Operations Integers
Comprehensive notes, formulas, and practice questions for Operations Integers.
Operations Integers
Operations on Integers
What you'll learn
- Add and subtract positive and negative integers using the number line and sign rules.
- Multiply and divide integers, including the sign rules: same signs → positive, different signs → negative.
- Evaluate expressions that mix more than one operation on integers (CBSE / NCERT Class 7).
- Check answers quickly by substituting back or using the inverse operation.
Key concepts
Level 1 — Core idea
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Integers recap — Integers are …−3, −2, −1, 0, 1, 2, 3… They include zero, all whole numbers, and their negatives. On a number line, moving right increases the value; moving left decreases it.
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Addition of integers
- Same sign: add the absolute values, keep the common sign.
Example: (−4) + (−7) = −11 and 5 + 8 = 13. - Different signs: subtract the smaller absolute value from the larger; the answer takes the sign of the number with the larger absolute value.
Example: (−9) + 4 = −5 because |−9| > |4|.
- Same sign: add the absolute values, keep the common sign.
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Subtraction of integers — Subtracting an integer is the same as adding its additive inverse:
a − b = a + (−b)
Example: 3 − (−5) = 3 + 5 = 8.
Level 2 — Process and representation
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Multiplication of integers
- (+) × (+) = (+), (−) × (−) = (+)
- (+) × (−) = (−), (−) × (+) = (−)
Example: (−6) × 4 = −24 and (−3) × (−5) = 15.
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Division of integers — Follow the same sign rules as multiplication. Division by zero is not defined.
Example: (−20) ÷ 5 = −4 and (−24) ÷ (−6) = 4. -
Order of operations — When several operations appear together, use BODMAS/PEMDAS: Brackets first, then multiplication and division (left to right), then addition and subtraction (left to right).
Worked example
Evaluate: (−8) + 12 − (−3) × 2
Step 1 — identify operations: addition, subtraction, multiplication
Step 2 — multiply first: (−3) × 2 = −6
Expression becomes: (−8) + 12 − (−6)
Step 3 — rewrite subtraction: (−8) + 12 + 6
Step 4 — add from left to right: (−8) + 12 = 4, then 4 + 6 = 10
Answer: 10
Evaluate: (−15) ÷ 3 + (−4) × (−2)
Step 1 — division and multiplication: (−15) ÷ 3 = −5; (−4) × (−2) = 8
Step 2 — add: −5 + 8 = 3
Answer: 3
Common mistakes
| Misconception | What students think | Scientific correction |
|---|---|---|
| Treating 3 − (−5) as 3 − 5 = −2 instead of **3 | Treating 3 − (−5) as 3 − 5 = −2 instead of 3 + 5 = 8. Always change subtraction to addition of the opposite. | Check the Key concepts and worked example for the NCERT-accurate version. |
| Forgetting that (−a) × (−b) is positive, not ne | Forgetting that (−a) × (−b) is positive, not negative. | Check the Key concepts and worked example for the NCERT-accurate version. |
| Assuming (−a) ÷ (−b) is negative; two negatives giv | Assuming (−a) ÷ (−b) is negative; two negatives give a positive quotient. | Check the Key concepts and worked example for the NCERT-accurate version. |
| Dropping the negative sign when the absolute value is l | Dropping the negative sign when the absolute value is larger on the left: (−12) + 5 = −7, not 7. | Check the Key concepts and worked example for the NCERT-accurate version. |
Quick check
- What is (−7) + 15? (Answer: 8)
- What is 6 − (−4)? (Answer: 10)
- What is (−5) × (−8)? (Answer: 40)
- Evaluate 2 − 3 × (−4) step by step. (Answer: 14)
Open the Practice tab for graded questions on operations with integers.
Key Takeaways (TL;DR)
- What you'll learn
- Key concepts
- Worked example
- Common mistakes
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