

11

Solve These Problems

• 3 + -5 =

• -4 + 7 =

• (+3) + (-4) =

• -6 + 7 =

• 5 + -9 =

• -9 + 9 =

-25 – 3 = 2

0

-4

1

-1

3

9 – 9 = 0 9 – 5 = 47 – 6 = 1

4 – 3 = 17 – 4 = 3

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6

When the number is positive countto the right.

When the number is negative countto the left.

+-

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

-

+3 + -5 = -2

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

-

+6 + -4 = +2

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

-

+3 + -7 = -4

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6-

+

-3 + +7 = +4

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6-

+

-5 + +3 = +4

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6-

+

-2 + +8 = +6

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6-

+

-5 + +2 = -3

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6-

-

-4 + -2 = -6

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6-

-

-2 + -3 = -5

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6-

-

-1 + -4 = -5

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6-

-

-4 + -1 = -5

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

+

4 + 1 = 5

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6+

+

3 + 2 = 5

One Way to Add Integers Is With a Number Line

0 1 2 3 4 5 6-1-2-3-4-5-6

+

+

1 + 5 = 6

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.”

When subtracting, change the subtraction to adding the opposite and

then follow your addition rule.Example #1: - 4 - (-7)

- 4 + (+7)Diff. Signs --> Subtract and use larger sign.

3Example #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

Operations with Integers

What is an Integer?• A whole number that is either greater

than 0 (positive) or less than 0 (negative)

• Can be visualized on a number line:

What is a Number Line?•A line with arrows on both ends that show the integers with slash marks

•Arrows show the line goes to infinity in both directions ( + and -)

•Uses a negative sign (-) with negative numbers but no positive sign (+)

with positive numbers

•Zero is the origin and is neither negative nor positive

What are Opposites?•Two integers the same distance from the

origin, but on different sides of zero

•Every positive integer has a negative integer an equal distance from the origin

•Example: The opposite of 6 is -6

•Example: The opposite of -2 is 2

What is Absolute Value?•Distance a number is from zero on a

number line (always a positive number)

•Indicated by two vertical lines | |

•Every number has an absolute value

•Opposites have the same absolute values since they are the same distance from zero

Example: |-8| = 8 and |8| = 8

•Example: |50| = 50 and |-50| = 50

What Can We Do to Integers?

•Integers are numbers, so we can add, subtract, multiply, and divide them

•Each operation has different rules to follow

Adding Rules – Same Signs•If the integers have the SAME signs:

ADD the numbers & keep the same sign!

•Positive + Positive = Positive Answer

•Negative + Negative = Negative Answer

• Examples: -3 + (-10) = ? ? = -13

• 6 + (8) = ? ? = 14

Adding (Same Signs) - Examples #1. -3 + (-10)

Step 1: 13 Add the #s

Step 2: -13 Keep same sign (Both #s are

negative – Answer is negative!)

#2. 6 + (8)

Step 1: 14 Add the #s

Step 2: 14 Keep same sign (Both #s are positive – Answer is positive!)

Adding Rules – Different Signs•If the integers have the DIFFERENT signs: SUBTRACT the numbers & use sign of the

BIGGER number!

•Bigger # is Positive = Positive Answer

•Bigger # is Negative = Negative Answer

• Examples: -13 + (7) = ? ? = -6

• 23 + (-8) = ? ? = 15

#1. -13 + (7)

Step 1: 6 Subtract the #s

Step 2: -6 Use sign of bigger # (Bigger # is negative - Answer is negative!)

#2. 23 + (-8)

Step 1: 15 Subtract the #s

Step 2: 15 Use sign of bigger # (Bigger # is positive - Answer is positive!)

Subtracting Rules•Put ( ) around second number & its sign

•Change SUBTRACTION sign to an ADDITION sign

•Change sign of 2nd number to its opposite

•Follow the rules for ADDITION:

-SAME signs: Add & keep the same sign -DIFFERENT signs: Subtract &

use sign of bigger # • Examples: -5 – -10 = ? ? = 5• 9 - 23 = ? ? = -14

Subtracting - Examples #1. -5 – -10 #2.9 - 23

Step 1: -5 – (-10) Insert ( ) 9 – (23)

Step 2: -5 + (-10) Change – to + 9 + (23)

Step 3: -5 + (10) Change 2nd sign 9 + (-23)

Step 4: 5 Follow adding rules -14 d

Multiplying Rules• Multiply the numbers like usual

•If the integers have the SAME signs: ANSWER will be POSITIVE

•If the integers have DIFFERENT signs: ANSWER will be NEGATIVE

• Examples: -3 · (-5) = ? ? = 15

• -9 · (-10) = ? ? = 90

• -7 · 7 = ? ? = -49

• 6 · -6 = ? ? = -36

Multiplying - Examples• #1. -3 · (-5) #2. -9 ·

(-10)

• 15 Multiply the numbers 90

• 15 Same signs = Positive Answer 90#3. -7 · 7 #4. 6 · -6

49 Multiply the numbers 36

-49 Different signs = Negative Answer -36

Dividing Rules• Divide the numbers like usual

•If the integers have the SAME signs: ANSWER will be POSITIVE

•If the integers have DIFFERENT signs: ANSWER will be NEGATIVE

• Examples: -33 ÷ (-3) = ? ? = 11

• -90 ÷ (-10) = ? ? = 9

• -20 ÷ 2 = ? ? = -10

• 6 ÷ -6 = ? ? = -1

Dividing - Examples• #1. -33 ÷ (-3) #2. -

90 ÷ (-10)

• 11 Divide the numbers 9

• 11 Same signs = Positive Answer 9 #3. -20 ÷ 2 #4. 6

÷ -6

10 Divide the numbers 1

-10 Different signs = Negative Answer -1

Mixed PracticeSolve the following problems:

-9 + - 9

-18

7 · -4

-28

-10 - (-19)

9

-35 ÷ -7

5

15 + -25

-10

-23 - 9

-32

•Created By –