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Cartesian Plane: Definition and Quadrants

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Cartesian Plane Definition

A Cartesian plane is a graph with one x-axis and one y-axis. These two axes are perpendicular to each other. The origin (O) is in the exact center of the graph. Numbers to the right of the zero on the x-axis are positive; numbers to the left of zero are negative. For the y-axis, numbers below zero are negative and numbers above are positive.

Simply put: a cartesian plane is just a number line with another number line at right angles.
cart (1)

The Cartesian plane matches a point on the plane with a pair of numbers located on the x and y axes. Each point on the plane has a unique set of numbers, called ordered pairs. The x is always listed first, followed by y. For example, the point (2,3) on the graph below is placed at x=2 and y=3.

The Cartesian plane showing several ordered pairs, which represent points on the graph.

The Cartesian plane showing several ordered pairs, which represent points on the graph.

The graph is named after the 16th century mathematician Rene Descartes.

Cartesian Plane Quadrants

The Cartesian plane is broken into four quadrants, I, II, III and IV. Quadrants are used extensively in trigonometry.

  • Quadrant 1 is at the top right.
  • Quadrant 2 is at the top left.
  • Quadrant 1 is at the bottom left.
  • Quadrant 1 is at the bottom right.

Cartesian plane showing the four quadrants. Kayau|Wimimedia Commons

Cartesian plane showing the four quadrants. Kayau|Wikimedia Commons

The picture above shows the position of a point in the third quadrant which is identified by the ordered pair (-1,-2). The first number in the set of ordered numbers listed (-1) is where the point is on the x-axis. In other words, the point is one space to the left of zero. The second number (-2) is where the point is on the y-axis. In other words, the point is two spaces below zero.

Fun fact: The invention of this system was revolutionary for its time. It gave us the first systematic link between algebra and Euclidean geometry*.

*Euclidean geometry is the technical name for the everyday geometry you learned in basic geometry classes.

Ordinate and Abscissa Definition

Ordinate and abscissa refer to ordered pairs on a Cartesian plane.

  • The abscissa is the x-value (the first number in an ordered pair).
  • The ordinate is the y-value (the second number in an ordered pair).

ordinate and abcissa

The above graph shows four ordered pairs:

  • For the point (-3,1), -3 is the abscissa.
  • For the point (2,3), 3 is the ordinate.
  • For the point (0,0), 0 is the abscissa and the ordinate.
  • For the point -1.5,-2.5, -1.5 is the abscissa and -2.5 is the ordinate.

The x-value (abscissa) is always written first in a coordinate system. If you forget which (X or Y) is called which (abscissa or ordinate), x comes before y in the alphabet and abcissa comes before ordinate.

Other Definitions and Uses

While Ordinate and Abscissa are generally used to describe the first and second points in a coordinate system, they are sometimes used to mean something slightly different.

  • Some dictionaries state that the terms are the distance between two points. For example, Merriam-Webster states an anscissa is “The horizontal coordinate of a point in a plane Cartesian coordinate system obtained by measuring parallel to the x-axis.” Use caution here, as this definition only works with positive numbers!
  • In the “old days,” the x-axis (i.e. the whole line and not any point on it) was called the abscissa and the y-axis was called the ordinate. For example, The Journal of Experimental Medicine from 1896 states that “…the abscissa is divided into six to eleven segments, each representing a period of one week, while the ordinate is divided into ten segments representing the abstract units 1 to 10.” In some sciences, like physics and astronomy, some scientists still use that definition.
  • The two terms can also be used in an oblique coordinate system. While the axes in a Cartesian plane are perpendicular, the axes in an oblique coordinate system are not.

If you’re really interested in reading more confusing aspects about ordinate and abscissa and the history behind the terms, I really recommend you take a look at Jason Dyer’s blog, where he goes all the way back to Fibonacci (c. 1170 – c. 1250) to find the words’ origins.

X Y Graph and the Cartesian Coordinate Graph

The Cartesian plane (or Cartesian coordinate graph) is practically the same thing as an X Y graph. An X Y graph is the basic graph, where you can make a simple plot of a series of coordinates, make a scatter graph or a line graph. A Cartesian coordinate graph is a more well defined X Y graph with quadrants (the four quarters of the graph).

Like a Cartesian plane, an X Y graph is a graph with two axes (axes is the plural of axis):

  • A horizontal line going from left to right (the X axis).
  • A vertical line going up and down (the Y axis).

x y graph

Plotting Points on an X Y Graph

To plot a point, move along the x axis to find the first coordinate (the first number), then move up or down to find the second coordinate. Draw a dot at that intersection. Watch the video to see how to plot two points on an X Y graph:

Points are always given to you as the x-axis number first, followed by the number on the y-axis. For example, (2,3) means 2 spaces on the x-axis and 3 on the y-axis.

A collection of points on an X Y graph is called a Scatter Plot because it looks like a scattered set of points. For an example of how to draw a series of dots on an X Y graph to make a scatter chart, see: How to Construct a Scatter Plot.

Make an X Y graph online: If you want to make sure you’ve drawn your points in the correct place, check out this online calculator from the Discovery Channel. Type in your points (make sure you use parentheses, like this: “(2,3)” and then click Draw them.The following image shows the point (2,3) drawn on the graph maker:
x y graph 3

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Cartesian Plane: Definition and Quadrants was last modified: October 12th, 2017 by Andale