Operations with Integers

What this chapter is about

In earlier classes, you learned about positive whole numbers and zero. Now you will work with integers, which include positive numbers, negative numbers and zero. Think of a thermometer showing 5°C above zero or 3°C below zero. The temperature below zero is written as −3. These negative numbers are part of the integer family.

This chapter teaches you how to add, subtract, multiply and divide integers. You already know these four operations for positive numbers. Here you will learn the rules that apply when one or both numbers are negative. This skill is needed in science (temperatures, altitudes, depths), in banking (deposits and withdrawals) and later in algebra.

After studying this chapter, you should be able to perform all four operations on any pair of integers, understand why the rules work, and check your answers using the number line.

Key ideas

• Integers are the set of numbers: …, −3, −2, −1, 0, 1, 2, 3, … They go on forever in both directions.
• On a number line, numbers to the right of 0 are positive and numbers to the left are negative.
• Adding a positive integer means moving right on the number line; adding a negative integer means moving left.
• Subtracting an integer is the same as adding its opposite: a − b = a + (−b).
• Multiplying two integers with the same sign (both positive or both negative) gives a positive product.
• Multiplying two integers with different signs (one positive, one negative) gives a negative product.
• Dividing integers follows the same sign rules as multiplication.
• Multiplying or dividing any integer by zero has special rules: a × 0 = 0, and division by zero is not allowed.

Formulas and facts to remember

• Rule: (+) + (+) = (+) · Meaning: Sum of two positives is positive.
• Rule: (−) + (−) = (−) · Meaning: Sum of two negatives is negative; add the numbers and keep the minus sign.
• Rule: (+) + (−) or (−) + (+) · Meaning: Find the difference and keep the sign of the larger absolute value.
• Rule: a − b = a + (−b) · Meaning: To subtract, add the opposite.
• Rule: (+) × (+) = (+) · Meaning: Positive times positive is positive.
• Rule: (−) × (−) = (+) · Meaning: Negative times negative is positive.
• Rule: (+) × (−) = (−) · Meaning: Positive times negative is negative.
• Rule: (−) × (+) = (−) · Meaning: Negative times positive is negative.
• Rule: Same sign rules apply to division. · Meaning: Divide the numbers, then apply the sign rule.
• Rule: a × 0 = 0 · Meaning: Any integer times zero is zero.
• Rule: Division by zero is undefined. · Meaning: You cannot divide any number by zero.

Worked examples

Example 1: Adding integers

Find −8 + 5.

Step 1: The signs are different. Take the difference of 8 and 5, which is 3. Step 2: The number with the larger absolute value is 8 (negative). So the answer carries a minus sign. Answer: −8 + 5 = −3.

Example 2: Subtracting integers

Find 4 − (−6).

Step 1: Change subtraction to addition of the opposite: 4 − (−6) = 4 + 6. Step 2: Both are positive, so add: 4 + 6 = 10. Answer: 4 − (−6) = 10.

Example 3: Multiplying and dividing integers

A shopkeeper records a loss of ₹15 each day for 4 days. What is the total change in money?

Step 1: A loss is shown as −15. Total change = (−15) × 4. Step 2: Different signs (negative × positive) give a negative product. Step 3: Multiply the numbers: 15 × 4 = 60. Answer: (−15) × 4 = −60. The shopkeeper lost ₹60 in 4 days.

Now divide: If a total loss of ₹60 happened over 4 days equally, loss per day = (−60) ÷ 4. Same rule: negative ÷ positive = negative. 60 ÷ 4 = 15. Answer: −15 per day.

Common mistakes

• Thinking −5 + (−3) = −2 because you subtract → add the numbers and keep the minus sign, so the answer is −8.
• Forgetting to flip the sign when subtracting a negative → a minus of a minus becomes a plus.
• Saying (−4) × (−2) = −8 → two negatives multiply to give a positive, so the answer is +8.
• Believing 0 ÷ 5 and 5 ÷ 0 are the same → 0 ÷ 5 = 0, but 5 ÷ 0 is not allowed.
• Mixing up rules for addition and multiplication → remember, addition uses the number line idea; multiplication uses the same-sign-positive, different-sign-negative rule.

Quick revision

• Integers include all positive and negative whole numbers along with zero.
• To subtract any integer, add its opposite.
• Same signs multiply or divide to give a positive result.
• Different signs multiply or divide to give a negative result.
• Anything multiplied by zero is zero; division by zero is never allowed.