MATHEMATICS - FORM 1

INTEGERS


       
                                 


INTEGERS

A) Understanding Integers

1. An integer is a whole number with the positive

    sign or negative sign, including zero which is

    neither positive nor negative.



2. Positive and negative numbers include decimals

    and fractions.

    For example :-

    (a) Positive numbers : +12, 8.5, + 11
                                                           2

    (b) Negetive numbers : -15, - 3.6, - 33
                                                            4

3. Positive integers can be written without the '+' sign.

    For example :-

    +4 is usually written as 4.

Worked Example 1  

   

From the numbers above, list all the

(a) integers,

(b) positive integers,

(c) negative integers.

Solution

(a) - 5, -2, 0, +8 and 18

(b) +8 and 18

(c) -5 and -2

B) Representing Integers on Number Lines

1. Integers can be represented on either a horizontal

    or a vertical number line.

2. Normally, a horizontal number line is used.

    For example :-

    

Worked Example 2

Draw a number line and mark the following numbers :-

- 15 , - 5, 10, 15

Solution



C) Comparing the Value of Two Integers

We can use the number line to compare the values

of integers. A given integer, x, is greater than all the

integers on its left but smalller than all the integers

on its right.



Worked Example 3

In each of the following pairs of integers, state the

integer that is greater in value.

(a)  - 3, 2      (b) - 6, - 3

Solution



Worked Example 4

State the integer with the smaller value : - 5 or 0



D) Arranging Integers in Order

1. In increasing order, the integers are arranged

    from the smallest to the largest.

    For example :-

         

2. In decreasing order, the integers are arranged

    from the largest to the smallest.

    For example :-

           

Worked Example 5

(a) Arrange the integers -6, 2, -1, +4, 0, - 2 in

    increasing order.

(b) Arrange the integers 3, 0, 5, - 5, 6, - 4 in

    decreasing order.

Solution

(a) - 6, - 2, -1, 0, 2, +4

(b)  6, 5, 3, 0, - 4, - 5

E) Determining the Largest Integer or the
Smallest Integer


Worked Example 6

Determine the largest integer and the smallest

integer in the following : 8, - 3, - 14, - 9, - 4, 7

Solution


F) Completing a Sequence of Integers

To complete a sequence of integers, it is necessary

to identify the pattern of the sequence.

Worked Example 7

   

State the integers that are represented by the letters

P, Q and R on the number line above.

Solution

   

Therefore, P = -8, Q = 0, R= 4

Worked Example 8

Complete the following sequence of integers.

- 12, - 9, - 6, __, __, __

Solution



G) Using Positive and Negative Integers
in Real- life Situations


Positive and negative integers can be used in

the following contexts in real-life situations.

(a) An increase or a decrease in value

     For example :-

     (i) +RM100 represents an increase of Rm100

        in price whereas -RM50 represents a decrease

        of RM50 in price.

     (ii) +60 sen represents a profit of 60 sen whereas

         - 30 sen represents a loss of 30 sen.

     (iii) +4 kg represents a mass gain of 4 kg whereas
 
           - 6 kg represents a mass loss of 6 kg.

(b) A value which is either greater than or less than

    zero.

     For example :-

     (i) +5oC represents 50C more than 00C whereas

        - 20C represents 20C less than 00C.

     (ii) As 0 m is the sea level, +3 m represents 3 m

         above  sea level whereas -10 m below sea

         level.

(c) A positive or a negative direction (i.e.opposite

    directions)

    For example :-

   (i) If +8 km represents 8 km to the east, then -7 km

      represents 7 km to the west.

   (ii) If a lift which moves 4 floors upwards is repre-

       sented by +4, then a lift which moves downwards

      is represented by - 3.

Worked Example 9 

Use positive or negative integers to represent the

following.

(a) A temperature rise of 180C

(b) 800 m below sea level

(c) A loss of RM70

Solution

(a) +180C      (c) -RM70

(b) - 800 m

ADDITION AND SUBTRACTION
 OF INTEGERS


A) Addition of integers

1. For a horizontal number line, the direction from left

   to right indicates the positive direction while the

   direction from right to left indicates the negative direction.

   For example :-

   

2. For a vertical number line, and upward direction

    indicates the positive direction indicates the negative

    direction.

    For example :-

   

3. We can add two or more integers using a number line.

    (a) When adding a positive integer, we move in the

         positive direction on the number line.

    (b) When adding a negative integer, we move in the

         negative direction on the number line.

    (c) Zero is usually taken as the starting point.

Worked Example 10

Solve the following by using the number lines.

(a) 3 + (+5)       (c) - 2+ (- 5)

(b) - 6 + 4         (d) 3 + (- 5)+(+8)

Solution





Worked Example 11

Solve each of the following.

(a) - 9 + (+5)         (c) - 6 + (- 2)

(b) + 7 + (- 5)

Solution




B) Problem Solving involving Addition of Integers

Worked Example 12

A submarine which is 9 km below sea level

moves up 3 km. What is the position of the

submarine now ?

Solution

1. Understand the problem

    Given information :

    The submarine is 9 km below sea level.

    It moves up 3 km.

    Find : Final position of the submarine

2. Devise a plan

    Use addition.

3. Carry out the plan

    - 9 + 4 = - 5

    Therefore, the submarine is 5 km below sea

    level now.

4. Check
  
           

C) Subtraction of Integers

1. Subtraction of integers is to find the difference

    between two integers.

2. We can use a number line to find the difference

    between two integers.

3. The difference can be determined by counting the

    number of units to be moved from the second integer

    to the first integer. The direction of the movement will

    determine the '+' or '_' sign in the answer.

Worked Example 13

Solve the following by using number line.

(a) 3 - 5            (c) - 2 - 4

(b) 8 - (+4)        (d) 0 - (- 7)

Solution




Worked Example 14

Find the value of each of the following.

(a) 8 - (+5)

(b) 6 - (- 3)

(c) - 9 - (- 9)

Solution



Worked Example 15

Find the value of each of the following.

(a) 30 - (+18) - (- 6)

(b) - 21 - (- 7) - (- 13)

Solution

(a) 30 - (+18) - (- 6) = 30 - 18 + 6

                             = 12 + 6

                             = 18

(b) - 21 - (- 7) - ( - 13) = - 21 + 7 + 13

                               = - 14 + 13

                               = -1

D) Problem Solving involving Subtraction
of Integers

A submarine is 8 m below sea level. An eagle is 3 m

above sea level directly above the submarine. What

is the distance between the submarine and the eagle?


Solution

1. Understand the problem

    Given information :

    A submarine is 8 m below sea level.

    An eagle is 3 m above sea level.

    Find : Distance between the submarine

    and the eagle 

2. Devise a plan

    Use subtraction.

3. Carry out the plan

    3 - (- 8) = 3 + 8 = 11

    Therefore, the distance between the submarine

    and the eagle is 11 m.

4. Check

   

E) Combined Operations of Addition and
Subtraction of Integers

Worked Example 17

Find the value of each of the following.

(a) 25 - (+30) + (- 12)

(b) - 12 - (- 26) + (- 15)

Solution

(a) 25 - (+30) + (- 12)

     = 25 - 30 - 12

     = - 5 - 12

     = - 17

(b) - 12 - (- 26) + (- 15)

     = - 12 + 26 - 15

     = 14 - 15

     = - 1

F) Problem Solving involving Addition and
Subtraction of Integers


Worked Example 18

A diver at 7 m below sea level swam up 3 m. A

turtle at that moment was 2 m below sea level.

What was the distance between them when the

turtle was directly above the diver?

Solution

1. Understand the problem

    Given information :

    The diver was 7 m below sea level at first.

    He swam up 3 m.

    The turtle is 2 m below sea level.

    Find : Distance between the diver and the turtle.

2. Devise a plan

    Use addition and subtraction.

3. Carry out the plan

    (- 7 + 3) - (-5) = (- 7 + 3) + 5

                        = - 4 + 5

                        = 1

    Therefore, the distance between the diver and


    the turtle was 1 m.

4. Check