Coordinates and the four quadrants

Finding a place by two numbers

How do you find a seat in a cinema? You are told a row and a seat number, such as "row 4, seat 7". Two numbers fix one place. How do you find a square on a chess board? A letter and a number, such as e4. The same idea fixes a point on a flat sheet of paper. It is called the Cartesian plane, after the French thinker René Descartes. A story says that he thought of it while watching a fly walk on the ceiling of his room. The place of the fly could be given by two distances from the walls.

Two number lines cross at a right angle. The horizontal one is the x-axis, and the vertical one is the y-axis. The point where they cross is the origin.

The ordered pair

We write the position of a point as $(x, y)$.

• The first number, $x$, is the x-coordinate, or abscissa. It tells how far the point is to the right (positive) or left (negative) of the y-axis.
• The second number, $y$, is the y-coordinate, or ordinate. It tells how far the point is above (positive) or below (negative) the x-axis.

The point $P(3, 2)$ is $3$ units to the right of the y-axis and $2$ units above the x-axis.

The order matters. The point $(3, 2)$ and the point $(2, 3)$ are different points. That is why the pair is called an ordered pair. Only when $x = y$ are the two the same.

A way to remember the order: go along the corridor first (the x direction), and then up the stairs (the y direction).

The four quadrants

The two axes cut the plane into four parts, which are called quadrants.

We go anticlockwise: I, II, III, IV. To remember the signs, start with both positive. Then $x$ turns negative, then $y$ turns negative, and then $x$ turns positive again.

Given only the signs, you can tell the quadrant without drawing. The point $(-8, 3)$ has a negative $x$ and a positive $y$, so it is in quadrant II. The point $(a, b)$ with $a < 0$ and $b < 0$ is in quadrant III.

Slide $x$ or $y$ to $0$. The readout drops to $0$ and the message appears: a point on an axis lies in no quadrant.

Points on the axes

A point on an axis does not lie in any quadrant.

• A point on the x-axis has $y = 0$: it looks like $(x, 0)$. Example: $(4, 0)$ and $(-3, 0)$.
• A point on the y-axis has $x = 0$: it looks like $(0, y)$. Example: $(0, 5)$ and $(0, -2)$.
• The origin is $(0, 0)$. It is on both axes.

A common mistake is to say that $(0, 4)$ is on the x-axis. It has $x = 0$, so it is on the y-axis.

The equation of the x-axis is $y = 0$, and the equation of the y-axis is $x = 0$. The points with $x = y$ lie on a slanting line through the origin, in quadrants I and III.

Where is the Cartesian plane used?

• Maps. Latitude and longitude are two numbers that fix a place on the Earth.
• Screens. Every pixel on a screen has an $(x, y)$ position.
• Graphs. A graph of marks against hours of study puts one number across and one up.
• Games. The position of a character is stored as coordinates.

Practise with new numbers

Try it in Python

This program says where a point is: in a quadrant, on an axis or at the origin.

Common mistakes

• Writing $(y, x)$ in place of $(x, y)$. The first number is always across.
• Placing a point on an axis in a quadrant. $(0, 4)$ is not in any quadrant.
• Mixing up the quadrants. They go anticlockwise from the top right.
• Forgetting the sign. $(-2, 5)$ is to the left of the y-axis.

Remember these

• A point is $(x, y)$: first the distance across, then the distance up.
• The order matters. $(3, 2)$ is not $(2, 3)$.
• Signs: I is $(+, +)$, II is $(-, +)$, III is $(-, -)$ and IV is $(+, -)$.
• On the x-axis, $y = 0$. On the y-axis, $x = 0$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. In which quadrant is $(-7, -2)$?

2. On which axis is $(0, -3)$?

3. In which quadrant is $(4, -9)$?

4. A point has $y = 0$ and $x = 6$. Where is it?