Directed Numbers

Directed Numbers 1

Addition and Subtraction

By Munashe-Liam Chifokoyo

Introduction to 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 (-).

For example:

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

We use our knowledge of the thermometer to explain directed numbers.

Directed Numbers

Looking at the thermometer, there are two sets of numbers; one set reading with a negative sign (-) and the other set reading without a sign meaning it is positive. As you go down towards the negative numbers, the temperature becomes colder and colder but as you go towards the positive numbers, the temperature becomes warmer and warmer.

Directed Numbers

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.

Directed Numbers-Number Line

If we look at the number line, we see 0 as the dividing point between positive and negative numbers. The numbers to the right of 0 are positive (+) numbers while the numbers to the left of 0 are negative (–) numbers.

Directed Numbers - Number Line

We normally use the number line to indicate directed numbers and it is also used to understand changes in temperature. Using the number line, we can determine the number of steps from one directed number to another.

For example :

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.

Example 1

The temperature has gone up by 9°C.

Note :

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.

How to add and subtract directed numbers?

• Rule 1: If the signs are the same then add the numbers and keep the original sign.

  
  

• Rule 2: If the signs are different then subtract: Big number minus small number. Keep the sign of the big number.

Directed Numbers

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

 

Addition and Subtraction of Directed Numbers

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.

Directed Numbers

+ (+) which means facing the positive direction and going forward means moving forward on the number line.

+ (+) = +

That means moving from left to right on the number line which is addition.

Therefore, + (+) = +.

Let's look at some examples

–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.

Directed Numbers

+ (–)  means facing the positive direction but moving backwards. This is actually going from right to left on the number line which is subtraction.

+(–) = –

Let's look at some examples

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 ! ! !

Directed Numbers

– (+) 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, - (+) = -

Directed Numbers

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

Directed Numbers

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.

Directed Numbers

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.

Directed Numbers

+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.

Directed Numbers

– (–) 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.

Directed

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.

Directed Numbers

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.

Directed Numbers

Let us look at some long examples asking to simplify

Simplify:

a) 18 – (–16) – 3 – (–5) + 2

b) –43 – (–19) – 21 + 25

Directed Numbers

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

Directed Numbers

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.