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Introduction to the x,y-Plane
    (The "Cartesian" Plane)
(page 1 of 3)

Sections: Introduction to the plane, Plotting points, The four quadrants

Before the end of the European Renaissance, math was cleanly divided into the two separate subjects of geometry and algebra. You didn't use algebraic equations in geometry, and you didn't draw any pictures in algebra. Then, around 1637, a French guy named René Descartes (pronounced "ray-NAY day-CART") came up with a way to put these two subjects together.

To explain Descartes' method, first think about using a street map. If you're trying to find a street that you've never been on before, you look for the street's name in the map's index. Suppose the index says that your target street is located at D12. This means that you go across the top of the map and find "D", and then you go down the side and find "12". You then trace down and across to find the box labelled as "D12", and look inside that box for the street you need. Somebody figured out that this was a handy way to specify the right general area on the map, telling you how far over (the D) and how far up or down (the 12) you needed to go. Descartes did something similar for mathematics.

You learned about the basic (counting) number line back in elementary school:

    basic number line

Later on, you were introduced to zero and negatives, which completed the number line:

    negatives, zero, and positives

Descartes' breakthrough was in taking a second number line, standing it up on its end, and crossing the first number line at zero:   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

    cartesian plane

The number lines, when drawn like this, are called "axes" (pronounced "ACK-seez"). The horizontal number line is called the "x-axis" ("eks-ACK-siss"); the vertical number line is the y-axis.


The arrows at the ends of the axes indicate the direction in which the numbers are getting larger. Therefore, only the axes should have arrows, and the arrows should be on one end only. Yes, many textbooks, especially in high school, draw the arrows incorrectly, as do many educators. (Don't point this out to them; for some reason, they're extremely touchy on this subject.) In "real life", not all axes point up or to the right; in the sciences, the x-axis can point left; in finance, the y-axis often points down. This information will be lost if you stick with the high-school way of splashing arrow-tips everywhere on every line.

The whole flat expanse, top to bottom, side to side, bursting outside of the box in the animation above and stretching off to infinity in all directions, is called the "plane". When you put the two axes in the plane, it is then called the "Cartesian" ("carr-TEE-zhun") plane. The name "Cartesian" is derived from the name "Descartes", after its creator, Rene Descartes. Please don't ask me to explain that derivation; I think it's Latin, and I can barely handle English.

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Cite this article as:

Stapel, Elizabeth. "Introduction to the x,y-Plane (The 'Cartesian' Plane)." Purplemath. Available from Accessed


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