CONNECT: Directed Numbers
SORTING THE POSITIVES FROM THE NEGATIVES
Firstly, why do we need negative numbers? Well, let’s say I owe you money –
I am in debt to you. How would we write this if we didn’t have negative
numbers? If I owe $2, then rather than writing “IOU $2”, we simply write “–2”,
or “-2”, or sometimes “-2” or “–2”!
Further examples of “everyday negatives”
NEGATIVE
“ZERO”
Downstairs
The landing between
stairs
0°C or 32°F
Below freezing (Brrr!)
You owe money!
You failed the subject
You feel lousy!
You dislike someone
Left
Down
You have no money but
you don’t owe any either!
Passmark (could be 50%,
for example)
You feel OK
You feel neutral towards
someone
Zero/0/Centre/Neutral
POSITIVE
Upstairs
Above freezing
You have money
You passed the subject
You feel great!
You really like someone
Neutral
When we talk in positives and negatives, direction is implied. Positives and
negatives go in opposite directions from each other.
You probably remember from school that there is a special line called the
Number Line. In the middle of the Number Line is zero (0). Positive numbers
increase in size (magnitude) from 0 to the right and negative numbers
increase in magnitude from 0 to the left:
1
Right
Up
The arrows at each end of the number line tell us that the line will continue
indefinitely in both directions. So, to the right of the number line shown, we
would continue with 4, 5, 6, … and to the left with -4, -5, -6, … We can also
use a vertical number line, with the positive numbers increasing upwards and
the negatives increasing downwards.
The opposite of 3 is -3 and the opposite of -516 is 516. (Note, we could also
write “+516” but generally we leave out the + sign.) In a pair of opposites,
each number is exactly the same distance from 0. For example, both 3 and -3
are 3 units from 0. Both 516 and -516 are 516 units from 0, but in opposite
directions. So their size, or magnitude, is the same, but their direction is
opposite.
A very quick exercise for you:
following numbers:
1. -5
7. 0
2. 13
3. -49
Write down the opposite of each of the
4. 2560
5. 3.87
6.
3¼
You can check your answers with those at the end of this resource.
Entering a negative number on a calculator
Use the (-) key. To enter -2, for example, you would press (-)
2 .
Calculating with positive and negative numbers
To calculate using positives and negatives, it may be easiest to think of
banking transactions. Positive numbers would represent money in the bank,
negatives would represent debt.
(a)
Addition
Keeping the idea of banking transactions in mind, adding a positive number
would mean you are depositing money into your account, while adding a
negative would mean you are withdrawing.
Example: 5 + 7.
We know this would be 12: (using banking: we start with a balance of ($)5
and deposit ($)7 to end up with ($)12.)
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Now, what about -6 + -3?
This means we start with a debt of ($)6 but want to withdraw a further ($)3 so
we would be a total of ($)9 in debt, that is, -6 + -3 = -9.
(Notice that when we add 2 positive numbers, we get a positive result, and
when we add 2 negatives we get a negative result. Both times the size (or
magnititude) of the answer is larger than the numbers we started with.)
Now, what about -6 + 2?
We are starting with a debt of ($)6 but depositing ($)2, so our debt will be
lessened. We will go up ($)2 to a debt of ($)4, so -6 + 2 = -4.
Last one: 5 + -7.
This would mean we would be trying to withdraw ($)7 when we only have ($)5
in the account. If our bank lets us do this, we would end up in debt ($2), that
is, 5 + -7 = -2.
(When we add a positive number and a negative number, the sign of the
answer will always be the same as the sign of the larger number we are
adding. For example, -3 + 9 will end up as + because 9 is larger than 3. But
3 + -9 will end up as negative. In both cases the size of the answer is 6.
-3 + 9 gives 6 and 3 + -9 gives -6.)
Here are some for you to try. Remember you can check your answers using
your calculator (this is a VERY good idea!) and also by looking at the answers
at the end of this resource.
1. -5 + 3
6. 4 + -8
2. 6 + 4
7. -5 + 5
3. -5 + -7
4. -4 + -2
8. -3 + 4 + -2
5. 5 + -2
(b) Subtraction
Think about 6 – 4 for example, especially in terms of the banking transactions.
Subtracting is the same as withdrawing. (Essentially, subtraction is the same
as adding a minus number.) 6 – 4 = 2, but what about 6 – 8? We are trying
to take (withdraw) ($)8 but we only have a balance of ($)6 to start with. So we
would be in debt ($)2, in other words, 6 – 8 = -2. Try these on your calculator
(just to make sure!). You would enter: 6 – 8 = .
What about -6 – 8? Here we would go further into debt. We would start with
a debt of ($)6, then withdraw a further ($)8, and end up with a debt of ($)14,
that is, -6 – 8 = -14.
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Subtracting a negative
Now, the tricky one! Let’s say we see 5 – -2. What’s going on here?! One
way to think about it is that the – - cancel each other, just like a “double
negative” in English. For example, I could say “he is not unlike my brother”; I
would mean that he is like my brother. “Not unlike” = “like”. The negatives
cancel! “He is not innocent” means “he is not not guilty” means “he is guilty”!
It is the same when we subtract a negative. So, 5 – -2 is the same as 5 + 2,
which gives 7. (We can also think back at our banking transactions. – -2
might be interpreted as cancelling a ($)2 withdrawal that you’ve previously
made so you are ($)2 better off.)
Try this calculation on your calculator. You need to be careful which “–“ you
use. To enter 5 – -2, use the following: 5 – (-) 2 = .
Here are some subtractions for you to try. Check them on your calculator and
with the answers at the end.
1. 3 – 2
6. -4 – 6
(c)
2. -3 – 2
7. 4 – -6
3. 3 – -2
4. -3 – -2
5. 4 – 6
8. -4 – -6
Multiplication
Perhaps the easiest way to think of multiplication is that you are simply
repeating a deposit or withdrawal a number of times.
Example, you know that 3 x 2 is 6; but think about what is happening in terms
of deposits here: you are making 2 deposits of ($)3, or you could even
interpret it as making 3 deposits of ($)2.
What about -3 x 2? We could say we are making 2 withdrawals of ($)3 and
would end up with a debt of ($)6, so -3 x 2 = -6.
Similarly, 3 x -2 can be interpreted as making 3 withdrawals of ($)2. Notice
that the order in multiplication does not matter (isn’t that lucky!) so we still get
-6.
Now, the tricky one. -3 x -2 = 6. The result is positive because this time we
are making 3 cancellations of withdrawals of ($)2 each so we are better off
($)6. (or the other way round: 2 cancellations of ($)3 each – we are still
better off ($)6!
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A general rule of thumb that you probably learned at school for multiplication
is: like signs give +, unlike signs give –. So,
+ x + = +;
– x – = +;
+ x – = –;
– x + = –.
Here are some for you to try. Please check them with your calculator, as well
as from the answers at the end:
1. 3 x 5
(d)
2. -3 x 5
3. 4 x -3
4. -5 x -2
Division
We can think of division as “how many?” For example, 6 ÷ 2 implies “how
many deposits of ($)2 are there in ($)6? There would be 3 deposits, so 6 ÷ 2
is 3.
-6 ÷ -2 implies “how many withdrawals of 2 in a debt of 6? There would be 3
again. So -6 ÷ -2 = 3.
What about 6 ÷ -2? How many withdrawals of 2 in a balance of 6? But this
doesn’t make sense unless the withdrawals were cancelled. Ahah!! That
means there must have been 3 cancellations of withdrawals, so 6 ÷ -2 is -3.
Similarly, -6 ÷ 2 implies “how many deposits of 2 in a debt of 6? Again it does
not make sense until we think about the cancellations. So, -6 ÷ 2 = -3.
Note that our answers are similar to multiplication: like signs give +, unlike
signs give –. So
+ ÷ + = +;
– ÷ – = +;
+ ÷ – = –;
– ÷ + = –.
Here are some for you to try. Again, please check them with the calculator
and with the answers at the end.
1. 12 ÷ 2
5
2. -8 ÷ 4
3. 10 ÷ -5
4. -15 ÷ -3
Raising a negative number to a power
(For more information you can refer to CONNECT: CALCULATORS: GETTING
TO KNOW YOUR SCIENTIFIC CALCULATOR and CONNECT: Powers and logs:
POWERS, INDICES, EXPONENTS, LOGARITHMS –THEY ARE ALL THE SAME!)
When we raise a number to a power, we are simply multiplying it by itself a
certain number of times, which is indicated by the power.
So, for example 32 = 9, and, as well, (-3)2 = 9. Notice that the brackets here
are essential. If we do not have brackets around the -3, we get -9 because
-32 means -(32) as the squaring takes precedence over the –. You must also
use brackets when entering (-3)2 into the calculator.
Examples:
(-5)2 = -5 x -5 = 25.
(-2)3 = -2 x -2 x -2 = -8.
(-3)4 = -3 x -3 x -3 x -3 = 81.
Try these (and check your answers):
1. (-5)3
2. (-4)2
3. (-1)5
4. 23 – 32
If you need help with any of the Maths covered in this resource (or any other
Maths topics), you can make an appointment with Learning Development
through Reception: phone (02) 4221 3977, or Level 3 (top floor), Building 11,
or through your campus.
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ANSWERS
Writing the opposites:
1. 5
2. -13
3. 49
4. -2560
5. -3.87
6. -3¼
7. 0 (the only number that is its own opposite).
Addition:
1. -2 (you are depositing 3 but your debt has not cleared yet so the answer is
still negative – you would have to deposit 5 to clear your debt.)
2. 10 (straightforward)
3. -12 (you are going further into debt)
4. -6 (you are going further into debt)
5. 3 (you are withdrawing 2 from a balance of 5)
6. -4 (you are withdrawing 8 from a balance of only 4)
7. 0 (you are clearing your debt!)
8. -3 + 4 gives 1, then 1 + -2 gives -1 so the answer is -1.
Subtraction:
1. 1 (straightforward)
2. -5 (you are going further into debt)
3. 5 (the two minuses cancel out)
4. -1 (we would get -3 + 2)
5. -2 (you are going into debt again)
6. -10 (you are going further into debt!)
7
7. 10 (the withdrawal has been cancelled, you are better off by 6)
8. 2 (the withdrawal has been cancelled, you are better off by 6)
Multiplication
1. 15 (straightforward – as long as you know your tables!)
2. -15 (5 withdrawals of ($)3 each)
3. -12 (4 withdrawals of ($)3 each)
4. 10 (5 cancellations of ($)2 withdrawals or vice versa)
Division
1. 6 (straightforward)
2. -2 (does not make sense without thinking about cancellations)
3. -2 (does not make sense without thinking about cancellations)
4. 5 (how many withdrawals of ($)3 made a debt of ($)15?)
Raising a negative number to a power
1. -125
8
2. 16
3. -1
4. 8 – 9 = -1