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.
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:
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.
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:
-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:
-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:
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.
We will now see some solved examples which will demonstrate the application of the above rules:
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 |
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 |
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.
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 + (+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 + (+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:
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.
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
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.
No comments:
Post a Comment