Integers

Syllabus tag: KCSE | Mathematics | Form 1 | Topic 6 Integers

Lesson objectives

By the end of this topic, you should be able to:

• Define integers and identify them on a number line.
• Perform the four basic operations on integers using the number line.
• Work out combined operations on integers in the correct order.
• Apply integers to real life situations.

Integers

Integers are the whole numbers, their negatives, and zero. They extend counting in both directions from zero.

a) On the number line

Positive integers sit to the right of zero, negatives to the left.

The number further right is always the larger. So −2 > −5, even though 5 > 2.

b) Adding and subtracting

Adding a positive moves right. Subtracting a positive moves left.

Adding a negative also moves left. So 2 + (−5) gives −3, exactly as 2 − 5 does.

Subtracting a negative moves right. So 4 − (−3) = 4 + 3 = 7.

Two signs together combine into one. Same signs give a plus; different signs give a minus.

c) Multiplying and dividing

Multiply or divide the numbers as usual, then decide the sign.

Two signs the same give a positive answer. Two signs different give a negative.

The same rule governs division. So −12 ÷ 3 = −4, and −12 ÷ −3 = 4.

d) Why two negatives make a positive

Look at a pattern.

3 × −2 = −6. Then 2 × −2 = −4. Then 1 × −2 = −2. Then 0 × −2 = 0.

Each step up the left factor adds 2 to the answer. Continue: −1 × −2 must be 2, and −2 × −2 must be 4.

The rule is not arbitrary. It is what keeps the pattern consistent.

e) Combined operations

Work in the usual order. Brackets, then multiply and divide left to right, then add and subtract left to right.

Take −8 + 12 ÷ (−4) × 3.

The division and multiplication come first: 12 ÷ (−4) = −3, then −3 × 3 = −9.

Then −8 + (−9) = −17.

Keep the sign attached to its number as you work. Losing a minus sign partway through is the commonest error here.

f) Where this is used

Temperature above and below freezing. Bank balances and overdrafts. Altitude above and below sea level. Profit and loss. Gains and losses in a game.

Words to know

• Integer -- a whole number or its negative, including zero.
• Number line -- a line on which numbers are marked at equal intervals in order.
• Negative integer -- an integer less than zero.
• Double sign -- two signs standing together, which simplify to one.
• Absolute value -- the distance of a number from zero, ignoring its sign.

:::checkpoint Check yourself

1. Which is larger, −9 or −4?
2. Work out −7 + 12.
3. Work out −6 × −4.
4. Work out −20 ÷ 5 + 3 × (−2). :::

Bridge to practice

The exercises begin with identifying and ordering integers, move through addition and subtraction on the number line and the double sign rule, and finish with the sign rules for multiplication, combined operations and real life applications. Write each step of a combined calculation on its own line; almost every error in this topic comes from working two steps at once.