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:
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| 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
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
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
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
(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
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:
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.
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.
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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