Most people are familiar with the Cartesian coordinate system, where points are graphed based on their “x” and “y” coordinates, but there is another important graphing system for the study of mathematics: the polar coordinate system. The Cartesian coordinate system was created in 1637 by mathematician René Descartes and the polar coordinate system was invented a few decades later by physicist Isaac Newton, which was then used by mathematician Jakob Bernoulli to solve calculus problems.

For those not familiar with the polar coordinate system, the first step in graphing it is to draw the axis, which is any point — now the origin — and then a ray that is drawn to the right. It looks like the right half of the x-axis. Points are then graphed based on an r value, which represents the distance from the origin to that point and a θ, or theta, value, which is the measure in degrees or radians of the angle formed by the origin pole and the line that connects said point to the origin. Like the Cartesian system, points are written as ordered pairs, but this time in the form of (r, θ) rather than (x, y).

An example of the polar coordinate system in action is the point (2, π). In this case, since r equals two, the point will be a distance of two from the origin and the angle between this point and the axis will be π radians, or 180 degrees. When graphing this, you would place a point that is two units directly to the left of the origin because that will form a straight line between the axis and the point, which has an angle of 180 degrees.

Beyond just graphing points, the polar coordinate system is also useful for graphing functions. A function is a relationship between a set of inputs and outputs. For example, you might be familiar with the function y = x^2, which takes the inputs of x and produces the outputs of y. Likewise, there are also polar coordinates, which are usually written as r equals something done to θ, such as r = sin(θ). The graph of this function on the polar coordinate system is a circle, and the polar graphing system is valuable because it allows mathematicians to graph important curves, such as cardioids and roses, that cannot easily be graphed by functions in the Cartesian system.

However, there is a problem with the polar coordinate system — the convention of writing points as (r, θ). I believe that it would make more sense to write polar points as (θ, r) because it fits more logically with the Cartesian points of (x, y). The main point behind my argument is that θ is more analogous to x, while r is more analogous to y. Therefore, polar points should be written as (θ, r) so they naturally flow with the (x, y) Cartesian points, making it less confusing.

The first reason why θ is similar to x lies in how polar functions are written. If you noticed earlier, the example of a Cartesian function that I gave was y = x^2. The x coordinate is the input, because something is being done to it by being squared, and y is the output. On the other hand, the polar function that I gave was r = sin(θ), which puts θ as the input and r as the output. Since x and θ are typically inputs of their respective type of function, and y and r are usually the outputs, it makes sense that they would be in the same spots of their ordered pairs.

Furthermore, to build off of the input and output similarities between the Cartesian and polar coordinates, it makes logical sense to have the input written before the output in the ordered pairs. Let’s say that you are tasked with finding a point that lies on the function, r = sin(θ). In order to solve this problem, you could pick an r value and then solve for θ, but it would be much easier to first choose a θ value because you do not have to take the inverse sine of both sides of the equation, making the problem much more complicated than it has to be. Since the θ value is determined first, it should be written first because it used to find the r value, not the other way around.

Finally, my last point is that using the convention (θ, r) is more useful when graphing polar points than the convention (r, θ). Let’s look back at the point (2, π). When graphing this point, the first piece of information that you need is the angle (θ), not the length (r). In order to graph (2, π), you must first travel π radians, or 180 degrees, from the axis, and the distance of two is valuable because you need to know how far out the point lies. As such, it would be more helpful if θ was listed before r because you need to know θ before the r value has any meaning.

Polar coordinates are a great way to graph unusual functions that would otherwise be impossible with the Cartesian system, but the convention of writing points as (r, θ) is flawed. It would be more logical and helpful to have polar coordinates written as (θ, r), because θ is more analogous to x and r to y. Having the input (θ) written before the output (r) makes logical sense, and θ is used before r when plotting points.

Elijah Engler is a freshman majoring in chemistry.