Directed Numbers

Numbers which have a direction and a size are called DIRECTED NUMBERS.

Once a direction is chosen as positive (+), the opposite direction is taken as negative (-).

If above zero degrees is positive (+), then below zero degrees is negative.

If north is positive (+), then south is negative (-).

If profit is positive (+), then loss is negative (-).

Directed numbers are numbers that have negative and positive signs attached to them. The set of all negative numbers and all positive numbers is called the set of INTEGERS. The number line and the thermometer are very much alike.

If the night temperature in a city was −4°C and by midday it rises to 5°C, counting from −4 to 5 on the number line gives the difference in the temperature.

When we want to write minus 5, minus 6 and minus 7 we must write it with the minus sign in front like this; –5, –6 and –7.

However, when writing positive numbers like plus 5, plus 6 or plus 7, we may use the plus sign +5,+6 or +7 but it is typical to write the numbers without the sign i.e. 5, 6 or 7.

Now let's work through some questions.

Now let us look at carrying out addition and subtraction of directed numbers involving more than one step.

When you have two signs next to each other, for example

+(+),

– (–),

– (+) and +(–),

let the first sign mean direction and the second sign mean movement.

Therefore: ‘+’ means facing the positive direction and '-' means facing the negative direction.

–2 + (+7) = –2 + 7 = 5

+3 + (+8) = +3 + 8 = 11

–9 + (+3) = –9 + 3 =–6

+4 + (+10) = 4 + 10 = 14

Note:

Write the MIDDLE STEP to help you understand what you are doing.

2 + (–5) = 2 – 5 = –3

11 + (–7) = 11 – 7 = +4

18 + (–11) = 18 – 11 = +7

–7 + (–5) = –7 –5 = –12

Note:

Don't forget the MIDDLE STEP ! ! !

– (+) which means facing the negative direction and moving forward resulting in a movement from right to left on the number line which is actually subtraction.

Therefore, - (+) = -

Let's look at some examples using the number line.

i. 3 – (+6) = 3 – 6 = –3

Using a number line, go to +3, face the negative direction and go forward 6 steps in the negative direction. That will give –3.

ii. –7 – (+8) = –7– 8 = –15.

Using a number line, go to –7, face the negative direction move forward 8 steps facing the ‘– ‘ direction. That will give –15.

+11 – (+2) = 11–2 = 9.

Using a number line, go to +11, face the negative direction move forward 2 steps facing the ‘– ‘ direction. That will give +9.

– (–) which means face the negative direction and go backwards.

We end up moving forward on the number line.

Facing the negative direction and going backwards is actually moving from left to right on the number line which is addition.

Let's look at the example below

3 – (–4) = 3 + 4 = 7

Using a number line, go to +3. Face the negative direction and go backwards 4 steps, that gives 7.

In conclusion :

+ (+)   = +

– (–)  = +

– (+)   = –

+ (–)   = –

When the two signs next to each other are the same, that is,  

+ (+) or – (–), it is a (+), so it is addition.

When the two signs next to each other are different, that is,

– (+) or + (–), it is a (–), so you subtract.

Let us look at some long examples asking to simplify

Simplify:

a) 18 – (–16) – 3 – (–5) + 2

b) –43 – (–19) – 21 + 25

Solution

a) 18 – (–16) – 3 – (–5) + 2

First, deal with the integers that have double signs in front of them. Next, group the positives together and the negatives together and simplify. It looks like this:

18 – (–16) – 3 – (–5) + 2

= 18 + 16 – 3 + 5 + 2  (Group positives)

= 18 + 16 + 5 + 2 –3

= 41 – 3

= 38

b) –43 – (–19) – 21 + 25

= –43 + 19 – 21 + 25   (Group like signs)

= –43 –21 + 19 + 25

= –64 + 44 = –20

NOTE: when moving numbers around, you must move each number together with the sign in front of it. If you move only the numbers and not their signs, you change their values and will end up with the wrong answer.

–64 + 44 = 44 – 64

Since 64 – 44 = 20, then 44 – 64 = –20.