Erklärus
Erklärus
9 years old · glasses · always has his notebook · can’t sleep for all his questions
Liri
Liri ⭐
A lost star · warm and golden · searching for the way home

The beginning – a night in November

Erklärus couldn’t sleep.

He never could – not because he had nightmares, but because he had questions. Always. Far too many questions for a single night.

He lay on his back, staring through his skylight into the black November sky – and counted stars. Not to get tired, but because he wanted to know: How many are really up there?

“One… two… three…” he murmured.

Then – just after midnight – there was a knock at the window. Not loud. More like someone knocking politely and hoping that somebody would open up.

Erklärus opened the window. On the windowsill sat a little star – about the size of an apricot, warm and golden, and trembling a little. Like someone who has travelled a long way and doesn’t know whether they are welcome.

Liri: “Sorry to disturb you. I’m lost. Can you help me?”

Erklärus blinked. He took off his glasses, cleaned them carefully and put them back on. The star was still there.

Erklärus: “You… speak?”
Liri: “All stars speak. It’s just that you humans mostly don’t listen – because you’re too busy. Mostly you let your mobile phones distract you.”

That was fair, thought Erklärus. He fetched his notebook and wrote on the first page:

“Tonight: met a star. Name: Liri. Goal: find the way home. Tool: mathematics.”
Liri: “I come from a small constellation – far, far away. But I don’t remember how to find it. I don’t even remember how many stars live there.”
Erklärus: “How many – that is a question for numbers. And numbers – that is mathematics. Let’s start right at the beginning.”
Liri: “How far back is the beginning?”
Erklärus: “At the very first number. At the number one.”
🌌 The real universe

Our solar system has 8 planets. Astronauts on the International Space Station see 16 sunrises in a single day, because they fly around the Earth so fast. How do we know? Because people began to count. Without numbers, humankind would never have built a spaceship, never named a star, never found its way through the universe.

What do these ten symbols mean?

Erklärus fetched a slip of paper and wrote ten symbols on it. Click on a number – and discover what it really means:

0Zero
1One
2Two
3Three
4Four
5Five
6Six
7Seven
8Eight
9Nine
← Click on a number!
Erklärus: “These are ten symbols. With these ten symbols you build all numbers – small ones, like your fingers. And bigger ones, like the stars in the Little Bear.”
Liri: “Only ten? For everything?”
Erklärus: “Only ten. The power isn’t in the symbols themselves – it’s in how you arrange them. Just as twelve notes make all the music in the world.”

Liri glowed a little brighter. She liked that – the idea that a few things together can make infinitely many.

The number line – where do numbers live?

“Look,” said Erklärus, and drew a long line in his notebook. “The natural numbers have an address – they stand in a fixed order. Click on a number!”

0
Zero?
1
Start
2
3
4
5
6
7
8
9
10
… ∞
← Click on a number!
Liri: “Why do mathematicians argue about zero?”
Erklärus: “Zero means: nothing. Some say: nothing isn’t a quantity, so it doesn’t belong among the counting numbers. Others say: ‘I have zero biscuits’ is still a statement. I say: zero belongs. After all, ‘no star found’ is also an answer.”
Liri: “Nothing is something too. I like that.”

What do we need natural numbers for?

Erklärus opened his notebook. He wrote down examples – from everyday life, from the universe, from his room:

In everyday life3 apples. 7 friends. 12 months in a year. 7 days in a week.
At school18 pupils in the class. 6 subjects. 5 school days a week.
In the universe8 planets. 2 moons of Mars. 1 Sun. 7 stars in the Plough.
In Erklärus’s room1 pair of glasses. 3 books. 1 notebook. 1 star on the windowsill.
Erklärus: “Without natural numbers you can’t describe anything that exists more than once. No recipe, no timetable, no galaxy.”
Liri: “And I couldn’t find my way home. Because I don’t know how many stars live there.”
Erklärus: “Exactly. So let’s start counting.”
❌ Common mistake in thinking

Many people think numbers are just symbols to learn by heart. You learn: “The sign 5 means five.” Done.

But that isn’t enough. A number describes a real quantity. The “5” means: five things really exist. When you count five stars, the five is not a mark on paper – it is the answer to a real question about the world.

Anyone who learns numbers as meaningless symbols will never truly understand mathematics. For those who understand numbers as descriptions of reality, the universe opens up.

🔭 Erklärus’s first task
Liri is pointing at stars in the Milky Way.
Click on each star one at a time – one after the other. Count out loud! How many stars can you see?
⭐ ⭐ ⭐ ⭐ ⭐ ⭐ ⭐
0
No star clicked yet
📓 Erklärus’s notebook – what you need to know

Natural numbers are the numbers 1, 2, 3, 4, 5 … They are positive and whole – no decimals, no negative numbers.

Zero is a special case: some people count it (ℕ₀ = {0, 1, 2, 3, …}), some don’t (ℕ = {1, 2, 3, …}). Both views have their merits.

The most important property: the natural numbers have no end. After every number comes a bigger one. This is called infinity – one of the most astonishing ideas in mathematics.

The natural numbers
ℕ = { 1, 2, 3, 4, 5, … }
With zero: ℕ₀ = { 0, 1, 2, 3, … }  ·  Positive · Whole · Infinitely many

The end of this night

Erklärus looked out of the window. The Milky Way stretched across the dark sky like a pale ribbon.

Erklärus: “Do you know what fascinates me most about numbers? They never stop. After every number comes a bigger one. The universe goes on into infinity – and so does mathematics.”
Liri: “Then perhaps my home isn’t so far away after all, if you count properly?”
Erklärus: “We’ll find out. Step by step. Number by number.”

Liri settled on Erklärus’s shoulder. She was warm. That surprised him. He closed his eyes and thought of the last thing he had seen before he drifted off.

Liri: “I remember a giant planet with rings. I flew past it and counted its moons, because they shone so brightly. Seven moons. Exactly seven, like your stars today.”
Erklärus: “A planet with seven moons. We’ll write that down – every memory is a point on your map home.”
“First lesson: numbers describe reality. They are not symbols – they are answers to questions about the world.”

Outside, the stars quietly followed their paths. Erklärus yawned for the first time. Perhaps, he thought, he might get a little sleep tonight after all.

🌟

Chapter 1 mastered!

Erklärus and Liri have made a start. Seven stars counted. The first question answered. The journey has begun.

Erklärus’s notebook – page 1:
“Natural numbers describe quantities. They start at 1 and never stop. Zero belongs too – because nothing is also information. Liri fell asleep on my shoulder. She is warm. Tomorrow we travel on.

Memory map, point 1: a giant planet with rings and seven moons, which Liri flew past.”
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