CBSE Class 7 Mathematics Operations with Integers Notes Set 01

Integers and the Number Line

Counting numbers 1, 2, 3, … are natural numbers. Adding 0 gives the whole numbers. Whole numbers together with the negatives of the counting numbers form the integers: … −3, −2, −1, 0, 1, 2, 3 … Positive integers lie to the right of zero on the number line and negative integers lie to the left. Every negative integer is smaller than 0 and smaller than every positive integer.

The absolute value of an integer is its distance from zero, so the sign is ignored. It is written with vertical bars: |−9| = 9 and |17| = 17.

Adding and Subtracting Integers

1. Same signs: add the numbers and keep the common sign. (−5) + (−7) = −12.

2. Different signs: subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value. 12 + (−16) = −4.

3. Adding a negative integer is the same as subtracting its positive. Subtracting a negative integer is the same as adding its positive. So 8 − (−3) = 8 + 3 = 11.

4. On a number line, adding moves you to the right and subtracting moves you to the left.

Properties of Addition and Subtraction

1. Closure: the sum and the difference of two integers are always integers.

2. Commutative: a + b = b + a, but a − b ≠ b − a in general.

3. Associative: a + (b + c) = (a + b) + c, but subtraction is not associative: a − (b − c) ≠ (a − b) − c.

4. Additive identity: a + 0 = a, so 0 is the additive identity.

5. Additive inverse: a + (−a) = 0, so −a is the additive inverse of a.

Multiplication and Division

Product or quotient of two integers with the same sign is positive. With unlike signs it is negative. In a product of several integers, an even number of negative factors gives a positive result and an odd number gives a negative result.

Properties of multiplication: closure, commutative (a × b = b × a), associative (a × (b × c) = (a × b) × c), a × 0 = 0, multiplicative identity 1 (a × 1 = a), and distributive: a × (b + c) = a × b + a × c and a × (b − c) = a × b − a × c.

Division is the inverse of multiplication. Integers are not closed under division because, for example, 7 ÷ 2 is not an integer. Division is not commutative or associative. For any non-zero integer a: a ÷ 1 = a and 0 ÷ a = 0. Division by zero is undefined.

Worked Examples

Example 1. The sum of two integers is −16. One of them is 12. Find the other. Other = −16 − 12 = −28.

Example 2. Verify a − (b − c) ≠ (a − b) − c for a = 9, b = 6, c = 4. Left side: 9 − (6 − 4) = 9 − 2 = 7. Right side: (9 − 6) − 4 = 3 − 4 = −1. Since 7 ≠ −1, the statement is verified.

Example 3. Use the distributive property: (−85) × 43 + 43 × (−15). Take 43 common: 43 × (−85 − 15) = 43 × (−100) = −4300.

Quick Check Questions