About Addition of Integers
Adding using a Number line
Integers are the set containing all whole numbers which include 0, 1, 2, 3 and so on, and also their opposites -1, -2, -3, so. We can use number lines to find the sum of integers. Adding two positive integers will result in jumping on the number line towards the right. Let us look at a few other things about plotting on number lines.
- When plotting on a number line, the length of the distance between zero and the number is the absolute value of the number it represents.
- When adding a positive number on a number line, you count up by moving to the right. When adding a negative number on a number line, you count back by moving to the left.
- An integer plus its opposite adds up to zero. The opposite of a number is called the additive inverse because the sum of two numbers is zero.
Let’s understand this with the help of an example.
Adding Integers Examples
Example 1: Find 5 + (-5)
Solution:
First, we draw a number line and we plot any of the addends. Let’s start off by plotting 5. So we count 5 units from zero and reach 5. To depict this we draw an arrow from zero to 5. Now we have to add -5. -5 is negative so we need to move to the left by \(\left | -5 \right | = 5 \) units. So starting from 5, we count back 5 units and we reach 0. To depict this, we draw an arrow from 5 to 0.
Hence we can say that sum of 5 + (-5)=0
Rules for Adding integers
Addition of integers with the same signs:
Find the absolute values of the addends and add them up. The sum will have the common sign.
For example: -3 + (-5) = -8
(Here, the common sign for both the integers is “-” and the common sign “-” is used for the sum)
Example 2: Find -5 + (-3).
Solution: -5 + (-3) = -8
In this we added the \(\left | -5 \right | = 5 \) and \(\left | -3 \right | = 3 \). So 5 + 3 = 8 and the common sign “-” is used for the sum.
Therefore, the sum is -8.
Addition of integers with different signs:
Subtract the integer with the lesser absolute value from the integer with the greater absolute value. Then use the sign of the integer with the greater absolute value.
For example:
-12 + 18 = 6
(Here 12 is the lesser absolute value and higher absolute value 18 has a “+” sign)
Example 3: Find 6 + (-11).
Solution: 6 + (-11) = -5
\(\left | -11 \right | = 11 \)
In this \(11> 6 \). so, subtract 6 from 11. Use the sign of -11 for the difference because it is of higher absolute value.
Therefore, the sum is -5.
Additive inverse property:
If the same numbers having different signs are added then it sums to 0.
For example: 7 + (-7) = 0
Solved Examples on Adding Integers
- Find -2 + (-3)
Solution: First draw an arrow from 0 to -2 to represent -2. Then draw an arrow 3 units to the left from 2 to represent adding -3.
So, -2 + (-3) = -5.
2. Find -2 + 7
Solution: Draw an arrow from 0 to -2 to represent -2. Then draw an arrow 7 units to the right from -2 to represent addition by 7.
So, -2 + 7 = 5.
3. Find -2 + 2
Solution: Draw an arrow from 0 to -2 to represent -2. Then draw an arrow 2 units to the right from -2 to represent the addition of 2.
So, -2 + 2= 0.
4. Find 7 + 13.
Solution: 7 + 13 = 20
In this we add 7 and 13 using the common sign “+” for the sum.
Therefore, the sum is 20.
5. Find -8 + (-5).
Solution: -8 + (-5) = -13
In this we added\(\left | -8 \right | = 8\) and \(\left | -5 \right | = 5\), and used the common sign “-” for the sum.
Therefore, the sum is -13.
6. Find -4 + 8.
Solution: -4 + 8 = 4
\(\left | -4 \right | = 4\)
In this case, \(8> 4\), so we subtract 4 from 8. Use the sign of +8 for the difference because it has higher absolute value. Therefore, the sum is 4.
7. Find 9 + (-10).
Solution: 9 + (-10) = -1
\(\left | -10 \right | = 10\)
In this case, \(10> 9\), so we subtract 9 from 10. Use the sign of -10 for the difference because it has higher absolute value.
Therefore, the sum is -1.
8. Jake was playing a game on his own and he played four rounds of the game. The scores for each round are given below. Find his total score after 4 rounds.
Rounds | Score |
|---|---|
1 | -50 |
2 | +60 |
3 | +80 |
4 | -60 |
Solution:
Total Score:
(-50) + 60+80+ (-60) = (-50) + 80 + 60 +(-60)
= (-50) + 80 + [60 + (-60)]
= (-50) + 80 + 0 (Additive inverse property)
= 30 (Added – 50 and 80)
So the total score is 30.
Suppose we need to add two integers ‘a’ and ‘b’ using a number line. First plot ‘a’ then plot ‘b’. The point ‘b’ is plotted by finding the absolute value of ‘b’ and counting \(\left | b \right |\) units from ‘a’ to the right or left depending on the sign of ‘b’. The point we reach after counting is the sum.
Yes, the sum of two negative integers is always negative.