Integers

Various Types of Numbers

Natural numbers The natural numbers are 1, 2, 3, 4, . . . They are also known as counting numbers.

Whole numbers The whole numbers are 0, 1, 2, 2, 3, 4, . . . It consists of all natural numbers together with 0 (zero).

Clearly, every natural number is a whole number but 0 is a whole number which is not a natural number.

Integers The integers are 0, 1, − 1, 2, − 2, 3, − 3, 4, − 4, . . . It consists of all natural numbers, 0 and negative of all natural numbers.

Following facts are important:

• Positive integers: 1, 2, 3, 4, 5, . . . are all positive integers (same as natural numbers).
• Negative integers: − 1, − 2, − 3, − 4, − 5, . . . are all negative integers.
• Zero is an integer which is neither positive nor negative.

Recall the rule for addition and subtraction of integers. On a number line when we

• add a positive integer, we move to the right.
• add a negative integer, we move to the left.
• subtract a positive integer, we move to the left.
• subtract a negative integer, we move to the right.

Note. When a positive integer is added to an integer, the resulting integer becomes greater than the given integer. When a negative integer is added to an integer, the resulting integer becomes less than the given integer.

State whether the following statements are correct or incorrect. Correct those which are wrong.

Example. When two positive integers are added we get a positive integer.

Solution. Correct. For example, (a) 46 + 73 = 119 (b) 113 + 72 = 185 etc.

Example. When two negative integers are added we get a positive integer.

Solution. Incorrect, since (– 6) + (– 17) = – 23, which is not a positive integer.

The correct statement is: When two negative integers are added we get a negative integer.

For example, (a) (– 46) + (– 73) = – 119 (b) (– 103) + (– 82) = – 185 etc.

Example. When a positive integer and a negative integer are added, we always get a negative integer.

Solution. Incorrect, since – 9 + 16 = 7, which is not a negative integer.

The correct statement is: When one positive and one negative integers are added, we take their difference and place the sign of the bigger integer. The bigger integer is decided by ignoring the signs of both the integers.

For example:

(i) (– 46) + (73) = 27

(ii) (– 103) + 82 = – 21

(iii) 16 + (– 33) = – 17

(iv) 115 + (– 101) = 14

Example. Additive inverse of an integer 8 is (– 8) and additive inverse of (– 8) is 8.

Solution. Correct. In fact, the additive inverse of any integer a is – a and additive inverse of (– a) is a.

Thus, we find that for any two integers a and b,

• a – b = a + additive inverse of b = a + (– b)
• a – (– b) = a + additive inverse of (– b) = a + b

Example. (–20) + 3 = 20 – 3

Solution. Incorrect, since L.H.S = (–10) + 3 = –7 and R.H.S = 10 – 3 = 7.

Therefore, (–10) + 3 ≠ 10 – 3.

However, the correct statement is (–20) + 3 = –20 + 3

Example. 8 + (–7) – (– 4) = 8 + 7 – 4

Solution. Incorrect, since

L.H.S = 8 + (–7) – (– 4) = 8 + (–7) + 4 = 1 + 4 = 5

R.H.S = 8 + 7 – 4 = 15 – 4 = 11

Therefore, 8 + (–7) – (– 4) ≠ 8 + 7 – 4.

However, the correct statement is 8 + (–7) – (– 4) = 8 – 7 + 4.

Exercise 1. Can you find a pattern for each of the following? If yes, complete them:

(a) 7, 3, – 1, – 5, _____, _____, _____.

(b) – 2, – 4, – 6, – 8, _____, _____, _____.

(c) 15, 10, 5, 0, _____, _____, _____.

(d) – 11, – 8, – 5, – 2, _____, _____, _____.

Exercise 2. In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jack’s scores in five successive rounds were 25, – 5, – 10, 15 and 10, what was his total at the end?

Exercise 3. At Srinagar temperature was – 5°C on Monday and then it dropped by 2°C on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday, it rose by 4°C. What was the temperature on this day?

Exercise 4. A plane is flying at the height of 5000 m above the sea level. At a particular point, it is exactly above a submarine floating 1200 m below the sea level. What is the vertical distance between them?

Exercise 5. Mohan deposits Rs. 2,000 in his bank account and withdraws Rs. 1,642 from it, the next day. If withdrawal of amount from the account is represented by a negative integer, then how will you represent the amount deposited? Find the balance in Mohan’s account after the withdrawal.

Exercise 6. Verify a – (– b) = a + b for the following values of a and b.

• (i) a = 21, b = 18
• (ii) a = 118, b = 125
• (iii) a = 75, b = 84
• (iv) a = 28, b = 11