Integers

What is an integer?

An integer is a whole number. It can be positive, negative or zero. There are no fractions or decimals in the family of integers.

These are integers: 7, 0, −3, 250, −1000.

These are not integers: ½, 2.5, −0.75.

You already know the positive integers. You use them every day: 3 oranges, 12 pupils, GH₵50. Zero is an integer too. It means "none".

The new idea in this book is the negative integers. These are numbers less than zero. We write them with a minus sign in front: −1, −2, −3 and so on. We say "negative three" or "minus three".

Why do we need numbers below zero? Because the world does not stop at nothing. A freezer is colder than 0 °C. A person can owe money, not just have money. A lift can go below the ground floor. Integers let us describe all of these clearly.

By the end of this book you will be able to place integers on a number line, put them in order, and add and subtract them with confidence.

Why we need numbers below zero

Imagine Ama is counting her coins. She has 5 cedis. She spends 5 cedis on kenkey and fish. Now she has 0. She cannot go lower than nothing in her hand.

But now imagine Ama borrows 3 cedis from her friend Abena to pay for a trotro. Ama has nothing in her pocket and she owes 3 cedis. Her true position is worse than zero. We say her money is −3.

Here is another example. Water freezes at 0 °C. A deep freezer in a chop bar is colder than that, perhaps −18 °C. Without negative numbers, we would have no way to write that temperature.

Negative numbers let us describe:

• cold below freezing
• money owed
• floors below the ground
• depth below the sea
• a loss in a game or business

The minus sign is not a mistake and it is not "wrong". It is simply a direction: below zero.

The number line

The best tool for understanding integers is the number line. It is a straight line with zero in the middle. Positive numbers go to the right. Negative numbers go to the left.

Every step along the line is one whole unit. From 0 to 1 is one step. From 0 to −1 is one step the other way.

The arrows at both ends tell you the line goes on for ever. There is no biggest integer and no smallest integer.

Two rules to remember:

1. Moving right makes a number bigger.
2. Moving left makes a number smaller.

So −1 is bigger than −4, because −1 is further to the right. This feels strange at first. Keep looking at the line until it feels normal.

Zero in the middle

Zero is special. It is neither positive nor negative. It is the point where we start counting in both directions.

Think of zero as a boundary:

• Above zero: temperatures that are warm enough for water to stay liquid.
• Below zero: temperatures where water freezes.
• Zero cedis: you have no money and you owe no money.
• Ground floor: the lift is neither up nor down.

Because zero is the boundary, the sign of a number tells you which side of zero it sits on. Every positive number is on the right of zero. Every negative number is on the left.

Try this Draw a number line from −10 to 10 in your exercise book. Mark zero clearly. Then put a dot at each of these numbers: 4, −6, 0, 9, −1, −9.

Which of your dots is furthest to the left? Which is furthest to the right? The one on the far left is the smallest number. The one on the far right is the biggest.

Opposites

Every integer has an opposite. The opposite of 5 is −5. The opposite of −12 is 12. The opposite of 0 is 0 itself.

On the number line, opposites are the same distance from zero but on different sides. 5 is five steps to the right of zero. −5 is five steps to the left.

This distance from zero has a name. It is called the absolute value. The absolute value of −5 is 5, and the absolute value of 5 is also 5. We write it with two bars: |−5| = 5.

Opposites are useful because they cancel each other out. If Kwame earns 20 cedis (+20) and then spends 20 cedis (−20), the two changes are opposites. Together they make zero. His money is back where it started.

Try this Write the opposite of each number: 8, −3, 15, −100, 0.

Which number has the larger absolute value, −7 or 4? (Answer: −7, because it is 7 steps from zero while 4 is only 4 steps away.)

Reading a thermometer

A thermometer is a number line stood on its end. Warm numbers are at the top, cold numbers are at the bottom, and zero is somewhere in the middle.

Ghana is warm, so you may never see the outside air drop below zero. But fridges and freezers go there all the time. Here are some typical temperatures:

• A hot afternoon in Tamale: 38 °C
• Room temperature in Accra: 28 °C
• Inside a fridge: 4 °C
• Ice: 0 °C
• Inside a freezer: −18 °C
• The coldest place on Earth (Antarctica): about −89 °C

Reading the thermometer, each small line is one degree. Below zero, the numbers grow as you go down: −1, −2, −3. The further down, the colder.

Lifts and floors

Some tall buildings in Accra and Kumasi have car parks under the ground. The lift buttons show this clearly.

The ground floor is 0. Floors 1, 2 and 3 are above it. Floors −1 and −2 are below it, under the ground.

Worked example Kofi gets into the lift on floor 2. He presses −1. How many floors does the lift travel?

From 2 down to 0 is 2 floors. From 0 down to −1 is 1 more floor. Altogether the lift moves 3 floors.

We can also say: the difference between 2 and −1 is 3.

Try this Abena is on floor −2. She goes up to floor 3. How many floors does she pass through? Draw a vertical number line to help you.

Recap: the first ideas

Let us pause and gather what we have so far.

• An integer is a whole number: positive, negative or zero.
• A negative number is less than zero and is written with a minus sign, like −4.
• The number line has zero in the middle, positives on the right and negatives on the left.
• Moving right makes a number bigger. Moving left makes it smaller.
• Every number has an opposite on the other side of zero. The opposite of 6 is −6.
• The absolute value is the distance from zero. |−6| = 6.
• Negative numbers describe cold, debt, floors below ground and depth below the sea.

Quick check

1. Is −7 an integer? Is 7.5 an integer?
2. Which is further from zero: −9 or 3?
3. What is the opposite of −20?
4. A freezer is at −12 °C. Is that warmer or colder than −5 °C?

Check your answers with a friend before moving on.

Above and below sea level

Geographers measure height from sea level. Sea level is zero. Mountains are above it (positive). Some land and all the sea floor are below it (negative).

Here are some heights, written as integers:

• Mount Afadja (Volta Region): +885 m
• Kumasi city centre: about +250 m
• Accra beach: 0 m
• Bottom of the Gulf of Guinea near the coast: about −50 m
• Deepest part of the Atlantic Ocean: about −8 000 m

Worked example A fishing boat's net hangs 30 metres below the surface. The seabed there is 45 metres deep. How far is the net above the seabed?

The net is at −30 m. The seabed is at −45 m. On the number line, −30 is 15 steps to the right of −45. So the net is 15 m above the seabed.

Having money and owing money

Money is the clearest everyday use of integers. Having money is positive. Owing money (a debt) is negative.

Kwame sells sachet water at the lorry station. On Monday he starts with nothing. He borrows GH₵20 from his aunt to buy a bag of water. His position is now −20: he has stock, but he owes 20.

He sells all the water for GH₵35. He pays his aunt back her 20. Now he has 35 − 20 = +15. He has moved from −20 to +15 on the number line.

Shops and businesses keep records in exactly this way. A profit is positive. A loss is negative. If a market woman's stall makes +120 on Tuesday and −40 on Wednesday (because her tomatoes spoiled), she has still made money overall across the two days.

Negative money is not shameful. It is just information about where you stand.

Which is bigger? Positive numbers

Before we order negative numbers, remember how positive numbers work. On the number line, 8 is to the right of 3, so 8 is bigger than 3.

We use two symbols:

• > means "is greater than". 8 > 3.
• < means "is less than". 3 < 8.

A useful trick: the symbol is like a hungry mouth. It always opens towards the bigger number.

Ordering positive numbers from smallest to largest is easy because bigger numbers are "further along". For example: 2, 5, 11, 40.

Now here is the question this book must answer: how do negative numbers fit into this ordering? Is −10 bigger than −2 because 10 is bigger than 2? Or is it smaller?

Think about temperature. Is −10 °C warmer or colder than −2 °C? It is colder. Colder means lower on the thermometer. So −10 is smaller than −2.

Turn the page to see why.

Ordering negative numbers

Look again at the number line. Going left means getting smaller. −2 is two steps left of zero. −10 is ten steps left of zero. So −10 is further left, and therefore smaller.

This is the rule: the bigger the number after the minus sign, the smaller the negative number is.

So:

• −1 is the biggest negative integer.
• −2 is smaller than −1.
• −100 is smaller than −10.
• Every negative number is smaller than zero.
• Every negative number is smaller than every positive number.

Worked example Put these in order from smallest to largest: 3, −7, 0, −2, 5.

Step 1: The negatives go first. −7 is further left than −2, so −7 comes first. Then −2. Step 2: Zero comes next. Step 3: Then the positives, in the usual order: 3, then 5.

Answer: −7, −2, 0, 3, 5.

Try this Order these from smallest to largest: −4, 9, −9, 1, −1, 0.

Using < and > with integers

Now we can compare any two integers with the "greater than" and "less than" symbols. Whenever you are not sure, picture the number line. The number further to the right is greater.

Examples:

• 4 > −4 (positive beats negative)
• −3 > −8 (−3 is to the right of −8)
• 0 > −1 (zero is to the right of every negative)
• −15 < −2 (−15 is far to the left)

A common mistake is to think −8 > −3 because 8 is bigger than 3. Remember that with negatives, a bigger number after the minus means further from zero on the cold side, so it is smaller.

Try this Copy and fill in each box with < or >:

1. −6 ☐ 2
2. −6 ☐ −9
3. 0 ☐ −12
4. −1 ☐ −2
5. 7 ☐ −7
6. −30 ☐ −3

Say each one out loud: "negative six is less than two", and so on. Speaking the statement helps you catch mistakes.

Worked example: ordering temperatures

A science class measured temperatures in different places and wrote them down in a muddle:

• Freezer in the canteen: −18 °C
• Classroom at noon: 31 °C
• Inside the fridge: 4 °C
• Ice-cream box at the shop: −6 °C
• Water with ice cubes: 0 °C

Question: List the places from coldest to warmest.

Step 1. Find the negatives: −18 and −6. The one with the bigger number after the minus is colder. −18 is colder than −6.

Step 2. Zero comes next: the ice water.

Step 3. Then the positives from smallest to biggest: 4, then 31.

Answer (coldest to warmest): Freezer (−18) → ice-cream box (−6) → ice water (0) → fridge (4) → classroom (31).

Check: On a thermometer, −18 is lowest and 31 is highest. The order matches the picture. Always check against the number line when you can.

Try this: ordering integers

Work through these in your exercise book. Draw a number line for any you find hard.

A. Smallest to largest

1. 6, −6, 0, 3, −3
2. −11, −2, −20, −5
3. 14, −1, 7, −14, 0, 2

B. True or false? 4. −5 > −2 5. 0 < −1 6. −100 < 1 7. −3 > −4

C. Word problems 8. Four towns recorded these night temperatures: Bolgatanga 19 °C, a town in Canada −12 °C, a town in Russia −25 °C, Accra 25 °C. Which was coldest? Which was warmest? 9. Three friends compare their money. Kofi has GH₵12. Ama owes GH₵5. Abena owes GH₵9. Write each as an integer, then say who is in the best position and who is in the worst. 10. Write three integers that are all less than −10, from largest to smallest.

Recap: ordering

Here is what you should carry with you from this section.

• On the number line, right is bigger, left is smaller. This never changes.
• All negative numbers are smaller than zero, and zero is smaller than all positive numbers.
• Among negative numbers, the one with the bigger digit after the minus is smaller. −9 < −4.
• The symbols: > means greater than, < means less than. The open mouth faces the bigger number.
• To order a mixed list: negatives first (most negative at the start), then zero, then positives.

Memory tip Think of the thermometer. Colder is lower. −20 °C is colder than −5 °C, so −20 is lower, so −20 is smaller.

Memory tip 2 Think of debt. Owing GH₵50 is a worse position than owing GH₵5. So −50 is smaller than −5.

If you can order integers, you are ready to add and subtract them. That is the next section.

Adding on the number line

Adding means moving along the number line. Start at the first number. Then:

• If you are adding a positive number, move right.
• If you are adding a negative number, move left.

Look at the picture. We start at −1. We add 3, so we jump three steps to the right: −1 → 0 → 1 → 2. We land on 2.

So −1 + 3 = 2.

Here is another. Start at −4 and add 2. Two steps right: −4 → −3 → −2. So −4 + 2 = −2. We are still on the negative side because we did not have enough steps to cross zero.

Every addition is just a start point and a jump. Keep this picture in your mind and the rules on the next pages will make sense.

Adding a positive number

Adding a positive number always moves you to the right, towards the warm side of the thermometer. It makes your answer bigger than where you started.

Examples

• 2 + 5 = 7 (start at 2, five steps right)
• −2 + 5 = 3 (start at −2, five steps right: −1, 0, 1, 2, 3)
• −8 + 3 = −5 (start at −8, three steps right: −7, −6, −5)
• −6 + 6 = 0 (opposites cancel)

Notice the last example. When you add a number to its opposite, you always land back on zero.

Worked example: temperature A freezer is at −18 °C. The power goes off and the inside warms by 10 degrees. What is the new temperature?

Start at −18. Warming means adding a positive: −18 + 10. Ten steps to the right of −18 is −8.

New temperature: −8 °C. Still below freezing, but warmer than before.

Try this −7 + 4, −3 + 9, −10 + 10, −1 + 1, −12 + 5.

Adding a negative number

Adding a negative number moves you to the left. It makes your answer smaller than where you started. Adding a negative is the same as taking away.

Examples

• 6 + (−2) = 4 (start at 6, two steps left)
• 3 + (−5) = −2 (start at 3, five steps left: 2, 1, 0, −1, −2)
• −4 + (−3) = −7 (start at −4, three steps left)
• 0 + (−9) = −9

We put the negative number in brackets so the two signs do not run into each other. "6 + (−2)" is clearer than "6 + −2".

Think of it with money. Ama has GH₵6. She "adds" a debt of GH₵2 (perhaps she buys something on credit). Her position is now 6 + (−2) = GH₵4.

Worked example: a second debt Kofi already owes GH₵4 (−4). He borrows another GH₵3. What does he owe now? −4 + (−3) = −7. He owes GH₵7.

Try this 5 + (−8), −2 + (−6), 10 + (−10), −1 + (−1), 0 + (−15).

Worked example: Kofi's water business

Kofi sells sachet water after school. Let us follow his money for one week. Having money is positive; owing money is negative.

Monday. Kofi has nothing. He borrows GH₵30 from his uncle to buy two bags of water. Position: 0 + (−30) = −30.

Tuesday. He sells water and earns GH₵25. Position: −30 + 25 = −5. He still owes 5.

Wednesday. He earns GH₵20. Position: −5 + 20 = +15. He is now in the positive.

Thursday. A bag of water bursts in the sun. He must buy another for GH₵15. Position: 15 + (−15) = 0. Back to nothing.

Friday. He earns GH₵28. Position: 0 + 28 = +28.

Kofi ends the week with GH₵28 after paying everything back. Every change was an addition: positive for money in, negative for money out.

Try this Make your own table for a friend who starts at −10 and has these changes: +6, −8, +15, −3. What is the final position?

Rule: same signs

You can always use the number line. But for bigger numbers, a rule saves time.

When both numbers have the same sign, add the sizes and keep the sign.

Both positive:

• 4 + 7 = 11 (add 4 and 7, keep positive)

Both negative:

• (−4) + (−7) = −11 (add 4 and 7, keep negative)
• (−20) + (−35) = −55
• (−100) + (−1) = −101

Why does this work? Two negatives both push you left. Two steps left and then seven steps left is eleven steps left in total. You never change direction, so the distances simply add up.

Money picture: owing GH₵20 and then owing another GH₵35 means you owe GH₵55 in total. Two debts make a bigger debt.

Try this

1. (−9) + (−6)
2. (−15) + (−15)
3. (−3) + (−40)
4. 18 + 7
5. (−250) + (−750)

Rule: different signs

When the signs are different, subtract the smaller size from the larger size, and take the sign of the number with the larger size.

Examples:

• 9 + (−4): sizes are 9 and 4. 9 − 4 = 5. The bigger size (9) is positive. Answer: 5.
• (−9) + 4: sizes are 9 and 4. 9 − 4 = 5. The bigger size (9) is negative. Answer: −5.
• (−3) + 10: 10 − 3 = 7. Bigger size is positive. Answer: 7.
• 6 + (−14): 14 − 6 = 8. Bigger size is negative. Answer: −8.

Why does this work? The two numbers pull in opposite directions. Part of the bigger one is cancelled by the smaller one. What is left over points in the direction of the bigger one.

Money picture: you have GH₵9 and owe GH₵4. After settling, you have GH₵5. But if you have GH₵4 and owe GH₵9, after settling you still owe GH₵5, so −5.

Try this (−12) + 5, 8 + (−3), (−2) + 11, 30 + (−45), (−7) + 7.

Try this: adding integers

Do these without a calculator. Use the number line for the first few, then try the rules.

A. Number line practice

1. −3 + 5
2. −6 + 2
3. 4 + (−7)
4. −1 + (−4)

B. Same signs 5. (−8) + (−9) 6. (−21) + (−19) 7. 35 + 65

C. Different signs 8. 13 + (−6) 9. (−13) + 6 10. (−50) + 80 11. 17 + (−40)

D. Words 12. The temperature inside a cold store is −10 °C. During a power cut it rises by 7 degrees. What is it now? 13. Abena owes GH₵18. She is given GH₵25 for her birthday and pays her debt. How much does she have? 14. A diver is at −12 m. She goes down a further 8 m. Where is she now? Write it as an integer. 15. A submarine at −40 m rises 40 m. Where is it?

Recap: adding integers

Adding is moving along the number line.

• Add a positive → move right (the answer gets bigger).
• Add a negative → move left (the answer gets smaller).

The two rules:

1. Same signs: add the sizes, keep the sign. (−4) + (−6) = −10.
2. Different signs: subtract the sizes, take the sign of the bigger size. (−4) + 6 = 2, but 4 + (−6) = −2.

Opposites cancel: 9 + (−9) = 0.

Watch out for these mistakes

• Writing (−4) + (−6) = −2. No: two negatives make a bigger negative, −10.
• Forgetting the sign in the answer. (−9) + 4 is −5, not 5.
• Mixing up "add a negative" with "subtract". They give the same answer, so you have not gone wrong, but be clear which you are doing.

One-minute test −5 + 8, −5 + (−8), 5 + (−8), 5 + 8. Answers: 3, −13, −3, 13.

Subtracting on the number line

Subtracting also means moving along the number line, but in the opposite direction to adding.

• Subtracting a positive number: move left.
• Subtracting a negative number: move right. (We will see why soon.)

Look at the picture. We start at 3. We subtract 4, so we move four steps to the left: 3 → 2 → 1 → 0 → −1. We land on −1.

So 3 − 4 = −1.

In Basic 4 you may have been told "you cannot take 4 from 3". Now you can. The answer is simply on the other side of zero.

Here is another: −2 − 3. Start at −2, move three steps left: −3, −4, −5. So −2 − 3 = −5. Taking away from a negative number makes it even more negative.

Subtracting a positive number

Subtracting a positive number moves you left. The answer is smaller than where you started. This is the subtraction you already know, extended past zero.

Examples

• 10 − 4 = 6
• 4 − 10 = −6 (start at 4, ten steps left)
• 0 − 7 = −7
• −3 − 5 = −8 (start at −3, five steps left)
• −20 − 30 = −50

A quick way: subtracting a positive is the same as adding a negative. 4 − 10 is the same as 4 + (−10). You can then use the different-signs rule: 10 − 4 = 6, bigger size is negative, answer −6.

Worked example: temperature drop A freezer is at −4 °C. It is switched to a colder setting and the temperature falls by 14 degrees. What is the new temperature?

Falling means subtracting: −4 − 14. Start at −4, go 14 steps left. −4 − 14 = −18 °C.

Try this 7 − 12, 0 − 3, −5 − 5, −1 − 9, 25 − 40.

Subtracting a negative number

This is the step that surprises everyone. Subtracting a negative number is the same as adding a positive number.

6 − (−2) = 6 + 2 = 8.

Why? Think about debt. Kofi has GH₵6 and owes a debt of GH₵2. His position is 6 + (−2) = 4. Now his aunt says, "Forget the debt, you need not pay." She has taken away the −2. Kofi's position goes up to 6. Removing a debt makes you richer.

Or think about the number line. Subtracting means going the opposite way to adding. Adding a negative goes left. So subtracting a negative goes right.

Examples

• 5 − (−3) = 5 + 3 = 8
• −4 − (−6) = −4 + 6 = 2
• −10 − (−10) = −10 + 10 = 0
• 0 − (−9) = 9

A memory trick: two minus signs next to each other turn into a plus. "− (−" becomes "+".

Try this 3 − (−4), −8 − (−2), −1 − (−1), 12 − (−12).

Worked example: the change in temperature

Subtraction tells us the difference between two numbers. This is very useful with temperatures.

Question 1. In the morning a freezer reads −18 °C. In the evening it reads −6 °C. By how much did it warm up?

Difference = later − earlier = −6 − (−18) = −6 + 18 = 12 degrees.

Check on the number line: from −18 up to −6 is 12 steps. Correct.

Question 2. The temperature in Accra at noon is 32 °C. Inside a cold store at the harbour it is −20 °C. What is the difference?

Difference = 32 − (−20) = 32 + 20 = 52 degrees.

Notice how large this is. When one number is above zero and the other is below, the gap is the sum of the two sizes, because you must cross zero in the middle.

Question 3. A town in Russia was −15 °C on Monday and −23 °C on Tuesday. Did it get warmer or colder, and by how much?

−23 − (−15) = −23 + 15 = −8. A negative change means it got colder, by 8 degrees.

Worked example: debts at the market

Ama's mother sells tomatoes at Makola Market. At the start of the week she keeps a record in a small book.

She owes the tomato farmer GH₵120 for a crate (−120). She has GH₵45 in her purse (+45).

Question 1. What is her overall position? −120 + 45 = −75. She is GH₵75 short of clearing her debt.

Question 2. She sells tomatoes for GH₵200. Now? −75 + 200 = +125.

Question 3. The farmer says the crate was smaller than agreed and cancels GH₵20 of the debt. What does cancelling a debt do? It is subtracting a negative: 125 − (−20) = 125 + 20 = 145.

Cancelling a debt is good news, and the maths agrees: her position goes up.

Worked example: the lift

Here is a problem with three steps, like a real journey.

A hospital in Kumasi has floors from −2 (lower car park) up to 6. Abena starts on floor −2.

Step 1. She goes up 5 floors to visit her grandmother. −2 + 5 = 3. She is on floor 3.

Step 2. She goes down 4 floors to the pharmacy. 3 − 4 = −1. She is on floor −1.

Step 3. She goes up 7 floors to the roof garden. −1 + 7 = 6. She is on the top floor.

Question. How many floors did Abena travel altogether? Add up the sizes of each journey: 5 + 4 + 7 = 16 floors. (Direction does not matter when we count the distance travelled.)

Question. What is the difference between where she started and where she finished? 6 − (−2) = 6 + 2 = 8 floors higher than she began.

Try this Kwame starts on floor 4, goes down 6, up 3, down 2. Which floor is he on?

Try this: subtracting integers

A. Subtract a positive

1. 5 − 9
2. −3 − 4
3. 0 − 11
4. −15 − 15

B. Subtract a negative 5. 8 − (−2) 6. −6 − (−9) 7. −4 − (−4) 8. 0 − (−30)

C. Mixed 9. 7 − 10 10. 7 − (−10) 11. −7 − 10 12. −7 − (−10)

D. Words 13. A freezer at −15 °C is turned down by another 6 degrees. New temperature? 14. The top of a hill is 320 m above sea level. A cave below it is at −40 m. How far is it from the cave to the top? 15. Kofi's position is −25 (he owes GH₵25). His friend cancels GH₵10 of it. New position? 16. The difference between two temperatures is 20 degrees. One is −8 °C. What could the other be? (There are two possible answers.)

Recap: subtracting integers

Subtraction moves you the opposite way to addition.

• Subtract a positive → move left. 3 − 5 = −2.
• Subtract a negative → move right. 3 − (−5) = 8.

The golden rule: subtracting a negative is the same as adding a positive. The two minus signs become a plus.

Subtraction also measures the difference between two numbers: later − earlier, or higher − lower. When the numbers are on opposite sides of zero, the difference is the sum of their sizes.

The four cases (using 7 and 3):

• 7 − 3 = 4
• 7 − (−3) = 10
• −7 − 3 = −10
• −7 − (−3) = −4

Common mistake: writing −7 − 3 = −4. No. Starting at −7 and going three more steps left gives −10. "Minus and minus" only turns into plus when the second minus is the sign of the number, as in −7 − (−3).

Word problems: temperature differences

Integers appear in weather reports around the world. Try these.

Problem 1. Yesterday a city in Norway had a high of 2 °C and a low of −9 °C. What was the range (the difference between high and low)?

2 − (−9) = 2 + 9 = 11 degrees.

Problem 2. The temperature at 6 a.m. was −3 °C. It rose by 8 degrees by noon, then fell by 12 degrees by midnight. What was the midnight temperature?

−3 + 8 = 5. Then 5 − 12 = −7 °C.

Problem 3. Four freezers in a shop read −12, −20, −9 and −16 °C. Which is coldest, and what is the difference between the coldest and the warmest?

Coldest: −20. Warmest: −9. Difference: −9 − (−20) = 11 degrees.

Your turn Thermometers in a laboratory read: −4, 7, −11, 0, 3. Find (a) the coldest, (b) the warmest, (c) the difference between them, (d) the reading that is closest to zero.

Word problems: the bank account

Banks and mobile money services use integers all the time. Money in is positive. Money out is negative. The balance is the running total.

Problem. Abena's mobile money wallet starts the week at GH₵40. Here are the transactions:

• Monday: received GH₵60 from her brother (+60)
• Tuesday: paid GH₵35 for school books (−35)
• Wednesday: paid GH₵80 for a textbook (−80)
• Thursday: received GH₵25 (+25)

Keep a running balance:

• Start: 40
• After Monday: 40 + 60 = 100
• After Tuesday: 100 − 35 = 65
• After Wednesday: 65 − 80 = −15 (she borrowed from the service's credit)
• After Thursday: −15 + 25 = 10

Abena ends the week with GH₵10.

Notice Wednesday. Her balance dipped below zero. Many banks charge a fee for this, so it is useful to spot it coming.

Your turn A shop's till starts at 0. Changes: +150, −90, −120, +200, −45. Find the balance after each change and at the end.

Integers in football

Football league tables use integers in the goal difference column (GD). Goal difference = goals scored − goals conceded.

If Hearts of Oak score 25 goals in a season and let in 18, their goal difference is 25 − 18 = +7.

If another team scores 12 and lets in 30, their goal difference is 12 − 30 = −18.

When two teams have the same points, the team with the higher goal difference is placed above. So a team on −3 is above a team on −10, because −3 > −10. Exactly the ordering rule you learned.

Worked example Three teams have the same points. Their goals are:

• Team A: scored 20, conceded 24
• Team B: scored 15, conceded 15
• Team C: scored 30, conceded 38

Goal differences: A = −4, B = 0, C = −8. Order from best: B (0), A (−4), C (−8).

Try this After a match, Team A wins 3–0. Update their goal difference. Team C loses 1–4. Update theirs.

Integers in a sequence

Counting with integers can go backwards past zero. Look at this sequence, going down by 3 each time:

10, 7, 4, 1, −2, −5, −8, …

It does not stop at 1. It carries on through zero into the negatives.

Here is one going up by 4, starting below zero:

−14, −10, −6, −2, 2, 6, 10, …

Worked example A sequence starts at −20 and goes up by 6 each time. What are the first six terms? Which is the first positive term?

−20, −14, −8, −2, 4, 10. The first positive term is 4 (the fifth term).

Worked example Find the missing numbers: 9, 5, 1, ☐, ☐, −11. Each step is −4. So after 1 comes −3, then −7, then −11. Missing numbers: −3 and −7.

Try this

1. Write the next three terms: 6, 3, 0, …
2. Write the next three terms: −9, −7, −5, …
3. A sequence goes down by 5 from 12. What is the first negative term?

Common mistakes and how to avoid them

Here are the errors teachers see most often. Check whether any of them are yours.

Mistake 1: "−8 is bigger than −3 because 8 is bigger than 3." Fix: picture the number line. −8 is further left, so it is smaller. Colder, deeper, more in debt.

Mistake 2: (−4) + (−5) = −1. Fix: same signs means add the sizes and keep the sign. −9.

Mistake 3: −4 − 5 = −1 or 1. Fix: start at −4 and go five more steps left. −9.

Mistake 4: 6 − (−2) = 4. Fix: subtracting a negative is adding. 6 + 2 = 8.

Mistake 5: forgetting the sign in the answer. (−9) + 2 is −7, not 7. Always ask: which side of zero did I land on?

Mistake 6: mixing up the sign of a number with the operation. In "5 − (−3)", the first minus is the operation (subtract) and the second minus is the sign of 3. The brackets are there to keep them apart. Use brackets whenever two signs would touch.

If in doubt, draw the number line. It is never wrong.

Puzzle: an integer magic square

In a magic square, every row, every column and both diagonals add up to the same number.

Here is one using integers. Read it row by row:

• Row 1: 1, −4, 0
• Row 2: −2, −1, 0
• Row 3: −2, 2, −3

Check the top row: 1 + (−4) + 0 = −3. Check the left column: 1 + (−2) + (−2) = −3. Check the main diagonal: 1 + (−1) + (−3) = −3.

Every line makes −3. Try the other rows, columns and the second diagonal yourself.

Your puzzle Copy this square and fill in the blanks so that every row, column and diagonal adds to 0:

• Row 1: 3, ☐, −1
• Row 2: ☐, 0, ☐
• Row 3: 1, ☐, −3

Hint: the middle row and middle column each need two numbers that are opposites. Start with the diagonals, which already have two numbers filled in.

Game: integer cards

This game helps the rules stick. You need a pack of cards or pieces of paper numbered from −10 to 10 (no zero needed, but you may include it).

How to play (two players)

1. Shuffle the cards and put them face down.
2. Player 1 turns over two cards and says the sum out loud. For example, −7 and 4: "negative seven plus four equals negative three."
3. Player 2 checks the answer. If it is right, Player 1 keeps the cards. If wrong, they go to the bottom of the pile.
4. Swap turns. The player with more cards when the pile runs out wins.

Harder version Turn over two cards and subtract the second from the first. For example, −2 and −9: "negative two minus negative nine equals seven."

Hardest version Turn over three cards and add all three.

Play for ten minutes a day for a week. By the end, the rules will feel natural and you will no longer need to count steps on the line.

Big recap and key words

Integer – a whole number: positive, negative or zero. Negative number – a number less than zero, written with a minus sign. Number line – a line with zero in the middle; right is bigger, left is smaller. Opposite – the same number on the other side of zero. The opposite of 4 is −4. Absolute value – the distance from zero. |−4| = 4. Greater than (>) and less than (<) – the open mouth faces the bigger number.

Ordering: negatives (most negative first), then zero, then positives.

Adding:

• same signs → add sizes, keep the sign
• different signs → subtract sizes, take the sign of the bigger size

Subtracting:

• subtract a positive → move left
• subtract a negative → move right (two minus signs become a plus)

Real life: temperature below freezing, money owed, floors below ground, depth below sea level, goal difference, profit and loss.

Whenever you are unsure, draw the line, put a dot on the start, and count the steps.

Self-test

Try all twenty without looking back. Then check your answers on the next page.

Ordering

1. Which is smaller, −6 or −2?
2. Put in order from smallest: 4, −4, 0, −9, 2.
3. True or false: −1 > −10.

Adding 4. −5 + 8 5. −5 + (−8) 6. 12 + (−20) 7. (−30) + (−30) 8. −9 + 9

Subtracting 9. 3 − 11 10. −3 − 11 11. 3 − (−11) 12. −3 − (−11) 13. 0 − (−6)

Problems 14. A freezer at −16 °C warms by 9 degrees. New temperature? 15. Ama owes GH₵14. She earns GH₵20. Her position? 16. A diver at −25 m rises 10 m. Where is she? 17. The difference between 15 °C and −15 °C? 18. Next two terms: 8, 3, −2, … 19. A team scores 22 and concedes 31. Goal difference? 20. Kofi is on floor −1, goes up 5, down 8. Which floor?

Self-test answers

1. −6 is smaller.
2. −9, −4, 0, 2, 4.
3. True.
4. 3
5. −13
6. −8
7. −60
8. 0
9. −8
10. −14
11. 14
12. 8
13. 6
14. −7 °C
15. +6 (she has GH₵6 after paying back)
16. −15 m
17. 30 degrees
18. −7, −12
19. −9
20. Floor −4

How did you do?

• 18–20: excellent. You are ready for multiplying and dividing integers in Basic 8.
• 13–17: good. Look again at the questions you missed and find the rule you slipped on.
• Below 13: go back to the number line pages and play the integer card game for a few days. Then try the test again.

Remember: nobody is born knowing this. It is practice that makes integers feel easy.

Learn more

• Khan Academy: Negative numbers — free videos and practice questions on the number line, ordering, adding and subtracting
• Maths is Fun: Positive and negative integers — clear explanations with pictures and a short quiz
• Purplemath: Negative numbers — the rules for signs explained step by step
• Britannica: Integer — a short reference article on what integers are

Watch more

Think about it

1. Where in your own town or home could you find a negative number? Think about fridges, lifts, loans and games.
2. Why is −1 bigger than −100, even though 100 is a bigger digit than 1?
3. If subtracting a negative is the same as adding a positive, what do you think multiplying two negatives might give? Make a guess and explain it.
4. A trader says, "My business made −GH₵200 this week." What does she mean, and is that the same as losing GH₵200?
5. Can you think of a situation where "zero" does not mean "nothing"?