−3 — The signs are different, so subtract: 8 − 5 = 3. The number with the larger absolute value is −8, so the answer takes its sign: −3.

12 — Negative × negative = positive, so (−4) × (−3) = +12.

−3 — Different signs give a negative quotient. 18 ÷ 6 = 3, so the answer is −3.

−8 — (−2)³ = (−2) × (−2) × (−2) = 4 × (−2) = −8. A negative number raised to an odd power stays negative.

Positive & Negative Numbers

Q: What is the absolute value of −7?

7 — |−7| = 7. Absolute value is the distance from 0, so you simply drop the sign.

Q: The temperature went from −3°C to +5°C. How much did it change?

8°C — Change = after − before = (+5) − (−3) = 5 + 3 = 8°C.

Q: Which of these is the smallest number?

−5 — The smallest number is the one farthest to the left on the number line: −5 < −1 < 0 < +2.

Q: A stock moved −¥200 on Monday, +¥150 on Tuesday, and −¥80 on Wednesday. What is the total change?

−¥130 — (−200) + (+150) + (−80) = −200 + 150 − 80 = −¥130.

Where this is used

Positive and negative numbers are the simplest tool we have for saying "more than" or "less than" some starting point. A winter morning at −5°C, a stock that closed ¥300 below yesterday, the Dead Sea sitting far below sea level — all of these are negative numbers. Finance, weather, maps, and even sports stats run on this one idea.

Start gently

There are numbers smaller than zero. Think of a thermometer: below 0°C you get −1°C, −2°C. All we have done is extend the number line to the left — everything else works exactly like the number line you already know.

Key points

What positive and negative numbers are

A number greater than 0 is a positive number; a number less than 0 is a negative number. You write a negative number by putting a minus sign in front of it.

The number line

Zero sits in the middle: positive numbers go to the right, negative numbers to the left. Each positive number has a matching negative number the same distance from 0 on the other side.

Absolute value

The absolute value of a number is how far it is from 0 on the number line. |+5| = 5 and |−5| = 5. Because it is a distance, an absolute value is never negative.

Rules for adding

Same signs: add the absolute values and keep the shared sign. Different signs: subtract the smaller absolute value from the larger one, then use the sign of the number with the larger absolute value.

Rules for multiplying and dividing

Two numbers with the same sign multiply or divide to a positive result. Two numbers with different signs give a negative result.

Think in stock prices and it sticks

If you own 3 shares and each one loses ¥200, your total is 3 × (−200) = −¥600. Multiplying by a negative number is just asking how many times a loss repeats.

See it drawn

Extend the number line to the left and negative numbers appear

Add 5 to −3 and you land on 2. Seen as movement left and right, arithmetic with negative numbers is nothing more than travel along the line.

Check yourself 10questions

Tap an option to check

Q1 What is the absolute value of −7?

Q2 (+5) + (−8) = ?

Q3 The temperature went from −3°C to +5°C. How much did it change?

Q4 (−4) × (−3) = ?

Q5 −18 ÷ (+6) = ?

Q6 Which of these is the smallest number?

Q7 A stock moved −¥200 on Monday, +¥150 on Tuesday, and −¥80 on Wednesday. What is the total change?

Q8 (−2)³ = ?

Q9 How many integers have an absolute value less than 4?

Q10 When is −a a positive number?

Jobs that use this

Financial Analyst $95k

Meteorologist $65k

Game Designer $75k

Geological Surveyor $60k

Financial Analyst$95k

Analyzes stocks, bonds, and currencies to guide investment decisions. Works with gains and losses — positive and negative rates of change — every single day.

Maths used: Positive & negative numbersPercentagesStatisticsCalculus (derivatives)

Meteorologist$65k

Builds forecasts from temperature, air pressure, and rainfall data. Below-zero temperatures and pressure differences are the basic arithmetic of the job.

Maths used: Positive & negative numbersGraphsStatisticsDifferential equations

Game Designer$75k

Designs scoring systems, difficulty balance, and character stats. Buffs and debuffs are literally just positive and negative values.

Maths used: Positive & negative numbersProbabilityGraph theoryOptimization

Geological Surveyor$60k

Surveys rock layers, mineral deposits, and groundwater. Elevation above and depth below sea level are recorded as positive and negative numbers all day long.

Maths used: Positive & negative numbersCoordinatesVectorsStatistics