Integers

Negative integers

Positive integers

Zero

… , -3 , -2 , -1 , 0 , 1 , 2 , 3 , …

1. An integer is a whole number that has a positive sign (+) or a negative sign (-) , including zero.

Understanding integersA

2. A positive integer is a whole number with a positive sign or without any sign.

Example

+2 , +7, 8 , 12.

3. A negative integer is a whole number with a negative sign.

Example

-3 , -10 , -20.

Solution (a) Positive seventy - nine

(a)Write +79 in words.

(b) Write negative one hundred and fifty in figures.

(b) -150

Example

Example

Solution The integers are -2 , 6 and 0.

State the integers from the list below.

25

-2 , + , 6 , -2.9 , 0 ,

143

Test Yourself

1. Write each of the following integers in words.

(a) -17 (b) +23 (c) +48

(d) -69 (e) -205 (f) +416

2. Write each of the following integers in figures.

(a) Negative forty - two(b) Positive nine(c) Positive sixty – eight(d) Negative two hundred and

seventy3. State the integers from the list below.

-9, ,6.2, 4, -78,- , -9.6

1

3

4

5

nted

Representing integers using a number line

B

Integers can be represented using a horizontal or a vertical number line

IncreasingDecreasing

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

Number line.

Positive integersNegative integersZero

-4 -3 -2 -1 0 1 2 3 4

Horizontal number line

4 3 2 1 0 -1 -2 -3 -4

Negative integers

Positive integers

Zero

Vertical number line

Example

(a) Use a number line to represent the integers from -5 to 3.

(b) Mark 6 , -3 , -1 and 4 on a number line.

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

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

Solution

A

B

Test Yourself

1. Use a number line to represent the integers from

(a) -3 to 4(b) -20 to -15

tol

Comparing two integersC

On a horizontal number line , an integer is always greater than the integers to its left and less than the integers to its right.

Example

-4 -3 -2 -1 0 1 2 3

-2 is greater than -4 but is less than 1.

Example

(a) Which integer is smaller , -5 or 3 ? (b) Which integer is greater , -2 or -8 ?

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

-8 -7 -6 -5 -4 -3 -2 -1 0

Solution

A

B

-2 is greater than -8.

-5 is smaller than 3.

Test Yourself

1. Copy and complete each of the following with ‘is greater than’ or ‘is less than’

(a) -4 +2(b) +8 -9(c) -3 -7(d) -10 -6(e) +9 -20

Arranging integers in orderD

1. We can arrange integers in increasing or decreasing order using a number line.

Example

(a) Arrange -4 , 6 , -3 , 5 , 0 and 1 in increasing order. (b) Arrange 6 , 0 , 4 , -2 and -4 in decreasing order.

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

Solution

Increasing order : -4, -3, 0, 1, 5, 6

(a)

(b) -4 -3 -2 -1 0 1 2 3 4 5 6 Decreasing order : 6, 4, 0, -2, -4

Test Yourself

1. Arrange each of the following sets of integers in increasing order.

(a) -5, -3, 0, -1, 2, -4(B) 8, -7, -5, 6, -9, 3

2. Arrange each of the following sets of integers in decrasing order.

(a) 9, -12, -6, 3, 7, -10(b) -11, -4, 8, -3, 5, -6

2. We can identify the largest integer and the smallest integer by arranging the given integers in order.

Example

Determine the largest integer and the smallest integer from the following set of numbers.

1, -2, 3, 0, -5

solution

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

The largest integer is 3 and the smallest integer is -5

Test Yourself

Determine the largest integer and the smallest integer from each of following sets of integers.

(a) -7, 5, -9, 3, 0, -2(b) 8, -12, 13,-15, 7, 11(c) -20, -15, -19, -7, -30(d) 5, -15, -20, 15, 10, -5

3. If the pattern of a sequence of integers is determined, we can find the missing terms in the sequence

Copy and complete the following sequence of integers.

, -5, 0, 5, ,

Example

Solution

, -5, 0, 5, , -10 10 15

+5 +5 +5 +5 +5

Test Yourself

Copy and complete each of the following sequences of integers.

(a) 9, 5, 1, , , (b) -12, , , 3, 8, (c) , , , -10, -4(d) -32, , , -23, -20, (e) , , -13, -4,

2.2 Addition and Subtraction of Integers

Addition of IntegersA

1. Addition of Integers is a process of finding the sum of two or more integers.

2. Addition an Integers to a positive integer can be represented using a number line by a movement towards the positive direction, which means from left to right.

-2 -1 0 1 2

+4For example, -2 + 4 = 2

3. Adding an integer to a negative integer can be represented using a number line by a movement towards the negative direction, which means from right to left

-2 -1 0 1 2

-4

For example, 2 +(-4)=-2

4. Integers with like signs are integers with the same sign.

For exampl

e 2 and 5

-8 and -12

5. Integers with unlike signs are integers with different signs.

For example

-4 and 10

3 and -9

1 +

Example

Simplify each of the following.

(a) 1 + (- 2)

-3 -2 -1 0 1 2

1 + (- 2) = -1

Solution

(b) -3 + 5

-3 -2 -1 0 1 2

Solution

-3 + 5 = 2

Evaluate 6 +(-4) + (-5).

Example

Solution

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

6 +(-4) + (-5) = -3

Example

The initial temperature of a cold storage was -2 °C. Two hours later, the temperature dropped by 3 °C. When the cold storage was switched off, the temperature rose by 7 °C. What was the new temperature of the cold storage?

Solution-2 +(-3)+7 =2

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

Therefore, the new temperature of the cold storage was 2 °C.

Test Yourself

1. Calculate each of the following.

1) 35 + (-18)

2) (-21) + 93) (-13) + 34

4) (-50) + (-28)5) (-39) + 16

2. Simplify each of the following.

1) (-27) + (-35) + (-14)

2) (-79) + 94 + (-32)3) 124 + (-215) + (-

18)4) (-116) + (-227) +

(-59) + (-32)5) (-358) + 256 + (-

78) + 496) 874 + (-359) + (-

291) + (-432)

Subtraction of IntegersB

1. Subtraction of integers is a process of finding the difference between two integers.2. The difference between two integers is the number of steps required to move from the second integer to reach the first integer on a number line.

((

• If you move to the right, you will get a positive integer.

For example, -3 – (-8) = 5

-8 -7 -6 -5 -4 -3

+5

• If you move to the left, you will get a negative integer.

For example, -8 – (-3) = -5

-8 -7 -6 -5 -4 -3

-5

3. To perform a subtraction involving three integers, always work out from left to right.

Example

Simplify 8- (-4) – 3.Solution 8- (-4) – 3 = 8 + 4 –

3 = 12 – 3 = 9

Example

In the morning, the temperature of a city was -3 °C. Its temperature then dropped by 5 °C in the afternoon. At night, its temperature dropped by another 4 °C. Find the temperature of the city at night.

Solution

-3 -5 -4 = -8 - 4 = -12

Therefore, the temperature of the city at night was - 12 °C

Test Yourself

1. Calculate each of the following.

1) (-3) - 52) 4 - (-16)3) 13 - (-7)4) (-63) - (-50)5) (-10) + (-13)

2. Simplify each of the following.

1) {(-4) - (-8)} + (-11)

2) 19 - (-25) + (-7)

3) (-12) - (-6) - 184) {15 + (-11) - (-

6)} + (-25)5) [(-10) + (-7)

–{ (-12) –( 9 ) + ( -6)} ] – {(-28) – (-25)}

and

2.3 Multiplication and Division of Integers

Multiplying integers

A

1. Rules for multiplication of two integers:1. Determine the sign of the

product. (+) × (+) = (+)(-) × (-) = (+)(+) × (-) = (-)(-) × (+) = (-)

2. Multiply the whole numbers.

(a)

Example

Find the product of the following.(a) -4 × 3

Solution

(a) -4 × 3= - (4 × 3)= -12

(b) -8 × (-7)

Solution

(b) -8 × (-7) = + (8 ×

7)= 56

2. When multiplying three integers, we work from left to right.

Example

Calculate the following.(a) -2 × 3 × (-5)

Solution

(a) -2 × 3 × (-5) = -6 × (-5)

= 30

(b) -6 × (-4) × (-3)

Solution

(a) -6 × (-4) × (-3)

= 24 × (-3)= -72

Example

The temperature in a refrigerator decreases 2 °C every hour. If the temperature now is 0 °C, find its temperature (a)3 hours later,(b) 4 hours earlier.

Solution

(a) -2 × (+3) = -6 °C Therefore, the temperature 3 hours later is -6 °C.

(b) -2 × (-4) = 8 °C Therefore, the temperature 4 hours earlier was 8 °C.

Test Yourself

1. Calculate each following.

1) 7 (-5)2) (-20) 493) (-13) (-28)4) (-35) 21 (-6)5) 111 (-25)

2. Solve the following.

1) {(-23) 41} - (-560) 2) (-36) (-11) + (-278) –

99 3) {(-64) + (-19)} (-12) - 487

4) {(-1,028) - (-457)} (-26)

5) (-109) {(-18) - (-32)} + (-84)

3. A submarine dived 3 m each minute. How deep can the submarine dive after 5 minutes?

4. The temperature at a highland resort

drops by 2 °C every hour. Find the total drop in temperature after 4 hours.

Dividing integersB

1. Rules for division of two integers:

1. Determine the sign of the quotient. (+) ÷ (+) = (+)

(-) ÷ (-) = (+)(+) ÷ (-) = (-)(-) ÷ (+) = (-)

2. Divide the whole numbers.

Example

Find the results of dividing the following.(a) -124 ÷ (-4)

Solution

-124 ÷ (-4) = 124 ÷ 4 = 31

(b)

Solution

-1506

-1506

= - 25

(c) 18 ÷ (-2) ÷ (-3)

Solution18 ÷ (-2) ÷ (-3)= -9 ÷ (-3)= 3

Example

Ake, sak and wach started a business together. In the first month, they made loss of 42510 Baht. Find the loss of each person if the loss is shared equally among them.Solution

-42510 Baht ÷ 3 = -14170 BahtTherefore, each person made a loss of 14170 Baht.

Test Yourself

1. Find the results of dividing the following.

1. (-16) (-1) 2. (-34) 2

3. 56 (-8) 4. (-220) 20

5. 132 (-11)6. {(-15) (-3)} - {180

(-90)}7. {(-450) 15} + {(-

208) (-13)}

2. The temperature of a cold storage room drops constantly by 28 °C in 4 hours. Calculate the drop in temperature each hour.

3. The water level in a reservoir decreases by 5 m in 4 days. Find the average decrease in the water level per day.

2.4 Combined Operations of Integers

Example

1. Solve each of the following.

(a) -234 + 56 - 23

-234 + 56 – 23 = -178 – 23 = -201

Solution

(b) 20 × 8 ÷(-4)

20 × 8 ÷(-4) = 160 ÷ (-4) = -40

Solution

(c) -(-3) × 8 + 25

-(-3) × 8 + 25 = 3 × 8 + 25 = 49

Solution

Example

The initial temperature in a freezer is 5 ˚C If the temperature decreases by 4˚C every minute , find its temperature after 6 minutes.

Solution

The change of temperature in the freezer = - 4 ˚C5 + 6×(-4) = 5 + (-24) = -19 ˚CTherefore, the temperature in the freezer after 6 minutes is -19 ˚C.

Test Yourself

1. Evaluate each of the following.

1) {(-15) (-3)} (-4)

2) {(-9) (-4)} {(-5) + 3}

3) {(-10) - (-11)} {(-12) - (-15)} (-1)

4) {(-3) (-20) (-15)} + (-9) - (-14)

5) {(-40) 8} (-16) {(-10) - (-9)}