Positive and Negative Numbers for Kids

DIAGNOSTIC TEST

Find the value of the following.

(a) 3-7 (b) (-3) - (-4)

(d) 2p + 3q ")

(c) 2-5-8 + 3

(e) 2p - 3q

(f) (-2) (+3)

given, p = -2, q = -3

(g) (-6)(- (h) (-2)

(i) 3x - 4x - 7 where x = (-3)

(j) -5(7-y) where y = -4

(k) ab cd where a = -2, b = 3 c = -5, d=4

- 4(m) 4. 7 - 5. 92

The answers are on the inside of the back cover. Check your work, and use the table below as a guide to the frame which would seem to be the most appropriate starting point for you.

If you have made a mistake in questions: Start at Frame

a b c

d e

f g h

i J k

1 m

1

16

19

24

27

N22378

]. ADDITION

Consider this problem:

You are on a ladder. If you ascend 3 rungs and then ascend 2 rungs what will be the final result? Obviously you will be 5 rungs higher up.

The problem and the solution can be written mathematically:

Ascending 3 rungs and then ascending up 2 rungs is equivalent to

(+3) + (+2)

If however you had first gone up 2 rungs and then 3 the result would be the same.

i.e. (+2) + (+3) = (+5)

So we get this rule for adding positive numbers: i ("

RULE EXAMPLES

Two or more positive numbers added together always give a positive number.

The order of addition does not matter.

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

(+4) + (+2) + (+3) = (+9)

(+2) + (+4) + (+3) = (+9)

(+4) + (+3) + (+2) = (+9)

ascending 5 rungs

(+5)

Note

We are using the symbol '+' to have two meanings.

In (+3), the '+' sign describes the 3. It indicates positive 3, and it distinguishes positive 3 from negative 3.

But in (+2) + (+3), we use the middle '+' sign to indicate the addition of the second number (+3) to the first (+2).

The operation of addition applies to the number after the symbol and this is always so for all the four basic operations, addition, subtraction, multiplication and division.

\

Complete these statements

a) (+3) + (+5) =

b) (+2) + (+7) + (+8) =

c) Does (+1) + (+3) + (+5) = (+5) + (+1) + (+3) ?

d) Does (+2) + (+4) + (+6) = (+6) + (+3) + (+3)?

a) H-3) + (+5) = (+8)

b) (+2) + (+7) + (+8) = (+17)

c) Yes. The order of addition does not matter.

d) Yes. Both are equivalent to (+12).

CM2

2. Going back to the ladder, it is obvious that the problem about ascending 3 rungs and then ascending 2 rungs would give the same numerical result if we changed the ascending to descending. The only difference would be one of direction . So:

Descending 3 rungs and then descending 2 rungs is equivalents to descending 5 rungs

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

Nor would it matter in which order we took the 2 and the 3. ;

i.e. (-2) + (-3) = (-5).

So we get the rule for adding negative numbers.

RULE EXAMPLE

Two (or more) negative numbers added together always give a negative number,

The order of addition does not matter.

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

(-2)+(-3)+(-7} = (-12)

(-3) + (-2) + (-7) = (-12)

(-7) + (-3) + (-2) = (-12)

(-5) + (0) = (-5) (0 is counted as either positive or negative)

I


a) (-2) + (-7) =

b) (-5) + (-2) + (-4) =

c) (+2) + (0) + (+3) =

d) (-2) + (0) H- (-3) =

e) Does (-3) + (-6) -f (0) = (-6) + (0) + (-3) ?

f) Does (-5) + (-7) + (-9) = (-6) + (-2) '+ (-13)?

a) (-9) d) (-5)

b) (-11) e) Yes. The order of addition does not matter.

c) (+5) f) Yes. Both are equivalent to (-21).

3. ADDITION OF POSITIVE AND NEGATIVE NUMBERS

Here again it helps to be on a ladder. Suppose you ascend 3 rungs and then descend 2. what now will be the final result? Obviously you will be one rung higher up. So:

Ascending 3 rungs and then descending 2 rungs is equivalent to ascending 1 rung

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

Now think about the order of ascent and descent. If you first descend 2 rungs and then ascend 3, you again finish 1 up. So;

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

But if we interchange the words 'ascending' and 'descending', the final result is also changed:

Descending 3 rungs and then ascending 2 rungs is equivalent to descending 1 rung

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

So if you ascend and then descend, or if you descend and then ascend, what determines the direction of your final position? Obviously the number of steps up as compared with the number of steps down. And it is the biggest number which wins'.

(+5) and then (-2) is equivalent to (+3). But (-5) and then (+2) is equivalent to (-3).

So when we add a positive number to a negative number, the numerical result is the difference in what we call their numerical values and its direction is the same as that of the bigger of the two directed numbers.

4. We can now 'interchange' the addition of positive and negative numbers.

There are three steps.

Step 1

Write down the numerical values of the signed numbers, with the biggest numerical value first.

Step 2

Subtract the smaller numerical value from the larger.

Step 3

Give the answer to Step 2, the sign of the number with the larger numerical value.

Illustrations are given in the table below. The expression to be simplified is in the first column. The directed number with bigger numerical value is underlined.

Expression

(-5) + (+7)(+3) + H:)

(-2) + (+5)(+1) + (-1)

(0) + (^5)(1) + (0)

(Steps 1 and 2)Numerical values in order of size, and the difference between them

7-5 = 24-3 = 1

5-2 = 31-1 = 0

5-0 = 57-0 = 7

(Step 3)Giving the sign of the number with the largest numerical value to the difference

(+2)(-1)(+3)( 0)(-5)(+7)

Try these for yourself

Expression

(-2) H- (+3) (+4) + (-1) (-6) + (+3)(+5) + (-2)

0 + (-6)(-7) + (+7)

Difference of numerical values

3 - 2 =

4-1 = 3

.... =3

Final signed value of the expression

( +...)(...3 ) / \

(...3 )

CM2

Expression

(-2) + (+3)(+4) + (-1)(-6) + (+3)(+5) + (-2)

0 + (-6)(-7) + (+7)


3-2 = 14-1 = 3

.6-3=35-2 = 3

6-0 = 6

7-7 = 0

Final signed value of the

expression

l+l)(+3)( -3) .( +3)( -6)( 0)

5. Here are some further examples to try.

\a) (+3) + el)d) (-8) + (+12)g) (-2) + (+5)

b) (+7) + (-9)e) (-8) + (+5)h) (+5) + (-5)

c) (-4) + (+4) f) (0) + (-8)

Expression

a) (+3) + (-1) = (+2)b) (+7) + (-9) = (-2)c) (-4) + (+4) = 0di (-8) + (+12) = (+4)e) (-8) + (+5) = (-3)f) 0 + (-8) = (-8;g) (-2) + (+5) = (+3)h) (+5) + (-5) = 0


3-1 = 2

9-7 = 2

4-4 = 0

12 - 8 = 4

8-5=3

8-0 = 8

5-2 = 3

5-5 = 0

Final signed value of the expression

(+2 )(-2 )(. 0 )(+4 )(-3 )(-8 )(+3 )

( o )

6. As we saw in Frames 1, 2 and S, we use the addition sign '+' to represent the words 'and then' in the context of two directed movements. We also saw that the final result of 'adding' two directed numbers is the same which ever of the two movements (or directed numbers comes first. In a nutshell we say 'the order of addition does not affect the result'. This important point is illustrated in this table:

Examples

(+4) + (-4)

(-4) + (+4)

(+7) + (-3)

(-3) + (+7>

(-7) + (+3)

(+3) + (-7)

Difference in numerical values

4-4 = 0

7-3 = 4

7-3 = 4

Result

(0)

(+4)

(-4)

Which pairs of the following additions are equal to each other ?

a) (44) + (-8) b) (+2) 4 (-5) c) (+8) + (-4)

d) (-5) + (+2) e) (+9) 4 (-11) f) (-8)+ (44)

g) (-9) + (+11) h) (-11) + (49)

a) and f) (44) + (-8) = (-8) 4- (44)

b) and d) (+2) + (-5) = (-5) 4 (4-2)

e) and h) (49) + (-11) = (-11) + (49)

7. MORE THAN TWO DIRECTED NUMBERS ADDED TOGETHER

We add all the positive numbers together, and then all the negative numbers together. Finally we add the totals like this:

a) (44) 4 (-2) 4 (-3) b) (43) 4 (-5) 4 (-1) 4 (+2) I____________1 I

(4-4) 4 (-5) I

(-D

Use the same technique to find the value of the following:

a) (45) 4 (-2) 4 (43)

b) (-1) + (48) 4 (4-9) 4 (-7)

C) (44) 4 (-11) 4 (45) 4 (-8)

(45) + (-6)|__________|

J(-1)

a) (46) b) (49) c) (-10)

a) (45) 4 (-2) 4 (43)

T(48)

-I-(-2)

c) {44)

T~(+6)

(-11) 4 (45) (-8)

b) (-1) 4 (48) 4 (49) 4 (-7)

(-8) 4 (417)

I(49)

(49) (-19)

~T(-10)

CM2

THE SUBTRACTION OF DIRECTED NUMBERS

If we are asked how much higher the Eiffel Tower is than St. Paul's Cathedral, we would ask ( or find out) their respective heights, and subtract the second number from the first. But, if we are asked how much higher is a temperature of 5 C above zero than a temperature of 3 C below zero, we would add 5 and 3. In the latter case we are saying that (+5) - (-3) = (+8). -

But we know that (+5) + (+3) = (+8).So-(-3) must be equivalent to+(+ 3). ;

Some people refer to this phenonemon as ' two minuses make a plus'. However the first minus sign is an operation arising from the practical situation of having to compare -3 C with +5 C. The second minus sign is an indication of direction; '3 C below zero'. So it is preferable to say that when we work with directed numbers we can change the operation of subtraction into the operation of addition providing we change the direction (i.e. the sign) of the number to be subtracted. This rule works equally well if the number to be subtracted is positive:

How much higher is 5 C above zero than 3 C above zero? Obviously 2 C higher.i.e. (+5) - (+3) = (+2)

But (+5) + (-3) = (+2) also So -(+3) is equivalent to+(-3).

Here the change is not necessary. However, it is so much more comfortable to add a positive than to subtract a negative that it is a good rule always to change the operation '-' into the operation '+' and change the sign of the number which follows the operation sign.

Rule

To subtract a positive number, add the corresponding negative number.

Examples

- (+4) = + (-4)

- (+3) = + (-3)

(+5) - (+4) = (+5) + (-4) = (+1)

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

\


a) (+6) - (+4) = (+6) + (....) = (..,

b) (+3) - (+5) = (+3) + (....) = (...

c) (-2) - (+4) = (-2) ....(....) =

d) (-5) - (+3) = ................

a) (+6) - (+4) = (+6) + (-4) = (+2)

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

c) (-2) - (+4) = (-2) + (-4) = (-6)

d) (-5) - (+3) = (-5) + (-3) = (-8)

9. Again.

Rule

To subtract a negative number, add the corresponding positive number.

Example

- (-4) = + (+4)

- (-3) = + (+3)

(+5) - (-4) = (+5) + (+4) = (+9)

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


a) (+6) - (-4) = (+6) + (....) = (.

b) (+3) - (-5) = (+3) + (....) = (.

.) c) (-2) - (-4) = (-2)

.) d) (-5) - (-3) = ....

a) (+6) - (-4) = (+6) + (+4) = (+10)

b) (+3) - (-5) = (+3) + (+5) = (+8)

c) (-2) - (-4) = (-2) + (+4) = (+2)

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

10. Here are two more worked examples with two subtractions

(1) (+2) - (+3) - (-2) (2) (-3) - (-4) -I

= (+2) + (-3) + (+2)

= (+4) + (-3)

= (+1)

= (-3)+(+4)+ (-5) = (+4) + (-8) = (-4)

\

Now two for you. Complete the following statements

a) (+4) - (+5) - (-2) = (+4) + (....) + (....)

b) (-3) - (-7) - (+5) = (-3) + (....) + (....)

a) (+4) - (+5) - (-2) = (+4) + (-5) + (+2)

= (+6) + (-5) = (+1)

b) (-3) - (-7) - (+5) = (-3) + (+7) + (-5)

= (-8) + (+7) = (-1)

ORDEE IN SUBTRACTION

Rule

We cannot change the order of the terms in subtraction.

Examples

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

But, if the order of the terms is changed

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

ere are some revision examples

I

I(+4) - (+5) + (-6) = (+4) + (-5) + (-6)

= (+4) + (-11) = (-7)


a) (-2) + (+3) - (+5) = (-2) + (....) + (....)= (....) + (....) = (.--.)

b) (+5) - (-8) + (-8) = (+5) .........= (....) + (....) = (....)

(-3) + (+8) - (-7) = (-3) -f (+8) + (+7)

= (-3) + (+15) = (+12)

Answers page 24

CM2

12. AK Complete these statements

a) (-1) + (+2) - (+5) = (-1) + (+2) + (....)

b) (+2) - (-3) + (-4) = (....)........

= (....) + (....)

= (....)

Answers page 24

13. THE SHORT FORM FOR EXPRESSING SIGNED NUMBEBS

So far we have put our signed numbers in brackets to distinguish them from the numbers we count with. But it is common practice to write them without brackets where misinterpretation is unlikely. Often, also, we omit positive signs that are not absolutely necessary.

Thus we have seen,

(+7) - (+5) - (+3) = (+7) + (-8) = (-1)

This could be written as

7-5-3 = 7-8 = -!

To evaluate expressions like, -2+5-3+7-8, we think of them as five directed numbers added together, so that we can re-order the numbers and place the positives with the positives and the negatives with the negatives.

30-2+5-3+7-8= 5+7-2-3- = 12 - 13 = - 1.

But to evaluate expressions like a - b when told that a = - 5 and b = - 3 needs care'. The danger is that of losing a minus sign.

If a = - 5 and b =-3. a-b=-5--3. And now we change - - 3 into +3.

Soa-b=-5+3=-2.

Evaluate the following

a) 3 - 5 + 6 =

b) -2 - 5 + 7 =

c) 8-8 + 0-1 =

d) -4 + 3-7 + 5 =

e) -8-5 + 6-4 =

f) -3+7-2+9- 15

a) 4, b) 0, c) 1, d) -3, e) -11, f) -4.

14. GOING ON DOING THE SAME THING

If we walk 5 miles east and stop for a rest; then walk another 5 miles east and stop for a rest; then walk another 5 miles east and stop for a rest, we shall have walked 15 miles east altogether. We could write this:

(+5) + (+5) + (+5) = (+15).

In algebra we write a + a + a. More briefly as 3a. Likewise (+5) + (+5) + (+5) can be written more briefly as 3(+5). But we must remember that 3(+5) = (+5 x 3) or (+15).

In the same way 4(+7) = (+28).

A similar result holds for negative numbers.

If we walk 3 miles west and then 3 miles west and then 3 miles west and then 3 miles west, we shall have walked 12 miles west.

That is to say (-3) + (-3) + (-3) + (-3) = (-12).

Or 4(-3) = (-12).

In the same way 7(-5) = (-5x7) or (-35).

Write down the values of the following:

a) 3(+2) b) 5(-3) c) l(+4) d) 0(-8) e) 2(0).

a) (+6) b) (-15) c) (+4) d) 0 e) 0.

15. As before, adding does not affect the sign. So

(+5) + 2 (+3) = (+5) + (+3x2) = (+5) + (+6) = (+11).

And (+5) + 2(-3) = (+5) + (-3x2) = (+5) + (-6) = (-1).

But, with subtraction, we still change the sign of the directed number and add the term, as before.

So (+5) - 2(+3) = (+5) + 2(-3) = (+5) + (-3x2) = (+5) + (-6) = (-1) and (+5) - 2(-3) = (+5) + 2(+3) = (+5) + (+3x2) = (+5) + (+6) = f1!)-

And here are two further examples with the numerical multiplication left out.

3(+5) - 2(+3) = (+15) + 2(-3) = (+15) + (-6) = (+9) 3(-5) - 2(-3) = (-15) + 2(+3) = (-15) + (+6) = (-9)


a) 2(+4) + 3(+5) d) 3(+3) - (+2)

b) 3(-2) + 4(-7) e) 4(-2) - (-8)

c) 5(-l) - 2(-8) f) 4(-8) - (-2)

a) (+8) + (+15) = (+23) d) (+9) + (-2) = (+7)

b) (-6) + (-28) = (-34) e) (-8) + (+8) = (0)

c) (-5) + 2(+8) =(-5) + (+16) = (+11) f) (-32) + (+2) = (-30)

16. EXTENDING THE WORK TO ALGEBRA

In algebra we meet expressions where letters of the alphabet stand for numbers, like this:

3L + 4S Now if L is standing for (+2) and S for (-3),

3L + 4S becomes 3(+2) + 4(-3) = (+6) + (-12) = (-6).

In such cases we say that we have substituted numerical values for L and S , and evaluated the algebraic expression.

10 CM2

Evaluate the algebraic expressions which follow by substituting the numerical values shown to the right of each expression.

\

a)

b)

c)

d)

2L +

3L -

5x +

3x -

3S

4S

2y

4y

(L

(L

(x

(x

= +3,

= +1,

= +2,

= -2.

S

s

yy

= +4)

= +2)

= +3)

= -3)

e)f)

g)h)

r+ 2q

3f + t

-3w - 4v

2a - 3c

(r =

19. MULTIPLICATION OF SIGNED NUMBERS

In previous frames we have found the value of terms such as 2(+3), 4(-2).

We must now consider expressions like

(-2)(-3) standing for (-2) x (-3)

and like (~2)(+4) implying (-2) x (+4).

How might these expressions arise in practice?

Suppose the temperature is rising at a rate of 2 C per hour, we could say that the velocity of the mercury in the thermometer is (+2) degrees/hour. By how many degrees will it have changed three hours from now? Obviously it will have risen by 6 C. The 6 comes from multiplying 2 (the speed) by 3 (the time). But the numbers all have a direction:

The velocity of the mercury is (+2)degrees Aour Three hours from now is (+3)hours Risen by 6 C is (+6)degrees So we see that (+2) x (+3) = +6

But suppose we had asked: How much higher was the temperature 3 hours ago? Well, the question is a silly one. If the temperature is rising 2 C per hour, it was 6 C lower 3 hours ago. In this case the directed numbers are:

The velocity of the mercury is (+2) degrees Aour Three hours ago is (-3) hours 6 C lower is (-6) degrees. So we see that (+2) x (-3) = (-6).

But suppose the temperature had been falling at a rate of 2 C per hour. In three hours time it will be 6 C lower.

Here the velocity of the mercury is (-2) degrees Aour In three hours time is (+3) hours 6 C lower is (-6) degrees And so we see (-2) x (+3) = -6.

And lastly, if the temperature is falling it will have been higher in past time.

So, if the velocity is (-2) degrees AourThree hours ago (-3) hours is was6C higher (+6) degrees

So (-2) x (-3) = +6

We summarise these results into two rules:

RULE 1 If two numbers of the same sign are multiplied together, the resulting number is positive.

(+2) (+3) = (+6)

(-2) (-3) = (+6)

RULE 2 If two numbers of opposite sign are multiplied together, the resulting number is always negative.

(+2) (-3) = (-6)

(-2)(+3) = (-6)

12 CM2

Complete the following

a) (+2) (+4) -.(+....)

b) (+2) (-4) = (-....)

c) (-2)(+4) = (-....)

d) (-2) (-4) = (+....)

e) (+3) (-5) = ..........

f) (+4) (-4) ...........

a) (+8)

b) (-8)

c) (-8)

d) (48)

e) (-15)

f) (-16)

g) (-3) (-6) =......

h) (+7)(0) = ........

i) (-8)(+2) = ........

j) (0)(-3) =............

k) (+4)(-7) ............

1\ / low 1 \

g) (+18)

h) (0)

i) (-16)

j) (0)k) (-28)

1) (+12)

20. MULTIPLYING MORE THAN TWO NUMBERS TOGETHER

If more than two numbers are multiplied together we can take them two at a time, e.g.

(+3)(-2)(+5) = (-6) (+5) = (-30)1 1

. (+3) (-2) (+5) (-4) = (-6) (+5) (-4) = (-30)(-4) = (+120) __f

Finding the sign of the final answer is easy. Every pair of negative signs becomes a positive sign, so only if odd negative signs exist will the final result be negative.

1 (odd) negative sign

(+ )(+ )(- ) *-(- )

2 (even) negative signs

1

3 (odd) negative signs

(+)(-)(-)(+)(+)(-) +~ H, and so on.

The only exception is when a zero is one of the factors. Then the final result is zero.


a) (+3) (-2) (+4) d) (-3)(-2)(+l)(-l)(-2)(+4)

b) (-2)(-l)(+4)(+6)(-2) e)

c) (+4)(-3)(+l)(+5)(+4) f) ( -

13

a) (-24) d) (-148)

b) (-96) e) (0) (As this is a product the zero gives the result immediately.)

c) (-240) f) (0)

21. POWERS OF NUMBERS

If a number is squared then its sign is repeated twice , and the result is a positive number, e.g.

(-2)2 = (-2)(-2) = (+4)

(+3)2 = (+3) (+3) = (+9)

For the cubes, the sign is repeated three times. and for negative numbers the result is a negative number, e.g.

(-2)3 = (-2)(-2)(-2) =,(-8)

(+3) 3 = (+3)(+3)

23. K Evaluate the following

a) (-2) (44) b) (-2) (-5) (-3)

What is the sign of the following? (Do not evaluate.)

c) (-4) , d) (-2) (+4) (-]

Answers page 24

23. AK Evaluate the following

a) (44) (-3) b) (+8)(-3)(-l)

What is the sign of the following? (Do not evaluate.}

. o) (-2)4 d) (-l) 3 (+5) 2 (-

Answers page 24

24. We can now substitute numbers into algebraic expressions involving products and powers.

Example 1

If a = (+3) and b = (-2)2

evaluate ab, b2

,* *(+3) (-2)

The sign of the result is positive (even number of negative signs)2

then (+3) (-2) = (+12)

Example 2

If a = (+3) b= (-2)2

evaluate a + b - ab

Substituting(+3) + (-2) - f+3) t-2)2

It is wise to work in small steps.

We calculate all multiplications before addition and subtraction,

From Example 1 we know that if a = (+3) and b = (-2)

ab2 = (+12).

2 2 Having calculated ab we can now write a + b - ab as

(+3) + 1-2} - 1+12)

which in turn can be written

1+3) + 1-2) + (-12) = (-11)

Example 3

Given a = (+3) b = (-2) , evaluate -4b(ab - 1) .

Remembering in such expressions brackets are always evaluated first. Also within the bracket multiplication proceeds addition and subtraction.

So, first, ab= (+3)(-2) = (-6)

Now (ab-1) = (-6) - (+1) = (-6) + (-1) = (-7)

Ignoring for the moment the minus sign in front of the 4, we work

4b(ab-l) =4(-2)(-7) =(+14) =(+56)

Finally -4b(ab-l) = - (+56) = (-56).

Evaluate the following for the values given.

a) a2 + b2 , a =(-2), b = (+1) d) 4ab2 c > a = (+2), b = (-3), c = (-1)2 2b) 3a(b-2a) 5 a = (-1), b = (-3) e) a - Sab + b , a = (+5), b = (-4)

c) 2xy-x + y3j x=(+2), y = (-2) f) a2 -4ab+4b2 , a = (-3),b = (-2)

a) a2 + b2 = (-2) 2 + (+1) 2 = (+4) + (+1) = (+5)

b) 3a(b-2a)First (b - 2a) = [(-3) - 2(-l)] = [(-3) + 2(+l) ] = [(-3) + (+2) ] = (-1)

Now 3a(b-2a) = 3(-l)(-l) = 3(+l) = (+3)

c) 2xy - x + y3 = 2(+2)(-2) - (+2) + (-2) 3 = 2(-4) + (-2) + (-8) = (-8) + (-2) + (-8) = (-18)

2 2d) 4ab c = 4(+2)(-3) (-1) = 4(+2)(+9)(-l) =4(+18)(-l) = 4(-18) = (-72)

e) a2 - Sab + fa2 = (+5) 2 - 3(+5)(-4) + (-4) = (+25) - 3(-20) + (+16) = (+25) + 3(+20) + (+16)= (+25) + (+60) + (+16) = (+101)

f) a2 - 4ab + 4b2 = (-3) 2 - 4X(-3) x (-2) + 4X(-2) 2 = 9 - 24 + 16 = 1

25. As we have seen, positive directional signs are often omitted in routine work, e.g.

2 Evaluate ab given a = 3 (i. e. (+3)),b=-2 (i.e. (-2))

ab2 = 3(-2) 2 = 12 (i. e. (+12))

Rework the solutions to Frame 24 omitting the directional '+' signs, and leaving out brackets where these are unnecessary.

a) a2 + b2 = (-2) 2 + I 2 = 4 + 1 = 5

b) (b - 2a) = -3 - 2 (-1) = -3 + 2(1) = -3 + 2 = -1 3a(b - 2a) = 3(-l)(-l) = 3(1) = 3

16 CM2

c) 2xy - x + y2 = 2(2)(-2) - (2) + (-2) = 4(-2) - 2 + (-8) = -8 - 2 - 8 = -18

d) 4ab c = 4(2)(-3) 2(-l) = 8(9) (-1) = 72(-l) = -72

e) a - Sab + b = (5) - 3(5)(-4) + (-4) = 25 - 15(-4) + (16) =25 + 60 + 16 = 101

O p p nf) a - 4ab + 4b = (-3) - 4 x (-3) x (-2) + 4 X (-2) = 9 - 24 + 16 = 1

26. K Evaluate the following expressions for the values given2 2

a) 3a + 2ab + b a = 2, b = -1

b) xy(l + 3x) x = -2, y = 4

Answers page 24

26. AK Evaluate the following expressions for the values given3 2

a) a + ah a = -1, h = -2

b) (x + y)(x-y) x =-3, y =-2

Answers page 24

27. USING SIGNED NUMBERS OTHER THAN WHOLE NUMBERS

For simplicity we have used whole numbers throughout this unit up to this point. But the principles we have been using apply to any signed number.

Examples - Addition and Subtraction

a) (+2.8) + (+5.6) = (+8.4)

b) (+4. 8) - (+3. 2) (Short form, 4.8 - 3.2 = 1.6

= (+4.8) + (-3.2)

= (+1.6)

c) (+2. 9) - (+5. 6) (Short form, 2. 9 - 5. 6 = -2. 7)

= (+2. 9) + (-5. 6) Note. The difference between 5. 6 and 2. 9 is 2. 7.

= (-2. 7)

d) (-2i) - (-if ) (Short form, -2^ - if = -4j )

= (-2*) + (-if)r' 1e) (+1 ) - (+4 ) (Short form ' X 3 " 4 6 = ~\ }

17


a) (+2.4) + (+3.2) e) (-2.4) - (-3. 2) i) (+2r )" Ci

b) (+2.4) + (-3.2) f) 2.4-(-3. 2) i) (-2- ) - (+3^)O

c) (+2.4) - (+3. 2) g) 3 2- k) (+2r ) - (+3^)23 d 2d) (-2.4) + (+3. 2) h) 4-37 1) (-4)- (-4)

a) (+5.6) e) (-2.4) + (+3.2) = (+0.8) i) 57D

b) (-0.8) (3.2) -2.4 = 0.8 f) 5.6 j) (-2^) + (-s|) = (-s|)c) (-0-8) g) 1- (1 r ) k) (+2r) + (-3-) = (-1-),

d) (+0. 8) h) -!-_ (3^ - 2\ = 1- )

28. FURTHER MULTIPLICATION

We now illustrate examples in the multiplication of signed numbers in fractional form.

a) (+2. 4) (+0.2) = (+0.48)

2 I 1 = _4 5 _4/ x 5

2c) (-1.1) =(+1.21) ( even number of negatives) ' ' e ' Rough working

1.2d) (-0.4)(-1.2)(-2. 5) = (-1.2) (odd number of negatives) x 0-4

0.48 x 2.5

1.2

e) Find the value of 4ab - (a + b) given a = 1. 2, b = -0. 8 Substituting (and using the short form) 4(1.2)(-0. 8) - (1.2 + (-0.8) ) = 4(-0. 96) - (0.4) = -3.84 - 0.4 = -4.24


a) (+2.7)(+0. 03) d) (+4.1)(-0.5)(-2. 2)

b) (-1.6) (-1.5) e) Find the value of 4 - 3ab2 , given a =-0.3 b = 1. 53 2

c) (-1-2) f) Evaluate 4 - 3(a - b ) using the values given in (e)

18 CM2

a) (2. 7 x 0. 03 = 0. 081) e) 4 _ 3ab2 = 4 _ 3(_ Q 3)(1 _ 5) 2 f) 4 - 3(a - b2 )

b) (1.6x1.5 = 2.4) =4-3 (-0.3) (2.25) = 4 - 3 (-0.3 - (1.5)2

c) (-12) 3 = 1728 =4-3 (-0.675) = 4 - 3 (-0.3 - 2.25) So (-1.2) 3 = (-1.728). = 4 + 3(+0.675) = 4 _ 3 ( _2 55)

= 4 + (+2. 025)= 6.025 =4+3(2.55)

d) 2. 2 x 0.5 = 1.1,1.1 x 4.1 =4.51 =4+ 7.65

= 11.65

29. DIVISION OF SIGNED NUMBERS

The rules given in Frame 19 for the multiplication of whole numbers can be extended to the operation of division.

jtULE 1 (Restated)If any two numbers of the same sign are multiplied or if one is divided by the other the resulting number is positive.

Examples

| .'I = (+2.0) and _'' = (+2.0)

BULK 2

If any two numbers of opposite sign are multiplied or if one is divided by the other the resulting number is negative.

Examples

= (- 2 -> and - = (-2 ->Evaluate the following

Ia)

e)

a)

e)

+50+5

+25+5

+10

+5

b) +25-5

f) +3.2-1.6

b) -5

f) +2.0

c) -25+5

g) -4.8+8.0

c) -5

g) -0.6

d) -25-5

h) -0.-0.

d) +5

h) +40

401

30. Using the short form of expressing signed numbers we have

: a\ 50 b^ 25 c) -25 ,_ d) -25 ,_ .*/ = 10 = -5 ~T~= -5 ~= 5, and so on.

19

31. FRACTIONS INVOLVING SEVERAL FACTORS

If several factors are involved in an expression the sign of the resulting number is found as in multiplication. The total number of negative signs is counted and if this is an odd number, the number is negative. An even number of negative signs gives a positive answer.

In there are three negative signs. Three is an odd number, so the result is2x3 negative. Its numerical value is the value of; = 1

(-6X-1)

(+2)(-3) In L there are six negative signs. Six is even, so the result is positive . Its

numerical value is~ = - x = 3 6 x I3 X 1

= (+3)

Some people find cancelling a source of error. You may prefer to work out the top (numerator) and bottom (denominator) of the fractions separately and then finally divide.

' ;, = -T' = (-1) (Taking account of signs at each stage.) --

(-6) (-If

(+18) (+6) = (+3)


a) (+2) (-2) (+3) (-4)

b) (+3) (-2) (-3)2 2 c) (+2) (-3)

(+4) (-8)

a) ("125 x 2

x 6125 ~ 3

(Five negative signs)

32. Evaluate the following

a) (+0.28) (-2.0) (-3.0) (-1.4)

b) 5.4 (-0.2) 1.8 (-6.0)

c) (-1.4) 0. 7(-2.0)

(-u. OP; __4i_2 _ _o 13 (+4.2) 42

-1.08-10.8

10.8 + 108 = 0.1 c)

+1.96 -1.4

19.614

= -1.4

33. K Work the following

ab a) y = whe re a = 2. 0, b = -4. 9 and c = 70Calculate the value of y.

20 I

b) Find t given2 mxJJ-IA. . _, _t = r- , where m = -1. 5

Q

c) If v = uw and u = -4. 0rt

find v.

x= -0. 6

w = -7.2 r = -2.4 and t = 5

CM2

Answer page 25

33. AK Work the following

a) Find v given

2 T .2 v=-Lt L = 5.4

b) If y = s (s - 3) and s = 2. 8, find the value of y.

c) IfH = Mrv2 andM=|, r = -^, v =-, find the value of H.

Answer page 25

When you are satisfied that you have understood the material in this unit, read carefully through the summary on page(s) 22 and 23 referring back, as necessary, to the relevant part of the text until you have understood it. When you are satisfied that you have understood all the material in the text, you should work through the post-test on page 26 without referring to the text. Next, check your answers to the post-test against those provided inside the back cover. Finally, discuss your work with your tutor, if you have one, and decide what to do next.

21

SUMMARY

THE USE OF THE '+' AND '-' SIGNS

Two of the uses are,

i). To indicate the way in which numbers should be combined together.

10 + 3 = 13

10 - 3 = 7

ii) To indicate some 'direction' to a numerical value.

Credit of 3 = +3

Debit of 3 = -3

(Both +3 and -3 are said to have a NUMERICAL value of 3)

Manipulation of signed numbers

ADDITION

1. If all numbers are positive (+) the result is positive

e.g. (+3) + (+2) = (+5) and

2. If all numbers are negative (-) the result is negative

e. g. (-3) + (-2) = (-5) and (-s|) + (-1 1 ) = (-l| )

3. Mixed numbers

(a) Two numbers

The difference between the numerical values is found and the resulting number has the sign of the number with the largest numerical value.

(+4) + (-5) = (-1) 5-4 = 1, (-5) has larger numerical value

1 1 4 1141(-1) + (+2 ) = (-*" ) 2r- - 1 = (+2) has larger numerical value266 6 2 6, 6

(b) More than two numbers

We add all the positive numbers together, and then all the negative numbers together, finally we add the totals.

(+3) + (-5) + (-1) + (+2)

SUBTRACTION

Subtraction of a signed number is equivalent to the addition of the number with its sign changed.

So -(+4) = +(-4) . And - (-3) = + (+3)

And (-7) - (+4) - (-3) = (-7) + (-4) + (+3) = (-8)

22

MULTIPLICATION

1. Just as b + b + b can be written 3b, so (-4>H-4)+(-4) can be written 3(-4). Its value is (-4 x 3) = (-12).

2. Multiplication of signed numbers

The RULES are

(I) Two signed numbers of the same sign yield a positive number.

(+3) (+2) = (+6); (-3)(-2) = (+6)

(II) Two signed numbers of opposite sign yield a negative number.

(+3)(-2) = (-6) ; (-3)

Answers to K aad AK Frames

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

= (-?) + (+3) = (-4)

b) (+5) - (-8) + (-8) = (+5) + (+8) + (-8)

= (+13) + (-8) = (+5)

t^.AK a) (-1) + (+2) - (+5) = (-1) + (+2) + (-5)

= (-6) + (+2) = (-4)

b) (+2) - (-3) + (-4) = (+2) + (+3) + (-4)

= (+5) + (-4) = (+1)

18.K a) 3(3) + 2(4) = 9 + 8 = 17

b) (-1) + 2(5) = -1 + 10 = 9

c) 4(-3) - (-4) = -12 + 4 = -

18.AK a) -(-2) + 3(0) = 2 + 0 = 2

b) 5(4) + 3(-l) = 20 - 3 = 17

c) 2(-l) + 3(0) + 4(-2) = -2 - 8 = -10

23,,K a) -8 b) -30

c) negative d) negative

23. AK a) -12 b) 24

c) positive d) positive

26. K a) 9 b) 40

a) 3(4) + 2(2)(-1) + (-1) 2 = 12 + (-4) + 1 = 9

b) (-2)(4) [l + 3(-2)J = -8(1 - 6) = -8(-5) = 40

20. AK a) -5 b) 5

a) (-1) 3 + (-l)(-2) 2 = (-1) + (-1)(4) = -1 - 4 = -5

b) [-3 - 2 ] [-3 - (-2)] = (-5)(-3 + 2) =

33. K a) y = -0.14

' 14 y = (2.0)(-4r9) = (- -~)

10

= (-0.14)

b) t = -0.108

t=(-1.5)(-0.6) 5

= (-0.108)

c) v = -2.4

v = (-4.(-2, 4) (5.0)

3.04. 0 x fr.-a ~2r4 x5.o

33. AK a) v=5.184 b) y = -1.568 c) H = - 32a) v =| (5.4)(-1.2) 2 = (3.6)(1.44) = 5.184

2 2b) y=(2. 8) (2.8 - 3) = (2. 8) (-0. 2) = -1.468

0' 464

25

POST-TEST

1. Find the value of

a) 5-8 b) -7 + 6 c) 2-8 + 4-9

d) -2. 74 + 4.26 e) 2. 74 - 4.26 f) 3^ - 4^D ^1

2. Given a = 3, b = 2, c = -6, x = -1, find the value of

a) a. b) be c) (x) d) acxb 2

e) ax + bx + c f) 4a - 3c g) 5(x + b)

h) (-2)(a-c) i) 5 ax j) -3 ax3bc 4b

2

26 CM2

ANSWERS TO THE POST-TEST

(8) (6=7

(8)- (f)

tot-)' (9-)) 9 (T)

81 = (T-M9-HS)

Positive and Negative Numbers for Kids

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