The Four Quadrants: Where Every Point Lives — Class 9 Mathematics Chapter 1

🖼 Image PendingA wide cinematic view of a 2-D plane painted in four softly-coloured quadrants, with two perpendicular axes glowing through them, evoking a divided yet unified mathematical universe

AI Generation Prompt

Ultra-wide cinematic banner (16:5 ratio). The Cartesian plane visualised as a vast, glowing landscape divided into four softly tinted quadrants — upper-right (warm gold), upper-left (cool blue), lower-left (deep red), lower-right (royal violet). Two glowing perpendicular axes intersect at a luminous origin in the centre. Faint constellation-like dot patterns suggest infinite points scattered across each region. The image conveys: every point in the plane belongs to one of four worlds, and the signs of its coordinates reveal which. Painterly cinematic illustration. Dark background. No text, no labels.

✦Think About It

On the previous page you met the Cartesian plane and learned that any point can be written as $(x, y)$ . Two numbers, two signs.

Now think: how many qualitatively different kinds of point are there? A point with both coordinates positive feels different from one with both negative, doesn't it? How many distinct regions does the plane split into?

Each coordinate can be positive, negative, or zero. How many combinations give you 'real' regions (excluding the axes themselves)?

✦लीलावती — भास्कराचार्य

On the Four Cardinal Directions

चत्वारि अंशानि भुवो विभक्ते

दिशि दिशि गणितस्य निर्णयः सिद्ध्यति॥

(catvāri aṃśāni bhuvo vibhakte / diśi diśi gaṇitasya nirṇayaḥ siddhyati)

'धरती चार हिस्सों में बँटी है — हर दिशा में, अपना अलग गणित है।'

'The earth is divided into four parts — in every direction, mathematics finds its own answer.'

Bhāskarāchārya (1150 CE) wrote about how surveying problems split into four directional cases. The four quadrants you are about to study are the same idea, written in modern notation: each direction has its own sign rule, and the same point looks different through each one.

The four regions of the plane

So far we have looked at points lying on the two coordinate axes. What about every point that is not on an axis?

The plane in which the axes are drawn is called the Cartesian plane — also known as the coordinate plane or the xy-plane. The two axes divide it into four regions, called quadrants. By convention they are numbered anti-clockwise, starting from the upper-right.

Each quadrant has a distinct sign pattern:

Quadrant I (upper-right): both $x > 0$ and $y > 0$
Quadrant II (upper-left): $x < 0$ and $y > 0$
Quadrant III (lower-left): both $x < 0$ and $y < 0$
Quadrant IV (lower-right): $x > 0$ and $y < 0$

The quadrant of a point is determined entirely by the signs of its two coordinates, nothing else. The size of $x$ and $y$ does not matter — $(0.1, 0.1)$ and $(99, 99)$ both lie in Quadrant I.

🖼 Image PendingA green-graph-paper Cartesian plane on a dark background. The four quadrants are labelled in their corners: Quadrant I (upper-right), Quadrant II (upper-left), Quadrant III (lower-left), Quadrant IV (lower-right). The x-axis is labelled with integers from −8 to 7; the y-axis from −5 to +4. Two points are plotted: Q(−5, 3) in Quadrant II and S(3, −5) in Quadrant IV. The origin O = (0, 0) is at the centre.

AI Generation Prompt

Cartesian plane diagram on green graph paper over a dark background. Two thick green axes meet at the 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 from −8 to +7 on the x-axis, from −5 to +4 on the y-axis. The four corners of the visible plane carry labels: 'Quadrant I' (upper-right), 'Quadrant II' (upper-left), 'Quadrant III' (lower-left), 'Quadrant IV' (lower-right). Two points are plotted as small dark dots with labels: 'Q(−5, 3)' in Quadrant II and 'S(3, −5)' in Quadrant IV. Style: clean educational mathematics illustration. Dark background, orange accent labels, clean technical illustration style.

📸 Fig. 1.4 — The Cartesian plane with the four quadrants labelled. Q(−5, 3) lies in Quadrant II; S(3, −5) lies in Quadrant IV.

The sign-rule table

Now you can read any point's coordinates with full confidence. In general, the coordinates of a point P in 2-D space are written as $(x, y)$ , where:

$x$ is the perpendicular distance of P from the y-axis, measured along the x-axis. We call $x$ the x-coordinate (sometimes called the abscissa).
$y$ is the perpendicular distance of P from the x-axis, measured along the y-axis. We call $y$ the y-coordinate (sometimes called the ordinate).

And note an important shortcut. Points on an axis have one coordinate equal to zero:

A point of the form $(x, 0)$ lies on the x-axis. (Its perpendicular distance from the x-axis is zero.)
A point of the form $(0, y)$ lies on the y-axis. (Its perpendicular distance from the y-axis is zero.)
The single point that lies on both axes is the origin $O = (0, 0)$ .

The axes themselves do not belong to any of the four quadrants — they are the borders between them.

Loading simulator…

Worked Example 1

SOLVED

In which quadrant does the point $Q(-5, 3)$ lie?

Solution

Step 1 — Read the signs.

The x-coordinate is $-5$ (negative); the y-coordinate is $+3$ (positive).

Step 2 — Match to the sign-rule table.

$x < 0$ and $y > 0$ ⇒ Quadrant II (upper-left).

Answer: Q lies in Quadrant II.

Worked Example 2

SOLVED

In which quadrant does the point $S(3, -5)$ lie?

Solution

Step 1 — Read the signs.

The x-coordinate is $+3$ (positive); the y-coordinate is $-5$ (negative).

Step 2 — Match to the sign-rule table.

$x > 0$ and $y < 0$ ⇒ Quadrant IV (lower-right).

Answer: S lies in Quadrant IV.

Worked Example 3

SOLVED

A point P has coordinates $(0, -7)$ . In which quadrant does P lie?

Solution

Step 1 — Check whether either coordinate is zero.

The x-coordinate is $0$ . So P lies on the y-axis — not in any quadrant.

Step 2 — Refine: which part of the y-axis?

Since $y = -7 < 0$ , P lies on the negative part of the y-axis (below the origin).

Answer: P lies on the y-axis, below the origin. It is not in any of the four quadrants — the axes themselves are the borders between quadrants.

Worked Example 4

SOLVED

The point $A(4, 7)$ lies in Quadrant I. If I flip the sign of only its x-coordinate, where does it go? What if I flip the sign of only its y-coordinate?

Solution

Step 1 — Flipping the x-sign.

$A(4, 7)$ becomes $A'(-4, 7)$ . Now $x < 0$ and $y > 0$ ⇒ Quadrant II.

Step 2 — Flipping the y-sign.

From $A(4, 7)$ we get $A''(4, -7)$ . Now $x > 0$ and $y < 0$ ⇒ Quadrant IV.

Step 3 — Pattern observation.

Flipping the x-sign of any point in Quadrant I sends it to Quadrant II. Flipping the y-sign sends it to Quadrant IV. Flipping both signs (the negative of the point) sends it to Quadrant III. Each sign-flip is a reflection across an axis.

Answer: The point migrates to Quadrant II (x-flip) and Quadrant IV (y-flip).

Logical ReasoningLevel 3 · Analysis

A friend claims: 'The points $(3, 7)$ and $(7, 3)$ are the same point — both pairs use the same numbers and both lie in Quadrant I.' Is this claim correct?

Practice Yourself — Quick-Fire Drill

Cover the answers and classify each point. (Answers below.)

$(2, 3)$
$(-1, 6)$
$(-4, -2)$
$(3, -8)$
$(0, 5)$
$(-9, 0)$
$(0, 0)$
$(-7, 7)$
$(11, -11)$
$(0.5, -0.001)$

Answers: 1. Q I · 2. Q II · 3. Q III · 4. Q IV · 5. y-axis (positive) · 6. x-axis (negative) · 7. origin · 8. Q II · 9. Q IV · 10. Q IV. If you scored 9+/10 you have it. If you scored less, the trick is the signs of the two coordinates — nothing else.

Ready to Go Beyond

In 3-D space (which you will study in Class 11), the three coordinate planes carve space into 8 octants, each with its own sign pattern $(\pm, \pm, \pm)$ . The pattern continues: $n$ axes split $n$ -D space into $2^n$ regions. Quadrants are the $n=2$ case of a much wider mathematical idea.

End-of-Chapter Q2Predicting the locus of H

NCERT

Point $W$ has x-coordinate equal to $-5$ . Can you predict the coordinates of point $H$ which is on the line through $W$ parallel to the y-axis? Which quadrants can $H$ lie in?

Quick Check

Q1.Point P has coordinates $(-2, -7)$ . Which quadrant?

🖼 Image PendingA wide cinematic view of a 2-D plane painted in four softly-coloured quadrants, with two perpendicular axes glowing through them, evoking a divided yet unified mathematical universe

AI Generation Prompt

✦Think About It

On the previous page you met the Cartesian plane and learned that any point can be written as $(x, y)$ . Two numbers, two signs.

Each coordinate can be positive, negative, or zero. How many combinations give you 'real' regions (excluding the axes themselves)?

✦लीलावती — भास्कराचार्य

On the Four Cardinal Directions

चत्वारि अंशानि भुवो विभक्ते

दिशि दिशि गणितस्य निर्णयः सिद्ध्यति॥

(catvāri aṃśāni bhuvo vibhakte / diśi diśi gaṇitasya nirṇayaḥ siddhyati)

'धरती चार हिस्सों में बँटी है — हर दिशा में, अपना अलग गणित है।'

'The earth is divided into four parts — in every direction, mathematics finds its own answer.'

The four regions of the plane

So far we have looked at points lying on the two coordinate axes. What about every point that is not on an axis?

Each quadrant has a distinct sign pattern:

Quadrant I (upper-right): both $x > 0$ and $y > 0$
Quadrant II (upper-left): $x < 0$ and $y > 0$
Quadrant III (lower-left): both $x < 0$ and $y < 0$
Quadrant IV (lower-right): $x > 0$ and $y < 0$

The quadrant of a point is determined entirely by the signs of its two coordinates, nothing else. The size of $x$ and $y$ does not matter — $(0.1, 0.1)$ and $(99, 99)$ both lie in Quadrant I.

AI Generation Prompt

📸 Fig. 1.4 — The Cartesian plane with the four quadrants labelled. Q(−5, 3) lies in Quadrant II; S(3, −5) lies in Quadrant IV.

The sign-rule table

Now you can read any point's coordinates with full confidence. In general, the coordinates of a point P in 2-D space are written as $(x, y)$ , where:

$x$ is the perpendicular distance of P from the y-axis, measured along the x-axis. We call $x$ the x-coordinate (sometimes called the abscissa).
$y$ is the perpendicular distance of P from the x-axis, measured along the y-axis. We call $y$ the y-coordinate (sometimes called the ordinate).

And note an important shortcut. Points on an axis have one coordinate equal to zero:

A point of the form $(x, 0)$ lies on the x-axis. (Its perpendicular distance from the x-axis is zero.)
A point of the form $(0, y)$ lies on the y-axis. (Its perpendicular distance from the y-axis is zero.)
The single point that lies on both axes is the origin $O = (0, 0)$ .

The axes themselves do not belong to any of the four quadrants — they are the borders between them.

Loading simulator…

Worked Example 1

SOLVED

In which quadrant does the point $Q(-5, 3)$ lie?

Solution

Step 1 — Read the signs.

The x-coordinate is $-5$ (negative); the y-coordinate is $+3$ (positive).

Step 2 — Match to the sign-rule table.

$x < 0$ and $y > 0$ ⇒ Quadrant II (upper-left).

Answer: Q lies in Quadrant II.

Worked Example 2

SOLVED

In which quadrant does the point $S(3, -5)$ lie?

Solution

Step 1 — Read the signs.

The x-coordinate is $+3$ (positive); the y-coordinate is $-5$ (negative).

Step 2 — Match to the sign-rule table.

$x > 0$ and $y < 0$ ⇒ Quadrant IV (lower-right).

Answer: S lies in Quadrant IV.

Worked Example 3

SOLVED

A point P has coordinates $(0, -7)$ . In which quadrant does P lie?

Solution

Step 1 — Check whether either coordinate is zero.

The x-coordinate is $0$ . So P lies on the y-axis — not in any quadrant.

Step 2 — Refine: which part of the y-axis?

Since $y = -7 < 0$ , P lies on the negative part of the y-axis (below the origin).

Answer: P lies on the y-axis, below the origin. It is not in any of the four quadrants — the axes themselves are the borders between quadrants.

Worked Example 4

SOLVED

The point $A(4, 7)$ lies in Quadrant I. If I flip the sign of only its x-coordinate, where does it go? What if I flip the sign of only its y-coordinate?

Solution

Step 1 — Flipping the x-sign.

$A(4, 7)$ becomes $A'(-4, 7)$ . Now $x < 0$ and $y > 0$ ⇒ Quadrant II.

Step 2 — Flipping the y-sign.

From $A(4, 7)$ we get $A''(4, -7)$ . Now $x > 0$ and $y < 0$ ⇒ Quadrant IV.

Step 3 — Pattern observation.

Answer: The point migrates to Quadrant II (x-flip) and Quadrant IV (y-flip).

Logical ReasoningLevel 3 · Analysis

A friend claims: 'The points $(3, 7)$ and $(7, 3)$ are the same point — both pairs use the same numbers and both lie in Quadrant I.' Is this claim correct?

Ready to Go Beyond

End-of-Chapter Q2Predicting the locus of H

NCERT

Point $W$ has x-coordinate equal to $-5$ . Can you predict the coordinates of point $H$ which is on the line through $W$ parallel to the y-axis? Which quadrants can $H$ lie in?

Quick Check

Q1.Point P has coordinates $(-2, -7)$ . Which quadrant?

Quadrant	x-sign	y-sign	Example point
I			(3, 4)
II	−		(−5, 3)
III	−	−	(−2, −7)
IV		−	(3, −5)

Quadrant	x-sign	y-sign	Example point
I			(3, 4)
II	−		(−5, 3)
III	−	−	(−2, −7)
IV		−	(3, −5)