Liri measured the temperature on an icy moon: minus 15 degrees. “How can a number be smaller than nothing?”
Erklärus: “Numbers don’t only carry on upwards – they also go below zero. These are the negative numbers.”
What lies below zero?
🔢 Natural numbers ℕ and integers ℤ
You already know the natural numbers (ℕ): 1, 2, 3, 4… They are always positive. Now you widen your view to the integers (ℤ) – these include the negative numbers and zero as well.
Erklärus: “The natural numbers are a subset of the integers. Every natural number is also an integer – but not the other way round. −5 is an integer, but not a natural number.”
Liri: “So ℕ is like a smaller spaceship that flies inside ℤ?”
Erklärus: “A perfect picture. And zero? It belongs to both – it is the border station.”
↔️ The number line – infinite in both directions
A number line has no beginning and no end. Zero lies in the middle, positive numbers grow to the right, negative ones to the left – like the universe itself, which expands in all directions.
📏 The number line from −5 to +5
● Negative numbers (left) · ● Zero · ● Positive numbers (right)
Liri: “So −5 is smaller than −2, even though 5 is bigger than 2?”
Erklärus: “Exactly. The further a negative number is from zero, the smaller it is.”
Which number is bigger: -3 or -8?
🌡️ Calculating with negative numbers
The temperature is -5°C and rises by 8°C. How warm is it then?
Erklärus: “You go 8 steps to the right on the number line, starting from -5. -5 + 8 = 3.”
Work out: -5 + 8 =
➕➖ Signs and operation signs
The operation sign tells you what you are doing (adding, subtracting…). The sign of the number tells you what the number is (positive or negative). When two signs stand next to each other, they are combined:
➕➖ Sign rule for addition & subtraction
Same signs → plus
+(+3) → +3
−(−3) → +3
Different signs → minus
+(−3) → −3
−(+3) → −3
⚛️ The antimatter connection
When an electron (matter, negatively charged) meets a positron (antimatter, positively charged), they wipe each other out – the result is zero, pure energy. In mathematics: (−3) + (+3) = 0. Equal amounts, opposite signs → annihilation.
Erklärus says: “Negative numbers weren’t invented – they were discovered, because the universe was already using them.”
✖️➗ Multiplication and division with negative numbers
The same rule applies here: same signs → positive result. Different signs → negative result.
| Sign A | Operation | Sign B | Result | Example |
|---|---|---|---|---|
| + | × or ÷ | + | + | 3 × 4 = 12 |
| − | × or ÷ | − | + | (−3) × (−4) = 12 |
| + | × or ÷ | − | − | 3 × (−4) = −12 |
| − | × or ÷ | + | − | (−3) × 4 = −12 |
Liri: “Minus times minus is plus – that sounds strange at first. But if antimatter meets antimatter, a double negative gives reality… that makes sense!”
Erklärus: “Exactly. Two negative charges that reverse each other – the result is positive. Mathematics and physics speak the same language.”
Work out: (-6) × (-3) =
🎮 Interactive – calculating on the number line
Enter a number and choose an operation. Watch how the result moves along the number line:
🎮 Number-line calculator
3 + 5 = 8
Start at 3, go 5 steps to the right → result: 8
🌌 Negative numbers in the universe
On the night side of some moons the temperature falls to -180°C.
Heights below sea level are often given with negative numbers, e.g. -30 m.
Without negative numbers, astronomers couldn’t sensibly give extreme cold in space.
❌ Common mistake
Many people think -8 is bigger than -3, because 8 > 3 – but with negative numbers it is the other way round!
Tip: look at the number line – further to the right is always bigger, whether the number is positive or negative.
↔️ Exercise 1 – Comparing
Which number is bigger: -10 or -2?
🌡️ Exercise 2 – Adding
Work out: -7 + 10
🌡️ Exercise 3 – Subtracting
The temperature is 2°C and falls by 6°C. How cold does it get?
There is a way even below zero
Liri wrote the icy temperatures of the moon onto her map – for the first time with minus signs. Suddenly she paused.
Liri: “Wait – these pairs of numbers that cancel each other out… +7 and −7, which make zero together… That is exactly how the energy balance of a starship was calculated at my home! Consumption and supply that balance each other out.”
Erklärus: “A real memory, Liri. If you know how to calculate with positive AND negative values, you can understand every balance – even the one that leads you home.”
Liri: “Now I know: even if a number is below zero, I don’t lose it. It simply has a place to the left of zero – just like a part of my memory that is slowly coming back.”
“Negative numbers lie to the left of zero. The further left, the smaller. -5+8=3 means: 8 steps to the right from -5. Minus times minus = plus.”
What you learned today
-5 + 8 = 3
Further left = smaller · Further right = bigger
🌟
Chapter 39 mastered!
Liri can now calculate with numbers below zero – even in icy cold.
Erklärus’s notebook – page 39:
“Negative numbers: to the left of zero, the further left the smaller. Calculating on the number line: plus = go to the right, minus = go to the left.”
“Negative numbers: to the left of zero, the further left the smaller. Calculating on the number line: plus = go to the right, minus = go to the left.”