Introduction to the Cartesian Coordinate System (Two Dimensions)

The Coordinate System: Axes, Origin, and Quadrants

The Cartesian Coordinate System, also known as the rectangular coordinate system, is a fundamental concept in mathematics and geometry used to precisely locate every point in a plane. It was developed by the French mathematician and philosopher René Descartes (hence the name 'Cartesian'). The system relies on two perpendicular number lines that intersect at a single point, establishing a grid or plane.

Coordinate Axes

The foundation of the Cartesian system lies in its two intersecting number lines, which are referred to as the coordinate axes. These axes provide the framework for specifying locations in the plane:

• The x-axis: This is the horizontal number line. Conventionally, positive values extend to the right from the point of intersection, and negative values extend to the left. It represents the horizontal position of a point.
• The y-axis: This is the vertical number line. Conventionally, positive values extend upwards from the point of intersection, and negative values extend downwards. It represents the vertical position of a point.

These two axes are mutually perpendicular, meaning they intersect at a right angle ($90^\circ$). The plane formed by these two axes is often called the Cartesian plane or the xy-plane.

Origin

The unique point where the x-axis and the y-axis intersect is called the origin. It serves as the central reference point of the coordinate system.

The origin is denoted by the letter 'O'. Since it is the zero point on both the x-axis and the y-axis, its coordinates are always $(0, 0)$. Any location in the plane is described by its directed distances from the origin along the x-axis and y-axis.

Quadrants

The two coordinate axes (the x-axis and the y-axis) divide the entire Cartesian plane into four distinct regions. These regions are called quadrants. They are conventionally numbered using Roman numerals (I, II, III, and IV) in a counter-clockwise direction, starting from the region in the upper right.

The location of a point within a specific quadrant can be identified by the signs of its coordinates $(x, y)$. The signs of the x-coordinate and the y-coordinate are consistent for all points within a given quadrant:

Points that lie directly on either the x-axis or the y-axis do not belong to any quadrant. A point on the x-axis has coordinates of the form $(x, 0)$, and a point on the y-axis has coordinates of the form $(0, y)$. The origin $(0, 0)$ lies on both axes.

Example: Identifying Coordinates and Quadrants

Answer:

The Cartesian Plane

The flat surface or space that contains both the x-axis and the y-axis of the Cartesian coordinate system is called the Cartesian plane. It is also frequently referred to as the coordinate plane or the xy-plane.

This plane represents a two-dimensional space, extending infinitely in all directions (up, down, left, and right). The Cartesian coordinate system provides a method to assign a unique ordered pair of numbers, known as coordinates $(x, y)$, to every single point located within this plane. This pair of coordinates acts like a precise address for each point.

The Cartesian plane is the fundamental setting for coordinate geometry, also known as analytic geometry. This branch of mathematics brilliantly connects algebra and geometry, enabling us to:

• Represent geometric shapes (like lines, circles, triangles) using algebraic equations.
• Visualize algebraic equations by drawing their corresponding graphs.
• Plot individual data points and analyze relationships between two variables graphically.

Since it is a two-dimensional system, each point in the Cartesian plane requires exactly two values (the x-coordinate and the y-coordinate) to specify its exact location.

Coordinates of a Point

In the Cartesian plane, the location of any specific point, let's call it P, is precisely identified by an ordered pair of numbers, conventionally written as $(x, y)$. This pair is called the coordinates of the point P.

The ordered pair $(x, y)$ consists of two components, each representing a distance and direction relative to the coordinate axes:

• x-coordinate (Abscissa): The first number in the ordered pair, denoted by $x$, is called the x-coordinate or the abscissa of the point. It quantifies the perpendicular distance of the point from the y-axis. This distance is measured horizontally along a line parallel to the x-axis (or projected onto the x-axis itself). A positive value of $x$ indicates the point is located to the right of the y-axis, while a negative value indicates it is to the left. If $x$ is zero, the point lies on the y-axis.

• y-coordinate (Ordinate): The second number in the ordered pair, denoted by $y$, is called the y-coordinate or the ordinate of the point. It quantifies the perpendicular distance of the point from the x-axis. This distance is measured vertically along a line parallel to the y-axis (or projected onto the y-axis itself). A positive value of $y$ indicates the point is located above the x-axis, while a negative value indicates it is below. If $y$ is zero, the point lies on the x-axis.

It is crucial to understand that the order of the numbers in the pair matters significantly. The point $(2, 3)$ is a completely different location in the plane than the point $(3, 2)$. The pair $(x, y)$ is always written with the x-coordinate first and the y-coordinate second.

Plotting Points

To visually represent or plot a point P with coordinates $(x, y)$ on the Cartesian plane, you can follow these steps:

1. Begin at the origin $(0, 0)$, which is the intersection of the x-axis and the y-axis.
2. Move horizontally along the x-axis. If the x-coordinate ($x$) is positive, move to the right $x$ units. If $x$ is negative, move to the left $|x|$ units. If $x$ is zero, remain on the y-axis.
3. From the position reached in step 2, move vertically parallel to the y-axis. If the y-coordinate ($y$) is positive, move upwards $y$ units. If $y$ is negative, move downwards $|y|$ units. If $y$ is zero, remain on the x-axis.
4. The final position you arrive at is the location of the point P$(x, y)$. Mark this point on the plane.

Answer:

Following the steps for plotting:

• Point A(3, 2): Start at the origin. Move 3 units right along the x-axis. From there, move 2 units up parallel to the y-axis. This point is in Quadrant I.
• Point B(-4, 1): Start at the origin. Move 4 units left along the x-axis. From there, move 1 unit up parallel to the y-axis. This point is in Quadrant II.
• Point C(-2, -3): Start at the origin. Move 2 units left along the x-axis. From there, move 3 units down parallel to the y-axis. This point is in Quadrant III.
• Point D(1, -4): Start at the origin. Move 1 unit right along the x-axis. From there, move 4 units down parallel to the y-axis. This point is in Quadrant IV.

Points on Axes

Points that lie on the coordinate axes have a special property regarding their coordinates:

• Any point that lies on the x-axis (excluding the origin) has its y-coordinate equal to zero. Its coordinates will always be in the form $(x, 0)$, where $x$ is a non-zero real number.
• Any point that lies on the y-axis (excluding the origin) has its x-coordinate equal to zero. Its coordinates will always be in the form $(0, y)$, where $y$ is a non-zero real number.
• The origin itself lies on both the x-axis and the y-axis. Its coordinates are $(0, 0)$. It is the only point where both coordinates are zero.