Chapter 4.4 - Subtraction of Negative numbers

In the previous sections we learned the rules for adding a negative number to another negative number. With that, the rules for addition are complete. In this section we will see subtraction.

Before starting the discussion on subtraction, let us once again see the features of addition. That is.,
• addition of a negative number to a positive number. And 
• addition of a negative number to another negative number. 

The fig.4.10 below shows this addition process in terms of x and y. 

Fig.4.10 Addition of a negative number to another number x

x is the number that we are having. It can be positive or negative. -y is the negative number that we are going to add to x. Let us see the various possibilities:
■ x is positive
• x is positive and numerically greater than y
In this case, -y will take away some portion from x. size of x will decrease but it will stay on the right side of zero on the number line. That is., the result will be positive, but lesser in size than x
• x is positive and numerically less than y.
In this case, -y will take away some portion from x. The size of x will decrease to such an extent that the result will be on the left side of zero. That is.,the result will be negative. We can say, the x which was on the positive side, will fall into debt.
■  x is negative
When x is negative, the -y has the same effect on x whether x is numerically greater than or lesser than y: 
x is already in debt. The -y will come and increase the debt.

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We have already seen in the previous section that the rules to find the result in all the above cases is the same: Use either Rule 1 for adding a +ve number and a -ve number. Or Rule 2 for adding two -ve numbers. Here we are not discussing about those rules. We have already discussed them in detail in the previous sections. Here, we want to know the effect that the -y cause on x, when an addition is carried out.

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So we see that, in any case we take, when a negative number ‘-y’ is added to another number (positive or negative), the result is a ‘loss’. So when an addition is carried out, -y is ‘bad news’. Then why don't we keep it away? As if we don't want any business with it? We can't. Because of the ‘+’ sign shown in red color in the fig.4.10. The ‘+’ sign tells us that we have to bring the ‘-y’ in. It is part of the calculations. We cannot discard -y, and so we cannot avoid the ‘losses’ that it will bring. The -y is a burden that causes loss.

But what if the ‘+’ sign shown in red color is a ‘-’ sign? The fig. will look as shown below:

Fig.4.11 Subtraction of a negative number from another number x

In that case, the opposite of ‘loss’ should happen. That is., we must receive 'gains'. The ‘-’ sign shown in red color will cause the subtraction of a ‘burden’ called ‘-y’. Subtraction of a burden is a ‘gain’.

■ If x is already positive, it will become more positive. because it gains by an amount y.
■ If x is negative 
     • If x is negative, it is already in debt. When subtraction is carried out, some burden equal to y will be removed. So that the debt decreases.
    • If x is negative, and y is numerically larger than x, all the debt will be paid off, and the result will become positive.
    • If x is negative, and y is numerically equal to x, it means that there is just sufficient quantity to pay off all the debt. The result will be zero. Being zero is better than being negative. So in this case also, a burden is removed.

Let us see some examples for the above cases:

Example 1: Let x = 2. It is positive. 
• We are going to remove a burden of -y = -5 from this 2. 
• How do we do it? By subtracting -5 from 2. That is., 2 - (-5). 
• Subtracting a 'burden which cause a loss of 5' is a gain of 5. So we can write:
• 2 - (-5) = 2 + 5 =7
Example 2: Let x = -7. It is negative. 
• We are going to remove a burden of -y = -4 from this -7. 
• How do we do it? By subtracting -4 from -7. That is., -7 - (-4). 
• Subtracting a 'burden which cause a loss of 4' is a gain of 4. So we can write: 
• -7 - (-4) = -7 + 4 = -3 (using the Rule 1 )
So we see that the debt of -7 has now decreased to -3. That is., to clear off the debt only 3 is required now in place of the previous 7. So a burden of 4 has been removed.
Example 3: Let x = -3. It is negative. 
• We are going to remove a burden of -y = -9 from this -3. 
• How do we do it? By subtracting -9 from -3. That is., -3 - (-9).
• Subtracting a burden which cause loss of 9 is a gain of 9. So we can write:
• -3 - (-9) = -3 + 9 = 6 (using the Rule 1)
So we see that the debt of -3 has been totally paid off. And an additional asset of 6 is created.
Example 4: Let x = -3. It is negative. 
• We are going to remove a burden of -y = -3 from this -3. 
• How do we do it? By subtracting -3 from -3. That is., -3 - (-3).
• Subtracting a burden which cause loss of 3 is a gain of 3. So we can write:
• -3 - (-3) = -3 + 3 = 0 (using the Rule 1)
So we see that the debt of -3 has been totally paid off. 

In all the above cases we can notice one thing: There is a ‘-’ sign in between the two terms. There is also another ‘-’ sign inside the parenthesis ‘( )’. These two combine together to form a ‘+’. 

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This gives us the general rule for subtraction. We will call it Rule 3:
When subtracting an negative number from any other number (positive or negative):
Combine the two ‘-’ signs into one ‘+’ sign. Then it will become an addition problem.

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We will see some solved examples which will demonstrate the above rule:

Solved example 4.10
Evaluate the following:
(i) 7 - (-4)              (iv) -794 - (-104)
(ii) -8 - (-9) (v) 401 - (-15)
(iii) 15 - (-21)        (vi) 386 - (-3)
Solution:
(i) Combining the two ‘-’ signs we get
 7 - (-4) = 7 + 4 = 11
(ii) Combining the two ‘-’ signs we get
 -8 - (-9) = -8 + 9 = 1 (using Rule 1)

In this way, all the remaining problems can be done:
(iii) 15 + 21 = 36 (v) 401 + 15 = 416
(iv) -794 + 104 = -690 (vi) 386 + 3 = 389

In the above questions, the second term which is negative, was put inside a parenthesis '( )'. This was for clarity. We may get questions with out such parenthesis. In such cases we must visualize as subtracting a negative quantity. The steps are the same. Let us do such problems:
Solved example 4.11
Evaluate the following:
(i) 12 - −5 (ii) -75 - −2

(iii) -89 - −4     (iv) 234 - −4
Solution
(i) 12 - −5 ⇒ 12 - (−5) = 12 + 5 = 17
(ii) -75 - −2 ⇒ -75 - (−2) = -75 + 2 = -73
(iii) -89 - −4 ⇒ -89 - (−4) = -89 + 4 = -85
(ii) 234 - −4 ⇒ 234 - (−4) = 234 + 4 = 238

So we have seen the details of addition as well as subtraction. We will now see some problems that involves both these operations:

Solved example 4.12
(i) -7 + 4 - (-3) + 2 +11
(ii) 15 - 11 - 12 - (-29) + 17
(iii) -231 -362 + 34 - (-56) +27 -15
(iv)  -19 - 41 + 23 - (-4) - 81 + 27 - (-90)

Solution:
(i) Let us group the terms into groups of twos:
 -7 + 4 - (-3) + 2 +11 
⇒  -7 + 4│ - (-3) + 2│+11 
⇒ -3│+ 3 + 2│+11 [∵ since - (-3) = +3]
⇒ -3 +5 +11
⇒ -3 + 5│ +11 = 2 +11 = 13

It may be noted that the vertical green line is drawn just after a term, before the sign of the succeeding term. In this way, the resulting number within each  group will have a 'sign' of it's own.

(ii)  Let us group the terms into groups of twos:
15 - 11 - 12 - (-29) + 17
⇒ 15 - 11│ - 12 - (-29)│ + 17
⇒ 4│-12 + 29│ + 17 
⇒ 4 +17 + 17 = 38
(iii)  Let us group the terms into groups of twos:
 -231 -362 + 34 - (-56) + 27 - 15
⇒  -231 -362│+ 34 - (-56)│+ 27 - 15
⇒  -593│ + 90│+ 12
⇒  -593 + 90│ + 12
⇒  -503 +12 =  - 491
(iv)  Let us group the terms into groups of twos: 
 -19 - 41 + 23 - (-4) - 81 + 27 - (-90)
⇒ -19 - 41│+ 23 - (-4)│- 81 + 27│ - (-90)
⇒ -60│ + 27│ - 54│ + 90
⇒ -60 + 27│ - 54 + 90
⇒ -33 + 36 =3
The image below shows the checks done for the above answers in a spreadsheet program

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With this we have completed the discussion on addition and subtraction of integers. In the next chapter section we will discuss about fractions.

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Chapter 4.3 - Addition of Negative numbers

In the previous sections we learned the rules for adding a positive number to a negative number. We learned it by the way of playing a game.  Let us continue the game: We had completed the third roll.

Fourth roll → dice give -3

Here also we have to proceed with care. We are already on the negative side at -2. That is., we are in a debt of 2. The result '-3' will take away 3 from us. That means we fall into more debt. In other words we move further towards the left by 3 units. Moving 3 units from -2 towards the left will bring us to -5. This is shown in the fig.4.9 below:

Fig.4.9

So we can write:

-2 + (-3) = -5

In this case, there is no need to check for various possibilities. Whenever a negative number is added to another negative number, the same thing happens: 
Already in debt. And more debt is coming. Thus the position which was already in the left of zero, moves further to the left as in fig.4.9 above 

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This gives us another general rule for addition. We will call it  Rule 2:

When two negative numbers are added together

■ First simply add their numerical values

■ Then give the '-' sign

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We will now see some solved examples which will demonstrate the application of the above rules:

Solved example 4.8

Evaluate the following:

(i) -8 + (-4)     (iv) -174 + (-87) 

(ii) -29 + (-7)    (v) -721 + (-329) 

(iii) -31 + (-8)     (vi) -17578 + (-19)

Solution:

Note that all the above questions contain two terms. Both are negative. All the questions have a '+' sign in between the two terms. So all of them are cases of addition of a negative integer to a negative integer.
(i) • Adding numerical values ⇒ 8 + 4 =12
    • Sign of the result is the '-' sign
So the answer is -12. So we can write: -8 + (-4) = -12
(ii) • Adding numerical values ⇒ 29 + 7 = 36
     • Sign of the result is the '-' sign
So the answer is -36. So we can write: -29 + (-7) = -36

Rest of the problems can be done in the same way:

(iii) -31 + (-8) = -39        (v) -721 + (-329)  = -1050 

(iv) -174 + (-87) = -261     (vi) -17578 + (-19) = -17594

In the above questions, the second term which is negative, was put inside a parenthesis '( )'. This was for clarity. We may get questions with out such parenthesis. In such cases we must visualize as adding a negative quantity. The steps are the same. Let us do such problems:

Solved example 4.9

Evaluate the following:

(i) -14 - 21      (ii) -183 - 92 

Solution:

(i)  -14 - 21 ⇒ -14 + (-21) = -35

(ii) -183 - 92 ⇒ -183 + (-92) = -275

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Thus, while learning how to play the game, we learned two rules for the addition of integers. There is of course one more rule. But we already know it, and there is nothing special about it:

The addition of two positive numbers:

We simply add the numbers and then give the '+' sign. (When the resulting sign is '+', it need not be specially indicated)

With the knowledge of the above rules for addition, we can now complete the game. The first player to reach 25 will win.

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With this we have completed the discussion on addition of integers. We can now add any two integers whether they be positive, negative or a combination of both. In the next section we will see subtraction.

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Chapter 4.2 - Adding a Positive number and a Negative number

In the previous sections we got a good understanding about positive and negative numbers. In this section we will discuss the addition of these numbers. For this we will see a simple game using the number line. It is played as follows:

We need two dice as shown in fig.4.7. One is an ordinary die with the numbers marked from 1 to 6 on the 6 sides. The other die does not have any numbers. Instead, it has symbols '+' and '-'. Three '+' signs and three '-' signs. 

Fig.4.7

The two are rolled together. On landing, the first die gives the numeric value and the second one gives the sign.
Examples:
• The first die lands with 4 on the upper face. And the second lands with a '-' sign on the upper face. Then the value that the player gets is -4
• The first lands with 2 on the upper face. And the second lands with a '+' sign on the upper face. Then the value is +2

With these rules finalized, we can start the game:
Initially, the player places his button at zero of the number line. Then he rolls the dice.
• If he gets say +2, he moves the button 2 units to the right. He reaches +2
• If he gets say -5, he moves 5 units to the left and reach -5.
Let us assume he reaches +2 and see the various possibilities when the game continues:

1. Initial position: 0
2. First roll → dice give +2 → New position: +2
3. Second roll → dice give +1 → New position: Add the dice result to the current position ⇒ +2 +1 = +3
4. Third roll → dice give -5

In the third roll, we get a result with a '-' sign for the first time in the game. So we have to move carefully. The current position is +3. We get a result of -5. In any game with dice, what ever result the dice give, is added to the current position. So we must add -5 to the current position.
This can be written as: +3 + (-5). 
The '-' sign indicates that 5 will be taken away from us. But we have only 3. We can give away all the 3 and reach back to zero. But still it is not sufficient. We have to give 2 more. So we move 2 units further to the left of zero to reach -2. This is shown in the fig.4.6 below:

Fig.4.6  Adding +3 and -5 gives -2

This means we are in a 'debt' of 2. Our resulting position is -2. 
So we can write: +3 + (-5) = -2. Note that the '+' sign in the middle is highlighted. This is to emphasize that we are adding two quantities. [Because what ever result is obtained from the dice is to be added to the current value]. The point to note is that, we are adding a negative quantity to a positive quantity.

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Let us analyse the above result: 
• The numerical value of the result is 2 ('numerical value' means the value of a number with out it's sign)
• It is obtained as: 5 - 3 = 2 ⇒ The larger numerical value - smaller numerical value
• The sign given to the result is the sign of the number with the larger numerical value. (here -5)

The above analysis gives us the steps to find the result when any positive integer is added to a negative integer. But before finalising the steps as rules, let us see a few more examples:

Let +6 be added to -4. What will be the result? Let us use the number line as shown in the fig.4.7 below:

Fig.4.7 Adding -4 and +6 gives 2

-4 indicates that there is already a debt of 4. +6 is added to this. 6 is larger than 4. So 4 out of the 6 will be used to pay off the debt. Then only 2 will remain. So the resulting position will be +2. So we can write:
-4 + (+6) = +2
Let us analyse the above result: 
• The numerical value of the result is 2 ('numerical value' means the value of a number with out it's sign)
• It is obtained as: 6 - 4 = 2 ⇒ The larger numerical value - smaller numerical value
• The sign given to the result is the sign of the number with the larger numerical value. (here +6)

The above analysis gives us the same steps as before. Let us see one more example.

Let +6 be added to -9. What will be the result? Let us use the number line as shown in the fig.4.8 below:

Fig.4.8 Adding -9 and 6 gives -3

-9 indicates that there is already a debt of 9. +6 is added to this. The debt is larger than what is newly acquired. So all the 6 is used up to pay the debt. Still a debt of 3 will remain. So the resulting position = -3. Thus we can write:
-9 + (+6) = -3
Let us analyse the above result: 
• The numerical value of the result is 3
• It is obtained as: 9 - 6 = 3 ⇒ The larger numerical value - smaller numerical value
• The sign given to the result is the sign of the number with the larger numerical value. (here -9)

The above analysis gives us the same steps as before. So we can finalize:

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The rules for the addition of any positive integer and a negative integer. We will call it Rule 1:
■ Numerical value of the result = The larger numerical value - smaller numerical value
■ Sign of the result is the sign of the number with the larger numerical value.

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We will now see some solved examples which will demonstrate the application of the above rules:

Solved example 4.6

Evaluate the following:

(i) 8 + (-3)     (iv) 122 + (-81) 

(ii) -36 + 4    (v) -879 + 329 

(iii) 21 + (-9)     (vi) 12541 + (-18)

Solution:

Note that all the above questions contain two terms. Out of the two, one is positive and the other is negative. All the questions have a '+' sign in between the two terms. So all of them are cases of addition of a positive integer to a negative integer.

(i) • Numerical value of the result = The larger numerical value - smaller numerical value = 8 -3 = 5

• Sign of the result is the sign of the number with the larger numerical value ⇒ sign of 8 which is '+'.

So the answer is +5. This can be simply written as 5. So we can write: 8 + (-3) = 5

(ii) • Numerical value of the result = The larger numerical value - smaller numerical value = 36 - 4 = 32

• Sign of the result is the sign of the number with the larger numerical value ⇒ sign of -36 which is '-'.

So the answer is -32. So we can write: -36 + 4 = -32

The other problems can be done in the same way:

(iii) 21 + (-9) = 12        (v) -879 + 329 = -509 

(iv) 122 + (-81) = 41     (vi) 12541 + (-18) = 12523

In the above questions, if the second term was negative, it was put inside a parenthesis '( )'. This was for clarity. We may get questions with out such parenthesis. In such cases we must visualize as adding a negative quantity. The steps are the same. Let us do such problems:

Solved example 4.7

Evaluate the following:

(i) 12 - 4      (ii) 179 -84 

Solution:

(i) 12 - 4 ⇒ 12 + (-4) = 8

(ii) 179 -84 ⇒ 179 + (-84) = 95

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So now we know the rules for adding a positive number to a negative number. We learned it by the way of playing a game. We deviated from the game for learning more about the rule. In the next section, we will continue the game.

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