Integers
Let’s observe some integers that are marked on the number line given below.
We can observe that an integer on a number line is always greater than every integer on its left. Thus, 4 is greater than -1 and -1 is greater than -4, 4>-1>-4.
In other way, we can say that an integer on a number line is always lesser than every integer on its right. Thus, -4 is less than -1 and -1 is lesser than 4, -4<-1<4.
Addition of integers on the number line
When we add a positive integer, we move to the right on the number line.
Let us add 2 and 3
Here, 3 is added to the given number i.e. 2, so we will make 3 jumps to the right of 2.
i.e. 1st jump – from 2 to 3,
2nd jump – 3 to 4
And 3rd jump – 4 to 5
Therefore, the sum of 2 and 3 is 5
When we add a negative integer, we move to the left on the number line.
Let us add 4 and -3
Here, -3 is added to the given number i.e. 4, so we will make 3 jumps to the left of 4.
i.e. 1st jump – from 4 to 3,
2nd jump – 3 to 2
And 3rd jump – 2 to 1
Therefore, the sum of 4 and -3 is 1
Subtraction of integers on the number line
When we subtract a positive integer, we move to the left on the number line.
Let us subtract 2 from 5 (i.e. 5-2)
Here, 2 is subtracted from the given number i.e. 5, so we will make 2 jumps to the left of 5.
i.e. 1st jump – from 5 to 4,
And 2nd jump – 4 to 3
Therefore, we get 5 – 2 = 3
When we subtract a negative integer, we move to the right on the number line.
Let us subtract -2 from 3 [i.e. 3-(-2) = 3+2]
Here, -2 is subtracted from the given number i.e. 3, so we will make 2 jumps to the right of 3.
i.e. 1st jump – from 3 to 4,
2nd jump – 4 to 5
Therefore, we get 3-(-2) = 5
Important Notes
1. Every positive integer is greater than every negative integer.
2. Zero is less than every positive integer and is greater than every negative integer.
3. When two positive integers are added we get a positive integer.
Example: 12+14 = 26 (If the signs are the same, we find the sum of the values of integers without sign, and then use the same sign as the integers have.)
4. When two negative integers are added we get a negative integer.
Example: (–6) + (–3) = – 9 (If the signs are the same, we find the sum of the values of integers without sign, and then use the same sign as the integers have.)
5. When integers with different signs are added, we find out the difference of the values of integers without sign (subtract lower value integer from greater value integer) and then use the sign of the integer with the greater value.
6. The additive inverse of any integer a is – a and additive inverse of (– a) is a. Example: Additive inverse of an integer 5 is (– 5) and additive inverse of (– 5) is 5.
7. For subtraction, we add the additive inverse of the integer that is being subtracted, to the other integer.
For example: 65 – 37 = 65 + additive inverse of 37 = 65 + (–37) = 28
Example 1: A monkey is on the third stair on a staircase and jump up by four more stairs. At which step will he be after he jumps back 2 stairs down?
Solution: He is currently at the third stair i.e. at (+3).
He goes up 4 stairs in the same direction.
3 + 4 = 7
Therefore, he is at 7th stair on the staircase.
Now, the monkey comes down by 2 stairs. Since he comes down in opposite direction i.e. downwards by 2 stairs (i.e. –2),
So, 7 + (–2) = 7 – 2 = 5. He is at 5th step now.
Example 2: Verify a – (– b) = a + b for the following values of a and b.
a = 20, b = 15
Solution: L.H.S = 20 – (– 15)
= 20 + 15
= 35
R.H.S = a + b
= 20 + 15
= 35
Here, L.H.S = R.H.S
Therefore, a – (– b) = a + b
Example 3: Use the sign of >, < or =.
(– 5) + 8 – (17) ___ 14 – 8 + (– 10)
Solution: L.H.S = (– 5) + 8 – (17)
= -22 + 8
= -14
R.H.S = 14 – 8 + (– 10)
= 14 -18
= -4
(– 5) + 8 – (17) < 14 – 8 + (– 10)
Example 4: Find the odd one out of the four options given below:
(a) (–2, – 5) (b) (+3, –10) (c) (–1, –6) (d) (–4, –5)
Solution: Here –2 + (–5) = –7,
+3 + (–10) = –7
– 1 + (–6) = –7
All the above pairs i.e. (–2, – 5); (+3, –10); (–1, –6) give same answer on adding, whereas – 4 + (– 5) = –9, gives a different answer. So, odd one out is (d).
Example 5: In an objective type test containing 20 questions. A student is to be awarded +3 marks for every correct answer, –2 for every incorrect answer and zero for not writing any answer. If Amit attempted 18 question correctly and 2 question incorrectly, then what is his total score?
Solution: Marks from 18 correct responses = (18 × 3) = 54
Marks obtained for 2 incorrect answers = 2 x (-2) = -4
Thus the number obtained = 54 - 4 = 50