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Numbers
Directed Numbers
Examples
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Practice Questions
 


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Numbers Units

Many of the numbers we use represent situations which have directions as well as size
The 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 are used in Mathematics, Engineering, Business and the Sciences.
For example: -15,  8,  100,  -100,  -3.5,  0.33,  -0.75   are directed numbers.
In the above example -15,  8,  100, -100 are called integers.
When writing positive numbers you can leave the positive sign and just write the number.
eg. +8  as  8

If  a directed number is a whole number, it is called an integer.

 

 

 

 

 

 

 

 

 

 

 

 

Example

Addition of Directed Numbers

Let's consider   -3  + + 4
In this problem  + and +  signs are side by side.There is no number in between them. So the two positive signs which are side by side gives a positive sign.
Remember this,

                             Two like signs give a positive sign
                                              
+ +  =  +

                       
    -3  + + 4  =  - 3  +  4
                                           
=    1

Sometimes directed numbers are written as



Let's consider          - 3   +  - 4
In this problem  positive (+) and negative (-) signs are side by side without a number in between them. Two unlike signs which are side by side gives a negative (-) sign.

Remember this:

                                Two unlike signs give a negative sign.
                                                      + -  =  -


                                -3   +  - 4  =  -3  - 4
                                                =   - 7

 

Subtraction of Directed Numbers

Let's consider -3 - - 4
In this problem the middle negative(-ve) signs are side by side without a number in between them. So the like signs which are side by side, always give a positive sign.

-3 - - 4   =   -3  +  4   =    1

This problem can also be written as

-3 -(- 4)   =   -3  +  4        =   1

and

 

Let's consider -3 - + 4
In this problem negative(-ve) and positive(+ve) signs are side by side without a number in between them.
That is two unlike signs are side by side, which gives a negative(-ve) sign.

-3 - + 4  =   -3  -  4      =   - 7

This problem can also be written as

and

-3  - (+ 4)   =   -3  -  4    =   -7

 

Multiplication of Directed Numbers

Let's consider    -3   -4

When multiplying directed numbers

Two  like signs always give a positive(+ve) sign   

Two unlike signs always give a negative(-ve) sign

(-ve)           (-ve)       =      (+ve)

-3     -4   =    12

-3     +4   =   -12

(-ve)       (+ve)     =     (-ve)

 

Dividing directed numbers

When dividing directed numbers

Two  like signs always give a positive(+ve) sign   

Two unlike signs always give a negative(-ve) sign

Let's consider  -3    -4 
Two like signs give a positive sign

Let's look at this problem.
-8 + (6 -3 - 2) -4 - -8
In this problem you can see all different operations, when we have more than one operation we have to follow the order of operations.
ie.    1. Brackets
       2. Division or multiplication from left to right
       3. Addition or subtraction from left to right

Let's do the brackets first  (6 -3 - 2), inside this bracket you can see the multiplication and the subtraction signs. Remember the order, we have to do multiplication first and then the subtraction

O.K        6 -3    = - 18  (multiplying two unlike signs,gives a negative sign.)
              Then we subtract 2
              (6 -3 - 2)  =  -18  -2  =  -20
Now our problem
    -8 + (6 -3 - 2) -4 - -8  =  -8 + (-20) -4 - -8
What's next?,  addition,subtraction or division.
Remember the order, division comes before addition and subtraction.
O.K         (-20) -4  =  +5 (dividing two like signs, gives a positive sign.)
Now our problem
    -8 + (6 -3 - 2) -4 - -8 
=  -8 + (-20) -4 - -8
=  -8 + +5 - -8
(two like signs without a number in between them gives a positive sign)
=  -8 + 5 + 8   now we are left with only addition and subtraction signs,so we can work out this problem from left to right.
=  -3 + 8
=  5

 

 

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Have a Go

Problem 1

(a) 6 +(+10)

(b) 5 - -3

(c) -4 -6

(d) +6 -2

 

See Solution

 

 

Problem 2

(a) 5 + - 18 - - 9

(b) 23 - (5 - - 7)

(c) 5 - 4 + - 8

(d) -25 [5 -2 -(-3 5)]

 

See Solution

 

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Practice Questions

Question 1
9 + ( - 3 -4 )

 

 

Question 2
- 3 - ( - 8 + 9 )

 

 

 

Question 3
- 2 - 2 -5

 

 

Question 4
8 - 4 -2

 

 

Question 5
6 - [15 - (- 3 - 2 )]

 

 

Question 6
- 1 [- 8 ( - 4 + 6 )]

 

 

 

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Solution 1
(a) 6 + (+10)  =  6 + 10               ( + + = +, Two like signs give a +)
                     
=  16

(b) 5 - -3        =  5 + 3                 ( - - = +, Two like signs give a +)
                     =   8

(c) -4 - 6    =   + 24                   [(-) (-) = +]
                     =   24

(d) + 6 -2   =   - 3                    [(+) (-) = - ]

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Solution 2
(a) 5 + - 18 - - 9   = 5 - 18 + 9     (+ - = - ,  - - = +, left to right )
                             = -13 + 9
                             = - 4

(b) 23 - (5- -7) = 23 -(5 + 7)
                                         (first, operations inside the brackets, - - = +)
                        
= 23 -(12)             ( then the brackets )
                         = 11

(c) 5 - 4 +  -8 = -20 + -8  (remember the order, first before +) 
                          = - 20 - 8    (+ - = -)
                          = -28

(d) -25 [ 5 -2 -(-3 5) ]  
( In this problem, ( )brackets are inside the [ ] brackets, so first you have     to do the inside brackets and then the outside brackets.)
    
= - 25 [5 -2 -( -15)]    ( -3 5 = -15  )
     = - 25 [ -10 - (-15) ]      
(5 -2 = -10)
     = -25 [-10 + 15]             ( - - = + )
     = -25 [5]
     = - 5                                  
( - + = - )

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