We know that a line consists of infinite number of points. When we draw a picture on a sheet of paper, we draw it such that it is at a fixed distance from a fixed point called the origin. The drawing is made up of many curves and straight line which consists of infinite number of points and each point is unique. The points with respect to the fixed origin can be represented in the form of ordered pairs (x, y), where x and y are real numbers. 

How do we divide the plane into quadrants? How  do we plot the points in the coordinate plane? What are the positive and negative sides of coordinate axis? Let us study this in detail in the following sections.

The points are represented on a number line or on a two dimensional graph sheet or in a three dimensional plane. The representation of points on the different planes are called cartesian planes which is named after the Greek Mathematician Descarte's Cartesian.

We know that the numbers are represented on the real number line. Therefore, the representation of points on the number line is called the one-dimensional plane.

Representation of points in ordered pairs (x, y) in graph sheets or checked sheets is called two dimensional planes. This contains a "Horizontal line" called x-axis and the "Vertical line" called the y-axis.

The two dimensional graph is divided into four parts, each one is called a quadrant, which is shown below. The following figure shows the different quadrants and the sign of the ordered pairs in each quadrant:

Cartesian Coordinates Definition
Representation of points in the form (x, y, z) in space is called the three dimensional planes. In the case of three dimensional plane, it is the intersection of three planes which are divided into 8 parts called octant.
In the cartesian coordinate system of two dimensions, we represent each point in the form of an ordered pair (x, y), where x is a point on the x-axis and y is the point on the y-axis.

The following graph shows the representation of the points (4,5), (6, 0) and (0, -3):

Cartesian Coordinate System Definition
To plot the point (4, 5), we look for the number 4 on the x-axis and the number 5 on the y-axis. The point where these two lines from x and y axis meet is the required point (4, 5).

To locate the point (6, 0), we look for 6 on the x-axis. Since y is 0 on the entire x-axis, the point (6, 0) is the point 6 on the x-axis.

To locate the point (0, -3), we should know that the x-coordinate is zero on the y-axis. Hence, we locate -3 on the y-axis, which has its coordinate (0, -3) on the two dimensional coordinate plane.
Coordinate means the pair of points which is located on the two dimensional plane. It contains the horizontal line called the x-axis and the vertical line called y-axis. 

The point of intersection of the x and y-axis is called the origin whose coordinate is (0, 0).

X and Y axis are two real number lines which have the common origin 0. The real numbers to the right of 0 (on the x-axis) are positive and those which are on the left 0 (on the x-axis) are negative. Similarly, the numbers which are above 0 (on the y-axis) are positive and those which are below 0 (on the y-axis) are  negative.

The following figure shows the positive and negative real numbers on the x and y axis of the co-ordinate plane:

Cartesian Coordinate Graph

Coordinates of Points on the x- and y-axis:


On the x-axis, y-coordinate is zero.
Hence, the coordinate form of the points on the x-axis is (x, 0).

On the y-axis, the x-coordinate is zero.
Hence, the coordinate form of the points on the y-axis is (0, y).

The following figure shows the points on the x and y axis:

Cartesian Coordinate Graphs

Location of Coordinates of the Form (x, y) on the Graph, where x $\neq$ 0 and y $\neq$ 0:


While locating the required points which are unique on the coordinate plane, we look for the corresponding real number on the x and y axis respectively. The intersection of the vertical line from the x-axis and the horizontal line from the y-axis is the required point in the coordinate plane.

The following figure shows the location of the points (3, 5), (-3, 5), (-4, -5) and (6, -4) which are in different quadrants.

Graph of Cartesian Coordinates
The cartesian coordinate plane is divided into four quadrants as shown in the graph below:
Cartesian Coordinates Quadrants
The signs of the coordinates in each quadrants are as follows :

First Quadrant: (+, +)

Second Quadrant: (-, +)

Third Quadrant: (-, -)

Fourth Quadrant: (+, -)


Let us identify the quadrant in which each of the following points are located:
  • (5, -6) - Since x-coordinate is positive and y-coordinate is negative, the point will be in the fourth quadrant.
  • (-2, 5) - Since x-coordinate is negative and y-coordinate is positive, the point will be in the second quadrant.
  • (5, 5) - Since x-coordinate is positive and y-coordinate is positive, the point will be in the first quadrant.
  • (-4, -7) - Since both x and y coordinates are negative, the point will be in the third quadrant.
Given below are some of the examples in cartesian coordinate system.

Solved Examples

Question 1: Write the ordered pairs for each of the following:
  1. x coordinate is 8 and the y coordinate is -4.
  2. y coordinate is 4 and the x coordinate is -5.
  3. The point which is in the second quadrant and is at a distance of 5 units from the x-axis and 5 units from the y-axis.
  4. The point on the y-axis which is at a distance of 4 units from the origin lying on the negative side of the y-axis.
  5. The point on the x-axis, which is at a distance of 3.5 units from the origin lying on the positive side of the x-axis.

Solution:
1. x coordinate is 8 and the y coordinate is -4.  Solution: (8, -4)
2. y coordinate is 4 and the x coordinate is -5. Solution: (-5, 4)
3. The point which is in the second quadrant and is at a distance of 5 units from the x-axis and 5 units from the y-axis. Solution: Since the point is in the second quadrant, the ordered pair should be ( -, + ). Therefore, (-5, 5).
4. The point on the y-axis which is at a distance of 4 units from the origin lying on the negative side of the y-axis. Solution: For any point on the y-axis, x-coordinate is 0. Therefore, the point is (0, -4).
5. The point on the x-axis, which is at a distance of 3.5 units from the origin lying on the positive side of the x-axis. Solution: The y-coordinate on the x-axis is 0. Therefore, the point on the x-axis is (3.5, 0).

Question 2: Identify the quadrant in which the following points are located:

  1. (2, -1)
  2. (8, 9)
  3. (-2, -1)
  4. (-3, 3)

Solution:
1. ( 2, -1)   According to the sign in each quadrant, the sign of the point (2, -1) is (+, -) which corresponds to 4th quadrant. Therefore, point will be in the 4th quadrant.
2. (8, 9) According to the sign in each quadrant, the sign of the point (8, 9) is (+, +) which corresponds to 1st quadrant. Therefore, the point will be in the 1st quadrant.
3. (-2, -1)
According to the sign in each quadrant, the sign of the point (-2, -1) is (-, -) which corresponds to 3rd quadrant. Therefore, the point will be in the 3rd quadrant.
4. (-3, 3)
According to the sign in each quadrant, the sign of the point (-3, 3) is (-, +) which corresponds to 2nd quadrant. Therefore, the point will be in the 2nd quadrant.

Question 3: Plot the points, A(1, 3), B(4, 3), C(5, 7) and D(8, 7) on the graph sheet and join the points to obtain the quadrilateral ABCD. Measure the length of each side of the quadrilateral. Write the special name of the quadrilateral.
Solution:
Cartesian Coordinate System Examples
In the above graph we, observe that the sides AB = BC = 3 units and AB // BC. Also, AD = BC = 5 units and AD // BC.
Therefore, the above quadrilateral ABCD is a parallelogram.