4.2 Integers & Directed Numbers

Directed numbers (positive and negative whole numbers) appear from Paper D onwards, and by Papers E-F ICAS expects you to combine them fluently in multi-step calculations. The topic looks simple but produces more sign errors than almost any other, precisely because students rush it. The reliable habit is to decide the sign of your answer first, then compute the size.

Negative Numbers in Context

Negative numbers describe positions below a zero reference point:

• Temperature: −4 °C means 4 degrees below freezing. If the temperature rises from −4 °C to 3 °C, the change is 3 − (−4) = 7 degrees.
• Floors and lifts: a car park on level −2 is two floors below ground. Going from −2 to the 5th floor is a rise of 5 − (−2) = 7 floors.
• Money: a bank balance of −$30 means you owe $30. Paying off a debt of $12 leaves a balance of −30 + 12 = −$18.
• Elevation: Death Valley sits at about −86 m relative to sea level.

These contexts are exactly how ICAS frames its questions, so practise translating the story into a number sentence before calculating.

The Number Line and Ordering

On the number line, numbers increase to the right and decrease to the left:

... −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5 ...

The critical fact: −7 < −2, because −7 is further left. Students who treat negatives like positives get this backwards, because 7 > 2. Think in terms of temperature or debt: −7 °C is colder than −2 °C; owing $7 leaves you worse off than owing $2.

Ordering example. Arrange from smallest to largest: 3, −5, 0, −1, 4.

Picture the number line: −5, −1, 0, 3, 4.

ICAS also asks for the difference between two directed numbers — always subtract the smaller from the larger (or compute the distance on the number line). The difference between −6 and 4 is 4 − (−6) = 10.

Adding and Subtracting Integers

Two ideas cover everything:

1. Adding a negative = subtracting a positive. 5 + (−3) = 5 − 3 = 2. On the number line, adding a negative moves you left.
2. Subtracting a negative = adding a positive. 4 − (−3) = 4 + 3 = 7. The two minus signs collapse into a plus — think of it as removing a debt, which makes you better off.

Worked examples.

• −6 + 4 = −2 (start at −6, move 4 right)
• −6 + (−4) = −10 (both moves are left)
• 3 − 8 = −5 (you cannot take 8 from 3 without going below zero)
• −2 − (−7) = −2 + 7 = 5
• −9 − 4 = −13 (moving further left)

A useful mental model for a − b: it is the gap from b up to a. For −2 − (−7), the gap from −7 up to −2 is 5. Positive gap means the first number is to the right.

Multiplying and Dividing Integers

Ignore the signs, multiply or divide the numbers, then apply the sign rule:

The rule is identical for multiplication and division: same signs positive, different signs negative. Two negatives make a positive; one negative makes a negative. For three or more factors, count the negatives: an even count gives a positive answer, an odd count gives a negative one. So (−2) × (−3) × (−5) = −30 (three negatives), while (−2) × (−3) × 5 = 30.

Mixed Operations

Apply the order of operations (brackets, then × and ÷, then + and −) exactly as with positive numbers:

Worked example 1. −3 + 4 × −2 = −3 + (−8) = −11. Multiplication first — do not be tempted to compute (−3 + 4) × −2 = −2, which is wrong and will be one of the distractor options.

Worked example 2. (−6 − 2) ÷ −4 = −8 ÷ −4 = 2. Brackets first; then same signs give a positive.

Worked example 3. The temperature at midnight was −3 °C. It fell 5 degrees by 3 am, then rose 12 degrees by noon. What was the noon temperature?

−3 − 5 + 12 = −8 + 12 = 4 °C.

Common Sign Traps in Multiple Choice

• −5² versus (−5)². Without brackets, the square applies before the negative: −5² = −25, but (−5)² = 25. ICAS sets this trap regularly.
• Reversed ordering. Options will include 7 where the answer is −7, so state the sign out loud before looking at the choices.
• Difference questions. "What is the difference between −8 and −2?" The answer is 6, not −6 or 10 — it is a distance, so it is positive.
• Debt logic. "Sam owes $45 and repays $20" gives a balance of −45 + 20 = −$25, not −$65; repayment moves the balance towards zero.

Before finalising any answer, do a sanity check: should it be positive or negative? Roughly how big? This two-second habit eliminates most of the errors this topic is famous for.