Chapter focus
Integers include positive whole numbers, negative whole numbers and zero. Their operations describe direction, change and quantities that can lie below or above a reference point.
These original explanations support the named Ganita Prakash Part II chapter. Use the textbook for its full number-line explorations and practice sequence.
1. Add and subtract integers
On a number line, adding a positive integer moves right and adding a negative integer moves left. Subtracting a number is the same as adding its opposite: a − b = a + (−b).
Examples: −4 + 7 = 3; 5 + (−8) = −3; −2 − 6 = −2 + (−6) = −8. For different signs, find the difference between magnitudes and keep the sign of the number with the greater magnitude.
2. Multiply and divide integers
For multiplication and division, like signs give a positive result and unlike signs give a negative result:
- (+) × (+) = +; (−) × (−) = +.
- (+) × (−) = −; (−) × (+) = −.
- The same sign rule applies to division by a non-zero integer.
Zero multiplied by any integer is zero. Division by zero is undefined.
Worked example
(−6) × (−4) = 24 because the signs are alike. Then 24 ÷ (−3) = −8 because the signs differ.
3. Use properties to simplify
Integer addition and multiplication are commutative and associative: changing order or grouping does not change the result. Subtraction and division are not commutative. Multiplication distributes over addition: a(b + c) = ab + ac.
Worked example
(−5) × (18 + (−3)) = (−5) × 15 = −75. Using distribution gives (−5 × 18) + (−5 × −3) = −90 + 15 = −75.
4. Interpret signs in context
Choose a reference point and explain what positive and negative mean. If a lift starts at floor 2 and descends 5 floors, its position is 2 + (−5) = −3 relative to the ground floor. In a money context, a gain may be positive and a debt negative. The sign depends on the stated convention.
For repeated changes, multiplication can represent equal groups: a loss of ₹40 each day for 6 days is 6 × (−40) = −₹240.
Common errors to avoid
- Subtracting a negative becomes addition: 7 − (−3) = 10.
- For addition with unlike signs, do not add magnitudes and keep the wrong sign.
- Two negative factors multiply to a positive result; this differs from adding two negative numbers.
- Never divide by zero.
Practice: calculate and interpret
1. Evaluate −7 + 12.
Answer: 5.
2. Evaluate 8 − (−5).
Answer: 8 + 5 = 13.
3. Calculate (−9) × 6.
Answer: −54; the signs differ.
4. Evaluate (−5) × (18 + (−3)).
Answer: (−5) × 15 = −75.
5. A diver is at −12 m and rises 7 m. What is the new position relative to sea level?
Answer: −12 + 7 = −5 m, still 5 m below sea level.
Chapter summary
Integers extend the number line in both directions from zero. Addition and subtraction model movement; multiplication and division follow sign rules. State what the signs mean in a context, apply operation properties carefully and check by estimating or reversing the operation.