Two Axes, Four Quadrants: The Cartesian Plane — Class 9 Mathematics Chapter 1

🖼 Image PendingA wide cinematic visualisation: two perpendicular axes of light meeting at a single bright origin point, arrows extending in all four directions into a softly star-lit darkness, faintly suggesting four quadrants

AI Generation Prompt

Ultra-wide cinematic banner (16:5 ratio). Two perpendicular axes drawn as glowing beams of light meet at a single bright origin point at the centre of the frame. The horizontal axis extends to the left and right, the vertical axis extends up and down. Each axis ends in a soft arrow of light. The four regions formed between the axes are subtly differentiated by faint pastel hues — suggesting the four quadrants without labelling them. The background is a deep starry darkness, with a faint scattering of mathematical glyphs (a small zero, a plus and minus sign) hovering near the origin. The image conveys: a flat universe organised by two simple lines and four equal regions. Painterly cinematic illustration, atmospheric and minimal. Dark background. No text, no labels.

✦Think About It

On a single straight road, one number locates any point — '20 metres from the bus stop' is enough.

Now stand against a flat wall and try to describe the position of a fly resting on it. '15 cm from the corner' is no longer enough — the fly could be anywhere on a quarter-circle of radius 15 cm.

How many numbers do you need to pin down the fly?

What if you measured along the floor and up from the floor separately?

✦ब्रह्मस्फुटसिद्धान्त — १८.३०

Brahmagupta on the Nature of Zero

धनयोरृणयोश्च योगो धनं ऋणयोः समैक्यं शून्यं।

(dhanayoḥ ṛṇayoḥ ca yogaḥ dhanaṃ, ṛṇayoḥ samaikyaṃ śūnyam)

'दो धन (positive) जोड़ो तो धन ही मिलेगा। दो ऋण (negative) जोड़ो तो ऋण मिलेगा। बराबर के धन और ऋण को जोड़ दो — तो शून्य बचता है।'

'A positive plus a positive is positive. A negative plus a negative is negative. A positive and an equal negative add to zero.'

Brahmagupta wrote these rules in 628 CE, in a single careful Sanskrit verse. They look almost too simple to deserve a formal verse — and yet without them, the picture you are about to draw on this page is impossible. The very point at which the two axes meet — the origin — is zero. The left half of the x-axis is the world of negative numbers. Every diagram in the rest of this chapter rests on Brahmagupta's careful sentence.

From one number line to two

In earlier classes you met the number line — a single horizontal line with 0 in the middle, positive numbers stretching to the right, and negative numbers stretching to the left. The number line is one-dimensional: each point on it is described by exactly one number.

The 2-D Cartesian coordinate system is what you get when you take two number lines and lay them at right angles to each other, crossing at zero on both. By convention:

The horizontal number line is called the x-axis.
The vertical number line is called the y-axis.
The point where the two axes meet is called the origin, written as O. Its coordinates are $(0, 0)$ .

Distances are measured along each axis in equal units — every step of one unit is the same length on the x-axis and on the y-axis. By convention:

Moving right of O on the x-axis or upward of O on the y-axis is counted as positive.
Moving left of O on the x-axis or downward of O on the y-axis is counted as negative.

That single set of choices — two perpendicular axes, an origin at zero, equal units, positive to the right and up — gives you the Cartesian plane. Every point in the flat plane now has a precise address. There is no spot, however small, that this system cannot describe.

🖼 Image PendingA green graph-paper grid on a dark background. Two thick green axes labelled x-axis (horizontal) and y-axis (vertical) meet at the origin O = (0, 0). Tick marks are drawn at every integer from −7 to +7 on the x-axis and from −5 to +5 on the y-axis. Four labelled points are plotted as small dots: E = (−2.9, 0) on the negative x-axis, B = (4.5, 0) on the positive x-axis, G = (0, −4.5) on the negative y-axis, and H = (0, 4) on the positive y-axis.

AI Generation Prompt

Clean Cartesian plane diagram drawn on green graph paper over a dark background. Two thick green axes meet at a clearly labelled origin O = (0, 0). The horizontal axis is labelled 'x-axis' to the right; the vertical axis is labelled 'y-axis' upward. Tick marks at every integer are visible from −7 to +7 on the x-axis and from −5 to +5 on the y-axis, with the integer numbers written near each tick. Four points are plotted as small bright dots, each with a clear label: 'E = (−2.9, 0)' on the negative x-axis at x = −2.9, 'B = (4.5, 0)' on the positive x-axis at x = 4.5, 'G = (0, −4.5)' on the negative y-axis at y = −4.5, and 'H = (0, 4)' on the positive y-axis at y = 4. Style: educational mathematical diagram, crisp lines, technical clarity. Dark background, orange accent labels, clean technical illustration style.

📸 The Cartesian plane with the origin O = (0, 0), and four example points: B = (4.5, 0), G = (0, −4.5), H = (0, 4), and E = (−2.9, 0).

Reading and writing coordinates

Look at the diagram above. To pin down any point in the plane, you need two numbers — and the order in which you give them matters.

The first number is always the x-coordinate — the perpendicular distance from the y-axis (i.e., how far left or right of O the point is). The second number is the y-coordinate — the perpendicular distance from the x-axis (how far up or down from O). The point is then written as $P = (x, y)$ .

Let's read off the four labelled points from the diagram:

B = (4.5, 0) — B is on the x-axis, 4.5 units to the right of O. So x = 4.5; y = 0.
G = (0, −4.5) — G is on the y-axis, 4.5 units below O. So x = 0; y = −4.5.
H = (0, 4) — H is on the y-axis, 4 units above O. So x = 0; y = 4.
E = (−2.9, 0) — E is on the x-axis, 2.9 units to the left of O. So x = −2.9; y = 0.

A few useful patterns fall out immediately:

A point of the form $(x, 0)$ — with $y = 0$ — lies exactly on the x-axis. If $x > 0$ , it's to the right of O; if $x < 0$ , it's to the left.
A point of the form $(0, y)$ — with $x = 0$ — lies exactly on the y-axis. If $y > 0$ , it's above O; if $y < 0$ , it's below.
The single point where both coordinates are zero — $(0, 0)$ — is the origin itself.

Shorthand. While writing coordinates, it is often convenient to drop the equals sign. Instead of writing $P = (x, y)$ , mathematicians simply write $P(x, y)$ . So $H = (0, 4)$ and $H(0, 4)$ mean exactly the same thing. The shorter form is what you will see most often when points are marked on a graph.

Loading simulator…

Worked Example 1.1

SOLVED

On the same Cartesian plane shown above, plot the point $Q = (-3, 2)$ . Then describe in plain words where Q lies relative to the origin O.

Solution

Step 1 — Read the coordinates.

The first number, $x = -3$ , tells us the horizontal distance from the y-axis. Since $-3$ is negative, Q lies 3 units to the left of the y-axis.

The second number, $y = 2$ , tells us the vertical distance from the x-axis. Since $+2$ is positive, Q lies 2 units above the x-axis.

Step 2 — Locate the point on the diagram.

Starting at the origin O:

Move 3 units to the left along the x-axis — you are now at $(-3, 0)$ .
From that spot, move 2 units straight up — you are now at $(-3, 2)$ .
Mark the point. That spot is Q.

Step 3 — Describe Q in plain words.

Q is 3 units to the left of O, and 2 units above O. It lies in the upper-left region of the plane (the region where x is negative and y is positive).

Answer: $Q(-3, 2)$ is 3 units left and 2 units above the origin.

Worked Example 1.2

SOLVED

A point lies on the x-axis and is 5 units to the left of the origin. What are its coordinates?

Solution

Step 1 — Use the rule for a point on the x-axis.

Any point that lies on the x-axis has $y = 0$ . (The y-coordinate measures distance from the x-axis itself, so a point on the x-axis is zero distance away.)

Step 2 — Find the x-coordinate.

The point is 5 units to the left of the origin. Distances measured to the left of O are negative, by convention. So $x = -5$ .

Step 3 — Write the coordinates.

Using the form $(x, y)$ :

$\text{Point} = (-5, 0)$

Answer: The coordinates of the point are $(-5, 0)$ .

Logical ReasoningLevel 3 · Analysis

Two friends, Aarav and Bina, both write down the location of a point P in the Cartesian plane. Aarav writes $(3, 7)$ . Bina writes $(7, 3)$ .

Aarav says: 'These are clearly the same point — both pairs use the same two numbers.'

Is Aarav right?

Remember

The four most important facts on this page:

The Cartesian plane is built from two perpendicular number lines — the x-axis (horizontal) and the y-axis (vertical) — meeting at the origin $O = (0, 0)$ .
Any point P in the plane is described by an ordered pair $(x, y)$ , where $x$ is the perpendicular distance from the y-axis and $y$ is the perpendicular distance from the x-axis.
Sign conventions: right of O and above O are positive; left of O and below O are negative.
Order matters. The pair $(x, y)$ is ordered — $(3, 7)$ and $(7, 3)$ are different points in the plane.

Ready to Go Beyond

Once you are comfortable with two axes, the door opens to several beautiful generalisations:

End-of-Chapter Q1Coordinates of the origin

NCERT

What are the x-coordinate and y-coordinate of the point of intersection of the two axes?

Quick Check

Q1.What are the coordinates of the origin of the Cartesian plane?

AI Generation Prompt

✦Think About It

On a single straight road, one number locates any point — '20 metres from the bus stop' is enough.

How many numbers do you need to pin down the fly?

What if you measured along the floor and up from the floor separately?

✦ब्रह्मस्फुटसिद्धान्त — १८.३०

Brahmagupta on the Nature of Zero

धनयोरृणयोश्च योगो धनं ऋणयोः समैक्यं शून्यं।

(dhanayoḥ ṛṇayoḥ ca yogaḥ dhanaṃ, ṛṇayoḥ samaikyaṃ śūnyam)

'A positive plus a positive is positive. A negative plus a negative is negative. A positive and an equal negative add to zero.'

From one number line to two

The 2-D Cartesian coordinate system is what you get when you take two number lines and lay them at right angles to each other, crossing at zero on both. By convention:

The horizontal number line is called the x-axis.
The vertical number line is called the y-axis.
The point where the two axes meet is called the origin, written as O. Its coordinates are $(0, 0)$ .

Distances are measured along each axis in equal units — every step of one unit is the same length on the x-axis and on the y-axis. By convention:

Moving right of O on the x-axis or upward of O on the y-axis is counted as positive.
Moving left of O on the x-axis or downward of O on the y-axis is counted as negative.

AI Generation Prompt

📸 The Cartesian plane with the origin O = (0, 0), and four example points: B = (4.5, 0), G = (0, −4.5), H = (0, 4), and E = (−2.9, 0).

Reading and writing coordinates

Look at the diagram above. To pin down any point in the plane, you need two numbers — and the order in which you give them matters.

Let's read off the four labelled points from the diagram:

B = (4.5, 0) — B is on the x-axis, 4.5 units to the right of O. So x = 4.5; y = 0.
G = (0, −4.5) — G is on the y-axis, 4.5 units below O. So x = 0; y = −4.5.
H = (0, 4) — H is on the y-axis, 4 units above O. So x = 0; y = 4.
E = (−2.9, 0) — E is on the x-axis, 2.9 units to the left of O. So x = −2.9; y = 0.

A few useful patterns fall out immediately:

A point of the form $(x, 0)$ — with $y = 0$ — lies exactly on the x-axis. If $x > 0$ , it's to the right of O; if $x < 0$ , it's to the left.
A point of the form $(0, y)$ — with $x = 0$ — lies exactly on the y-axis. If $y > 0$ , it's above O; if $y < 0$ , it's below.
The single point where both coordinates are zero — $(0, 0)$ — is the origin itself.

Loading simulator…

Worked Example 1.1

SOLVED

On the same Cartesian plane shown above, plot the point $Q = (-3, 2)$ . Then describe in plain words where Q lies relative to the origin O.

Solution

Step 1 — Read the coordinates.

The first number, $x = -3$ , tells us the horizontal distance from the y-axis. Since $-3$ is negative, Q lies 3 units to the left of the y-axis.

The second number, $y = 2$ , tells us the vertical distance from the x-axis. Since $+2$ is positive, Q lies 2 units above the x-axis.

Step 2 — Locate the point on the diagram.

Starting at the origin O:

Move 3 units to the left along the x-axis — you are now at $(-3, 0)$ .
From that spot, move 2 units straight up — you are now at $(-3, 2)$ .
Mark the point. That spot is Q.

Step 3 — Describe Q in plain words.

Q is 3 units to the left of O, and 2 units above O. It lies in the upper-left region of the plane (the region where x is negative and y is positive).

Answer: $Q(-3, 2)$ is 3 units left and 2 units above the origin.

Worked Example 1.2

SOLVED

A point lies on the x-axis and is 5 units to the left of the origin. What are its coordinates?

Solution

Step 1 — Use the rule for a point on the x-axis.

Any point that lies on the x-axis has $y = 0$ . (The y-coordinate measures distance from the x-axis itself, so a point on the x-axis is zero distance away.)

Step 2 — Find the x-coordinate.

The point is 5 units to the left of the origin. Distances measured to the left of O are negative, by convention. So $x = -5$ .

Step 3 — Write the coordinates.

Using the form $(x, y)$ :

$\text{Point} = (-5, 0)$

Answer: The coordinates of the point are $(-5, 0)$ .

Logical ReasoningLevel 3 · Analysis

Two friends, Aarav and Bina, both write down the location of a point P in the Cartesian plane. Aarav writes $(3, 7)$ . Bina writes $(7, 3)$ .

Aarav says: 'These are clearly the same point — both pairs use the same two numbers.'

Is Aarav right?

Ready to Go Beyond

Once you are comfortable with two axes, the door opens to several beautiful generalisations:

End-of-Chapter Q1Coordinates of the origin

NCERT

What are the x-coordinate and y-coordinate of the point of intersection of the two axes?

Quick Check

Q1.What are the coordinates of the origin of the Cartesian plane?