➕Mathematics of Positive and Negative Numbers, Made Easy for Students

Positive and negative numbers are numbers above and below zero. You use them more than you think: money you have and money you owe, a freezer's temperature, how far above or below sea level a place is. This lesson goes through them one step at a time, with practice as you go and answers at the end.

By the end of this lesson you will be able to

Say what positive and negative numbers are.

Find them on a number line.

Add, subtract, multiply and divide them.

Use them in everyday problems with money and temperature.

What are positive and negative numbers?

Positive numbers are bigger than zero, like 1, 5 or 10.5. Sometimes they're written with a plus sign (+3), but usually with no sign at all.

Negative numbers are smaller than zero, like −1, −5 or −10.5. They always have a minus sign.

Zero is in the middle. It is neither positive nor negative.

The number line

Picture a ruler that stretches both ways from zero. Positive numbers are on the right. Negative numbers are on the left. The further a number is from zero, the bigger its size, whichever side it's on.

Try it: Activity 1

Draw a number line from −6 to 6. Mark these numbers on it: −3, 2, −5, 0, 4. Which are positive? Which are negative? How far is each one from zero?

Where you meet them in real life

Money: if you receive ₵20, that's +20. If you owe a friend ₵15, that's −15.

Temperature: a hot afternoon in Tamale might be +38°C. Inside a freezer it can be −18°C.

Height: a hill 300 metres above sea level is +300. A diver 50 metres below the surface is at −50.

Time: two years from now is +2; three years ago is −3.

Try it: Activity 2

Write each of these as a number with the right sign, and say why: a submarine dives 200 metres below sea level; you put ₵50 into your MoMo wallet; a freezer is at 8 degrees below zero; a hiker climbs 300 metres up a hill.

Adding

Adding is moving along the number line. Adding a positive number moves you right. Adding a negative number moves you left.

Positive + positive: 3 + 2 = 5. Start at 3, move 2 to the right.

Negative + negative: −3 + (−2) = −5. Start at −3, move 2 to the left.

When one number is positive and the other negative, use this rule:

1Ignore the signs and look at the sizes. For −7 + 4, the sizes are 7 and 4.

2Take the smaller size from the bigger one: 7 − 4 = 3.

3Give the answer the sign of the number with the bigger size. Here that's −7, so the answer is −3.

Another one: 6 + (−2). Sizes 6 and 2. 6 − 2 = 4. The bigger size belongs to 6, which is positive, so the answer is 4.

Check with the number line: for −7 + 4, start at −7 and move 4 to the right. You land on −3.

Your turn: what is 8 + (−5)? Check it on a number line.

Subtracting

Here's the trick: subtracting a number is the same as adding its opposite. The opposite of 4 is −4, and the opposite of −3 is 3. So rewrite the sum as an addition, then use the adding rules.

6 − 4 = 6 + (−4) = 2

−6 − (−4) = −6 + 4 = −2

5 − (−3) = 5 + 3 = 8

−5 − 3 = −5 + (−3) = −8

Notice the third one: taking away a negative makes the answer bigger. That surprises a lot of people at first, but the number line shows it clearly. Start at 5 and move 3 to the right.

Your turn: what is −8 − 2? Rewrite it as an addition first.

Multiplying and dividing

These are the easiest, because it's all about the signs:

Same signs give a positive answer: 3 × 2 = 6, and −3 × −2 = 6.

Different signs give a negative answer: 3 × −2 = −6, and −3 × 2 = −6.

Division follows exactly the same rule: −8 ÷ 2 = −4 (different signs), and −8 ÷ −2 = 4 (same signs).

Try it: Activity 3

Work these out and explain the sign of each answer: −4 + 6; 7 − (−2); −3 × 5; −12 ÷ −3.

Absolute value

The absolute value of a number is just its distance from zero, so it's never negative. It's written with two straight lines: |−5| = 5 and |3| = 3.

If there's a sum inside the lines, work it out first. For |−7 + 2|: −7 + 2 = −5, and |−5| = 5.

Try it: Activity 4

Find |−8|, |3 − 7| and |−2 + (−4)|. Why is the answer never negative?

Real-life problems

Example: a submarine is 50 metres below sea level (−50). It rises 20 metres, then dives 30 metres. Where does it end up?

Start: −50

Rise 20: −50 + 20 = −30

Dive 30: −30 + (−30) = −60

It finishes at −60, which means 60 metres below sea level.

Try it: Activity 5

Kofi has ₵40. He buys airtime on credit worth ₵60, then receives ₵35. How much does he have now?

A freezer is at −10°C. It warms up by 15°C, then cools down by 5°C. What is the temperature now?

A trader's profit changes by −2, then +5, then −3 cedis over three days. What is the total change?

What to remember

Positive numbers are above zero; negative numbers are below zero.

Use a number line to see what happens when you add or subtract.

To subtract, add the opposite.

For multiplying and dividing: same signs give positive, different signs give negative.

Absolute value is distance from zero, so it's never negative.

Answers

Your turns: 8 + (−5) = 3, and −8 − 2 = −8 + (−2) = −10.

Activity 1: 2 and 4 are positive; −3 and −5 are negative; 0 is neither. Their distances from zero are 3, 2, 5, 0 and 4.

Activity 2: the submarine is −200, because it is below sea level; the MoMo deposit is +50, because money comes in; the freezer is −8, because it is below zero; the hiker is +300, because they go up.

Activity 3: −4 + 6 = 2 (6 is bigger, so the answer is positive); 7 − (−2) = 7 + 2 = 9; −3 × 5 = −15 (different signs); −12 ÷ −3 = 4 (same signs).

Activity 4: |−8| = 8; |3 − 7| = |−4| = 4; |−2 + (−4)| = |−6| = 6. A distance can't be negative, so the answer never is.

Activity 5: Kofi has 40 − 60 + 35 = 15 cedis. The freezer ends at −10 + 15 − 5 = 0°C. The trader's total change is −2 + 5 − 3 = 0.

Want more practice? Try the Number work quiz or the Reasoning puzzles. Both adjust to your level as you go.