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Cartesian Plane

The French mathematician Rene Descartes first introduced the Cartesian plane. The word ‘Cartesian’ comes from the Latin word Cartesium. The idea to explain the Cartesian plane was developed by Rene Descartes and Pierre de Fermat in 1637. The Cartesian plane explains the position of a point or the object on the surface by using the two intersecting lines. The developing of the Cartesian planes helped Isaac Newton to develop the calculus a lot. There are some other coordinates systems developed by the Rene Descartes, as polar coordinate system, spherical coordinate system and cylindrical coordinate system.

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One Dimension Cartesian Plane

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For the Cartesian coordinate system in one dimension, draw a straight line and choose a point O as the origin at the middle of the line. The part of the line segment, which is in the right of the origin O is considered as positive, while the line segment which is in the left of the origin O is considered as the negative. So give the sign as ‘+’ and ‘-’ to any point as we locate it on the number line. The line, which is chosen for locating the points in one dimension, is called the number line.

Origin:

The centre point from which the distances are marked is called the origin. In two- dimensional planes the X-axis and Y-axis crossed the point is called as origin.

Cartesian Plane Quadrants:

The axes X –axis and Y-axis split the plane into four parts. These four parts are called the quadrants. Quadrants are denoted as I, II, III, and IV in anticlockwise direction. The axes in the plane are called as Cartesian axes and the plane is known as Cartesian plane.

Two Dimension Cartesian Plane

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Two dimensional Cartesian coordinate system is represented by X-Y plane. The two mutually perpendicular lines represent X-Y plane.The points are defined as the ordered pair and written in the parenthesis. It has two perpendicular lines. One of the line is called the X-axis and the other is Y-axis. X-axis is horizontal line and Y-axis is the vertical line. The point where the two axes meets is called the origin O. For any given point P, let x and y be the corresponding number lines and the coordinates are written as (x, y), where x is called the abscissa and y is called the ordinate.

Graph of cartesian


Cartesian Plane Quardants

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The two axes divide the plane into four regions called the quadrants:

cartesian plane quadrants

The ray OX is taken as positive x-axis, OX| as negative x-axis, OY as positive y-axis and OY| as negative y-axis. The quadrants are thus characterized by the following signs of abscissa and ordinate:

I) Quadrant x > 0 , y > 0 or (+,+)

II) Quadrant x < 0 , y > 0 or (-,+)

III) Quadrant x < 0 , y < 0 or ( -,-)

IV) Quadrant x > 0 , y < 0 or (+,-)

Further if the abscissa of a point is zero, it would lie somewhere on the y-axis and if its ordinate is zero it would lie on x-axis. Thus by simply looking at the coordinates of a point we can tell in which quadrant it should lie, e.g , the points (3,4) , (1,-2) , (-2,-3) , (-4,5) lie respectively in I , IV , III and II quadrants.

Absolute Cartesian system limits to I quadrant with all positive points and (0,0) and origin where as the Cartesian system is of all the 4 quadrants.

Definition

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The Cartesian plane consists of two perpendicular and directed lines whose intersection point is the zero point for both the lines. The horizontal line is known as X-axis and the vertical line is known as Y-axis. The coordinate point (x, y) on the Cartesian plane says that the horizontal distance of the point from the origin is x, and the vertical distance is y.
If the sign of x is positive the point is on the right of the origin, else it is on the left. Similarly, if the sign is positive for y the point is y points above the origin else it is y points below it.

3D Cartesian plane

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The 3D Cartesian plane has one more axis perpendicular to the normal Cartesian plane. For the XY plane, there is an axis Z which is perpendicular to the XY palne. Any point lying on this plane is defined by set of three points (x, y, z) where x defines the position along the X-axis, y defines the position along the Y-axis and z defines the position along the Z-axis. The given graph shows the 3D Cartesian plane with three axes X, Y and Z.




Graph

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Each axis of the Cartesian plane will have points marked on equal intervals. To plot a point (x, y) on a XY-plane we first move x points left or right, according to the sign and then y points up or down dependent on the sign. For example: To plot a graph of the point P(-2, 4) on a Cartesian plane, we first need to move 2 points left from the origin.

Then, we will move 4 points upwards along the Y-axis. This is the position of the point P.



Cartesian Representation of Complex Numbers

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A complex number is a combination of real and imaginary numbers. We know that an imaginary number cannot be represented on a number line or Cartesian plane. Hence, they are represented on a complex plane. A complex plane is just like any other two-dimensional Cartesian plane, but here one axis will represent the real part of the number and the other axis will represent the imaginary part of the number. For a complex number a + ib, the point represented on the complex plane will be (a, b).
For example: Take a number 2 + 3i. The given graph will be as given here.



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