Directed Numbers

By studying this lesson you will be able to

• identify what directed numbers are,

• add integers using the number line, and

• add directed numbers without using the number line.

8.1 Identifying directed numbers

The thermometer in the figure has been calibrated with the values 10, 20, 30, ... in one direction to indicate the temperatures that are greater than 0o C, and with the values -10, -20, -30, ... in the opposite direction to indicate the temperatures that are less than 0o C.

Let us now consider the number line given below.

The positive whole numbers marked to the right of the position indicating zero on the number line are defined as positive integers and the negative whole numbers marked to the left of the position indicating zero are defined as negative whole numbers.  

{..., -3, -2, -1, 0, 1, 2, 3, ...} is the set consisting of all the integers.

Any positive number can be marked on the above number line to the right of the position indicating and any negative number can be marked to the left of the position indicating 0, taking into consideration the magnitude of the number.

All the numbers that are written with a positive or negative sign to indicate not only their magnitude but also one of two directions which are opposite to each other are defined as directed numbers.

Note

•  When a sign is not written in front of a number, it is considered as a positive number.

8.2 Adding directed numbers which are integers by using the number line

Let us consider adding directed numbers which are positive integers by using the number line.

•  The sum of two positive integers

Let us find the value of (+2) + (+1) using the number line.

Find each of the following sums using the number line.

(i) (+2) +(+3)                           (ii) (+3) + (+3)                       (iii) (+4) + (+1)                          (iv) (+5) + (+3)

8.2 Adding directed numbers which are integers by using the number line

Let us consider adding directed numbers which are positive integers by using the number line.

• The sum of two negative integers

Let us consider adding directed numbers which are negative integers by using the number line.

Let us find the value of (-2) + (-1) using the number line.

Find the value using the number line.

(i) (- 4) + (-1)                  (ii) (- 2) + (- 2)                 (iii) (- 2) + (- 3)

(iv) (- 1) + (- 3)               (v) (- 3) + (- 3)                 (vi) (- 4) + (- 2)

• The sum of a positive integer and a negative integer

Now let us consider adding a positive integer and a negative integer.

Let us find the value of (+5) + (-2) using the number line.

Find the value using the number line.

(i) (+3) + (-1)           (ii) (-4) + (+6)          (iii) (-7) + (+2)

(iv) (+2) + (-5)         (v) (+1) + (-1)          (vi) (-3) + (+3)

8.3 Adding integers without using the number line

• Finding the sum of two integers

Let us consider the examples related to adding two positive integers that were studied in the previous section.

Using the number line we obtained previously that,

(+2) + (+1) = (+3) and

(+3) + (+2) = (+5),

                                                        (+2) + (+1) = (+3)                                                   (+3) + (+2) = (+5)

                                                               2 + 1 = 3                                                                    3 + 2 = 5

Let us now reconsider the examples related to adding two negative integers that were studied in the previous section.

Using the number line we obtained previously that

(-2) + (-1) = (-3) and

(-3) + (-2) = (-5)'

Let us consider (-2) + (-1) = (-3)

•  Without considering the signs of the two directed numbers, obtain their sum. 

         2 + 1 = 3

•  Then write the answer with the negative sign. Therefore the answer is -3.

• When adding two negative directed numbers, add the two numbers without considering the negative sign and then write the answer with the negative sign.

Simplify.

(i) (+3) + (+8)                (ii) (-7) + (-3)                   (iii) (+12) + (+4)  

(iv) (-9) + (-16)              (v) (-20) + (-13)              (vi) (+17) + (+13)

(vii) (-11) + (-29)          (viii) (+2) + (+8)               (ix) (-3) + (-10)

•  Finding the sum of a positive integer and a negative integer

Using the number line we obtained previously that,

(+5) + (-2) = (+3) and

(-5) + (+2) = (-3)'

We can find the sum of a positive integer and a negative integer as follows.

When adding two directed numbers of opposite signs (positive and negative), obtain their difference without considering the signs, and write the answer with the sign of the directed number which is further away from 0 on the number line.

(1) Evaluate the following.

(i) (+7) + (-2)                        (ii) (-10) + (+4)                   (iii) (-3) + (+6)

(iv) (-5) + (+9)                     (v) (-11) + (+4)                    (vi) (-4) + 0

(vii) (+9) + (-8)                   (viii) (+7) + (-15)                  (ix) (+5) + (-6)

(x) (-7) + (+5)                     (xi) (+8) + (-10)                    (xii) (-9) + (+4)

8.4 Adding directed numbers

We have so far considered the addition of directed numbers which are integers. Now let us consider the addition of any two directed numbers.

The methods that were used above to add integers are used here too.

(2) The ground floor of a building has been named Floor 0 and the floors above it have been named 1, 2, 3, ... respectively, while the floors below it have been named -1" -2" -3" '' respectively.

                (i) If a person in Floor 7 climbs up a further 5 floors, which floor will he be in?

               (ii) If a person in Floor -1 descends a further 2 floors, which floor will he be in? 

               (iii) If a person in Floor 8 descends 3 floors, which floor will he be in?

               (iv) If a person in Floor 2 descends 4 floors, which floor will he be in?

(3) The temperature at 6.00 a.m. in Moscow on a certain day was recorded as - 4.7o C, while the temperature at 4.00 p.m. on the same day was increased by 120 C. Find the temperature in Moscow at 4.00 p.m.