Cartesian plane — axes, quadrants, ordered pairs: Foundations

Before you can trust a single line of the parent note, you must be able to read every mark on the page. Below is every symbol and idea the topic uses, built in an order where each one only leans on the ones before it. Nothing is assumed.

1. A number line — the seed of everything

The picture: a ruler that runs forever in both directions. Each spot on the ruler is a number.

Why the topic needs it: the whole Cartesian plane is just two number lines crossed at right angles. If you cannot picture one, you cannot picture the plane.

2. The symbols $x$ and $y$ — names for unknown positions

The picture: think of $x$ as a slider that can slide left or right, and $y$ as a slider that can slide up or down. A letter is just a labelled dial.

Why these letters? Descartes used letters near the end of the alphabet ( $x, y, z$ ) for unknown quantities. The topic keeps $x$ first because "across" is read first, like text left-to-right.

3. Perpendicular — the right angle that makes it all work

The picture: the corner of a sheet of paper. That exact corner is $9 0^{\circ}$ . Look for the little square drawn in a corner — that symbol means "right angle here".

Why the topic needs it: the two axes must be perpendicular so that "across" and "up" are completely independent. Moving across never changes your height, and moving up never changes your across-position. This independence is what lets ONE address $(x, y)$ pin down exactly one point.

4. The two axes and the origin

The picture: a big plus-sign $+$ made of two rulers. Where they overlap is home base: the origin.

5. The ordered pair $(x, y)$ — the address itself

The picture: a house address "Street 3, Floor 5". Swap them and you get a different house. $(3, 5)$ and $(5, 3)$ are two different points.

Why "ordered"? Because $(3, 5) \neq = (5, 3)$ . The comma-inside-brackets is a promise: across first, then up.

6. Reading signs: what $<$ , $>$ , $=$ mean here

Why the topic needs it: the four quadrants are defined entirely by these sign combinations. If you can read $>$ , $<$ , $=$ , you can name any region.

The figure above says it all directly: read each quadrant off the picture as a plain sentence.

7. Absolute value $∣ \cdot ∣$ — distance, ignoring direction

The picture: stand on the number line at $a$ and walk back to zero — count the steps. You never count negative steps.

Why the topic needs it: to measure the length of a horizontal or vertical gap. Using the subscript notation from Section 2, the distance across from a first point to a second is $∣ x_{2} - x_{1} ∣$ — the size of the gap between their across-values, whichever point sits further left.

8. The square root and Pythagoras' rule

Why the topic needs it: the straight-line distance between two points is the hypotenuse of a right triangle we can build from their coordinates. Let's build it step by step.

Formula Building the distance formula from scratch

Take a first point $A (x_{1}, y_{1})$ and a second point $B (x_{2}, y_{2})$ . We want the straight-line distance $d$ between them.

Step 1 — make a corner point. Drop straight down from $B$ and straight across from $A$ ; where they meet is a helper point $C (x_{2}, y_{1})$ . What it looks like: the orange (across) and green (up) lines in the figure meet at $C$ , forming an L-shape. Because one line is horizontal and one is vertical, the corner at $C$ is a right angle — exactly what Pythagoras needs.

Step 2 — measure the horizontal leg $A C$ . $A$ and $C$ share the same up-value $y_{1}$ , so the gap between them is purely across: $A C = ∣ x_{2} - x_{1} ∣$ .

Step 3 — measure the vertical leg $C B$ . $C$ and $B$ share the same across-value $x_{2}$ , so the gap is purely up: $C B = ∣ y_{2} - y_{1} ∣$ .

Step 4 — apply Pythagoras. The distance $d = A B$ is the hypotenuse, so $d^{2} = A C^{2} + C B^{2} = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} .$ (The absolute-value bars vanish because squaring already throws away the sign.)

Step 5 — undo the square with a square root: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} .$ Read it aloud: the gap across, squared, plus the gap up, squared, then square-root the total. Every symbol here is one you now own — see Distance formula for further practice.

How these foundations feed the topic

The list below reads as an "is needed for" chain: each idea on the left must be understood before the idea it points to on the right. It ends at the topic itself.

Number line → variables $x$ and $y$ , and the idea of a right angle.
Variables $x, y$ → the two axes and the origin → the ordered pair $(x, y)$ .
Ordered pair → sign reading ( $>, <, =$ ) → the four quadrants.
Ordered pair + absolute value → square root and Pythagoras → the distance formula.
Axes, quadrants and distance together → the Cartesian plane topic.

Everything downstream — Midpoint formula, Section formula, Slope of a line, Equation of a line, Area of triangle, Colinearity, and later Polar coordinates, the Complex plane and Vectors — is built on top of these same eight bricks. Return to the parent Cartesian plane once every symbol above reads instantly.

Equipment checklist

Cover the right side and answer aloud. If any line stumps you, reread its section.

What does a point on a number line represent?

A single number — the line is the numbers.

What does a negative number mean directionally?

Steps in the opposite direction from positive, not merely "smaller".

What does the subscript in

x_{2}

tell you?

Which point the across-value belongs to (here, the second point) — it doesn't change what

x

means.

What is a degree, and how many make a right angle?

A unit of turning; a full turn is

36 0^{\circ}

and a right angle is

9 0^{\circ}

(a quarter turn).

(x, y)

, which coordinate comes first and what does it measure?

x

first — horizontal (across) distance from the origin.

Why must the two axes be perpendicular?

So across and up are independent — one never affects the other, giving each point a unique address.

Where is the origin and what are its coordinates?

Where the axes cross;

O (0, 0)

What sign pattern defines Quadrant III?

x < 0

and

y < 0

(both negative).

Why does

(0, 5)

belong to no quadrant?

x = 0

puts it on the y-axis, which is a boundary, not a region.

What does

∣ x_{2} - x_{1} ∣

measure?

The length of the horizontal gap between the two points' across-values, always positive.

In Pythagoras' rule, what is

h

The hypotenuse — the longest side, opposite the right angle; here it is the distance between the two points.

Why does the distance formula contain a square root?

It undoes the squaring in Pythagoras' rule to recover the actual length

d

Recall Fast self-quiz

Sign of $x$ in Quadrant IV? ::: Positive ( $x > 0$ , $y < 0$ ). $∣ - 7∣ =$ ? ::: $7$ . Is $(4, - 5)$ the same point as $(- 5, 4)$ ? ::: No — order matters, different points.