Lesson 1: Integers and the Number Line

It is 6 a.m. in Chicago and the weather app on your phone says −4°F. That little minus sign is doing a big job: it tells you the temperature is 4 degrees below zero, not above it. By lunchtime the temperature has risen 9 degrees. Where does it end up? Questions like this are exactly what integers are for. Integers are the whole numbers and their opposites: …, −3, −2, −1, 0, 1, 2, 3, … In this lesson you will place integers on a number line, compare them, and add and subtract them with confidence — the foundation for almost everything else in this course.

The number line: a map for integers

Picture a straight line with 0 in the middle. Positive numbers stretch out to the right, growing bigger as you go. Negative numbers stretch out to the left, and here is the part that surprises people: as you move left, the numbers get smaller, even though the digits get bigger. The number −8 sits further left than −3, so −8 < −3. A good way to feel this: −8°F is a colder, harsher day than −3°F. A debt of $8 is worse than a debt of $3. On the number line, "bigger" always means "further to the right", never "bigger digits".

Integers describe any situation with two opposite directions: temperatures above and below zero, money you have and money you owe, floors above and below ground level, yards gained and lost in football, and heights above and below sea level. Whenever a situation has opposites like these, integers are the right tool.

Opposites and absolute value

Every integer has an opposite: the number the same distance from 0 but on the other side. The opposite of 7 is −7. The opposite of −12 is 12. Zero is its own opposite.

The absolute value of a number is its distance from 0 on the number line, and distance is never negative. We write it with two bars: |−8| = 8 and |8| = 8. Both 8 and −8 are exactly 8 steps from 0, so they share the same absolute value. Think of absolute value as answering the question "how far?" while ignoring the question "which direction?"

Worked example 1. A scuba diver is 25 feet below the surface, written as −25. A seagull is flying 25 feet above the surface, written as +25. Their heights look different, but their absolute values are equal: |−25| = |25| = 25, because both are 25 feet from the surface.

Worked example 2. How far apart are −3 and 5 on the number line? Count the steps, or subtract and take the absolute value: |5 − (−3)| = |8| = 8. The two numbers are 8 units apart.

Adding integers

Adding integers is movement on the number line. Adding a positive number moves you right; adding a negative number moves you left.

When the signs are the same, add the absolute values and keep the sign. When the signs are different, subtract the smaller absolute value from the larger one, and take the sign of the number with the larger absolute value.

Worked example 3. Compute −6 + 9. The signs differ. |9| − |−6| = 9 − 6 = 3, and 9 has the larger absolute value and is positive, so −6 + 9 = 3. On the number line: start at −6, hop 9 steps right, land on 3 — the weather question from the start of the lesson works the same way: −4 + 9 = 5, so by lunchtime it is 5°F.

Worked example 4. Compute −4 + (−7). The signs are the same, so add the absolute values, 4 + 7 = 11, and keep the negative sign: −4 + (−7) = −11. owing $4 and then owing $7 more means owing $11 altogether.

Subtracting integers

Subtraction has one golden rule: subtracting a number is the same as adding its opposite. So turn every subtraction into an addition and use the addition rules.

Worked example 5. Compute 4 − (−3). Change it to 4 + 3 = 7. This is why a temperature of 4°F feels 7 degrees warmer than −3°F: the gap between them is 4 − (−3) = 7 degrees.

Worked example 6. Compute −2 − 6. Change it to −2 + (−6) = −8. And in a football game, a team that gains 12 yards, then loses 7, then loses 15 finishes at 12 − 7 − 15 = −10: they end up 10 yards behind where they started.

In the real world: your bank account

Suppose your account holds $25 and you buy a $40 game. Your new balance is 25 − 40 = −$15. The negative balance is the bank's way of saying "you owe us $15". If you then deposit $20, your balance becomes −15 + 20 = $5. Banks, elevators (basement floors are often numbered −1, −2), and thermometers all speak integer — now you do too.

Common mistakes

• Thinking −9 is greater than −2 because 9 is greater than 2. Wrong: −2 is further right on the number line, so −2 > −9. Picture debt: owing $2 is better than owing $9.
• Dropping the sign when signs match. −5 + (−3) is −8, not 8. Same signs: add and keep the sign.
• Writing |−6| = −6. Absolute value is a distance, and distances are never negative: |−6| = 6.
• Forgetting to flip on subtraction. 3 − (−4) becomes 3 + 4 = 7, not 3 − 4 = −1.

Exam Experience

1. What is the value of |−12|? A) −12 B) 0 C) 12 D) 24

2. A lift starts on floor 3, travels down 5 floors, then up 2 floors. On which floor does it stop? A) −4 B) 0 C) 4 D) 10

3. Which list shows the numbers in order from least to greatest? A) −1, −5, 0, 3 B) −5, −1, 0, 3 C) 0, −1, −5, 3 D) 3, 0, −1, −5

4. (Constructed response) The table shows the noon temperatures, in °C, for one week: Monday −2, Tuesday 3, Wednesday −5, Thursday 0, Friday −1. (a) Which day was warmest and which was coldest? (b) How many degrees warmer was the warmest day than the coldest day? (c) Find the mean (average) of the five temperatures.

Answer key and explanations

• 1. C — Absolute value is distance from 0, and −12 is 12 units from 0.
• 2. B — 3 − 5 + 2 = 0. The lift ends on floor 0 (the ground floor).
• 3. B — On the number line, −5 is furthest left, then −1, then 0, then 3.
• 4. (a) Warmest: Tuesday (3°C); coldest: Wednesday (−5°C). (b) 3 − (−5) = 8 degrees. (c) (−2 + 3 − 5 + 0 − 1) ÷ 5 = −5 ÷ 5 = −1°C.

Scoring guide for question 4 (3 points)

• 1 point for correctly naming Tuesday as warmest and Wednesday as coldest.
• 1 point for the difference, 8 degrees, with a subtraction that adds the opposite (3 − (−5) = 8) or equivalent reasoning.
• 1 point for the mean, −1°C, from a correct total of −5 divided by 5.

Going further: chains of integers

Real problems rarely involve a single step. The good news: chains of additions and subtractions are handled by working steadily left to right, combining as you go.

Worked example 7. In a board game, Priya's score changes like this: +15, −20, +8, −5, +12. Starting from 0, her running score goes 15, then −5, then 3, then −2, and finally 10. Her final score is 0 + 15 − 20 + 8 − 5 + 12 = 10 points.

Worked example 8. Evaluate −9 + 14 − 6. Take it in stages: −9 + 14 = 5 (different signs, subtract and keep the sign of the bigger). Then 5 − 6 = −1.

Worked example 9. City A is at −8°C. City B is 11 degrees warmer, so it is −8 + 11 = 3°C. City C is 6 degrees colder than A: −8 − 6 = −14°C. The gap between B and C is 3 − (−14) = 17 degrees — warmer-minus-colder always needs that "subtract a negative" move.

Quick practice

Try these, then check the answers.

• 1. −7 + (−5)
• 2. 6 − (−9)
• 3. |−20| − |4|
• 4. A lift starts at floor −2 (the car park), rises 9 floors, then descends 4. Where does it stop?

Answers. 1. −12 (same signs: add and keep negative). 2. 15 (subtracting a negative becomes adding). 3. 20 − 4 = 16. 4. −2 + 9 − 4 = floor 3.

Investigation: one day in numbers

Follow a single day through the eyes of integers. (a) At midnight the temperature is −6°C. It rises steadily by 3 degrees per hour for 4 hours, reaching −6 + 4 × 3 = 6°C at 4 a.m. — note how the multiplication happens before the addition, exactly as lesson 3 will formalise. Then a front arrives and the temperature falls 8 degrees by breakfast: 6 − 8 = −2°C. The day's total swing, from its coldest (−6) to its warmest (6), is 6 − (−6) = 12 degrees. (b) A hiker's altimeter reads +38 m at a cliff-top lighthouse and a dive boat reports a wreck at −24 m. The vertical distance between the lighthouse and the wreck is 38 − (−24) = 62 m — a subtraction of a negative, doing real work in the real world. (c) A bank account is overdrawn at −$12. A deposit of $50 arrives, then a $65 payment leaves: −12 + 50 − 65 = −$27. Three situations, one toolkit: place everything on the number line, and let the signs carry the meaning. When a problem mixes rises and falls, gains and losses, heights and depths, it is asking for integers — recognise the costume and the method is already chosen.

Challenge corner

The highest point on Earth, Mount Everest, stands at +8,849 m above sea level. The deepest point, the Mariana Trench, lies at −10,935 m. The total vertical distance between them is 8,849 − (−10,935) = 19,784 m — nearly 20 km of integer subtraction. Now your turn to reason: a walker starts at sea level, descends 15 m into a cave system, climbs 22 m, then descends 9 m more. Her position is −15 + 22 − 9 = −2 m: still just below where she started, despite the big climb in the middle. Chains of integers reward those who keep a running total rather than guessing from the largest number — write the running total down, step by step, every time.