The address of a point

Liri had hundreds of single memory points on her map – but no way of describing one particular point exactly.

Liri: “How do I tell you exactly where a point is, without pointing at it with my finger?”
Erklärus: “With a coordinate system. Every point gets its own address – two numbers that fix it unambiguously.”

🧭 Structure – axes, origin, quadrants

The coordinate system is made of two number lines that stand at right angles to each other. They cross at the origin – the point (0, 0). This creates four regions called quadrants.

I II III IV x y 1 2 3 4 -1 -2 -3 -4 1 2 3 4 -1 -2 -3 -4 O (0, 0) A (3, 2) B (−2, 3) C (2, −3)
The four quadrants
Quadrant I ✦ we start here
x > 0, y > 0
top right
Quadrant II
x < 0, y > 0
top left
Quadrant III
x < 0, y < 0
bottom left
Quadrant IV
x > 0, y < 0
bottom right
Liri: “Four regions – that suddenly looks like a lot!”
Erklärus: “Don’t worry. Quadrants II, III and IV need negative numbers – you’ll get to know those properly later. To begin with we stay entirely in quadrant I, where x and y are both positive. That is enough to understand the whole principle.”

➕ The x-axis and the y-axis (in quadrant I)

A coordinate system has two axes: the horizontal x-axis and the vertical y-axis. They meet at the origin.

x y 0 1 2 -1 1 2 P (2, 2)

A point is written as (x, y): first the steps to the right, then upwards. The point above is P(2, 2) – 2 to the right, 2 up.

Plotting a point – from the address to the point
Erklärus: “Plot the point P(3, 4). What do you do?”
Liri: “First walk: 3 steps to the right along the x-axis. Then climb: 4 steps up. That is where P is!”
1. Start at the origin (0, 0)
2. x = 3 → 3 units to the right
3. y = 4 → 4 units up
→ mark the point P(3, 4) ✓
Reading a point – from the point to the address
Erklärus: “And the other way round: you see a point on the graph. How do you find its coordinates?”
Liri: “I draw a line straight down to the x-axis – that gives me the x value. Then a line sideways to the y-axis – that gives me the y value. Like crosshairs!”
A point is 3 to the right and 1 up. How do you write that? (x, y)

🗺️ Liri’s interactive star map

Enter coordinates and see where the point lands – or click straight into the coordinate system!

Click in the coordinate system or enter x and y.
🌌 Coordinates in the universe

Astronomers use their own coordinate system in the sky to find stars exactly – with “right ascension” and “declination” instead of x and y.
GPS satellites give every place on Earth two coordinates (latitude and longitude) – just like a point in the coordinate system.
Without coordinates you couldn’t tell another person exactly where a particular star or place is.

❌ Common mistake

The order is often swapped: (3, 1) means first 3 to the right, THEN 1 up – not the other way round!

Memory aid: “Along the corridor (x), then up the stairs (y)” – first walk to the stairs, then climb up.

📍 Exercise 1
A point is 4 to the right and 2 up. Write the coordinates.
📍 Exercise 2
How many steps to the right does the x-coordinate show for (5, 3)?
📍 Exercise 3
How many steps up does the y-coordinate show for (5, 3)?

Every point now has an address

Liri drew a grid over her whole star map. From now on she could name every memory point exactly.

Liri: “My home is at (7, 4) – now I can even write it down instead of just pointing.”
“Coordinate system: x-axis horizontal, y-axis vertical. Point (x, y): first x steps right/left, then y steps up/down.”
What you learned today
P (x, y)
First x (right/left), then y (up/down)
🌟

Chapter 28 mastered!

Liri can now name every point on her map exactly, using coordinates.

Erklärus’s notebook – page 28:
“Coordinate system: x-axis (horizontal), y-axis (vertical), origin in the middle. A point is written as (x, y) – first right/left, then up/down.”
← Back to overview 28 – Coordinate System