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.
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.
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.
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:
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 ‘+’.
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.
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
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 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.
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.
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 |
■ 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 ‘+’.
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.
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
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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