Coordinates

Learn the basic concepts of the coordinate plane, understanding the x-axis, y-axis, origin, and how to represent coordinates.

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How to Convey Position to Someone?

There are Steve and a Zombie. Steve needs to communicate the Zombie's position to either escape from it or defeat it. What's needed to accurately convey a position to someone who is in a different location from you?

To identify a position on a plane, you need to define a reference point and specify the direction and distance from that point. In elementary school, you learned to identify positions where both $x$ and $y$ values are positive, but in middle school, the $x$ -axis and $y$ -axis are extended to include negative ranges.

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History of Coordinates

What are coordinates in the first place?

Here, we'll briefly explain the coordinates drawn on a plane (Cartesian coordinate system). Simply put, it's a method to represent "where you are" on a plane.

Two lines are drawn, a horizontal $x$ -axis and a vertical $y$ -axis, with evenly spaced markings on these lines. Using these $x$ and $y$ values, we can represent a position like "3 across, 2 up" as $(3, 2)$ . These numbers "3, 2" are the "coordinates" of that point, allowing us to identify that point on the plane.

Who established coordinates?

The planar coordinate system with perpendicular $x$ and $y$ axes was established by the 17th-century French mathematician René Descartes (1596 - 1650). He wrote a book called "Discourse on the Method" in 1637, where he presented the concept of coordinates.

Actually, there were ideas similar to coordinates before Descartes. However, they only represented one direction. Descartes developed this further by using two directions ( $x$ -axis and $y$ -axis) and extending $x$ and $y$ to include negative values. In other words, he made it possible to deal with all real numbers.

This is called the Cartesian coordinate system , named after his achievements.

Significance of Coordinates

Now, let's look at why the Cartesian coordinate system is so amazing.

Normally, lines and circles are created by "drawing a line" or "using a compass". However, with the Cartesian coordinate system, we can represent these shapes in the form of "equations".

For example, the equation of a line can be written as follows:

$$y=2x+1$$

This equation tells us the rule for drawing a "line" on paper. In other words, if we decide on a value for $x$ , the corresponding $y$ value is determined by this equation, so connecting these points creates a line.

There's also an equation for a circle:

$$x^2+y^2=25$$

This represents a circle with center $(0, 0)$ and radius $5$ . By finding combinations of $x$ and $y$ that satisfy this equation, we can draw a perfect circle.

Thanks to the Cartesian coordinate system and equations, we can now tackle more complex shapes and problems. For example, we can represent ellipses (elongated circles) and parabolas (shapes like the trajectory of a thrown ball) with equations.

In this way, in the world of mathematics, we can not only draw shapes but also "calculate" them using equations. This created a new field of mathematics called "analytic geometry" and became a tool for solving various problems like the ones we're studying now.