The Number Game

We all are well familiar with the number line. We define it as the linear arrangement of numbers, bearing every number, from - ∞ to ∞. But the big question, "Is that it?" is something which deserves an answer. For a simple instance, how do we value √-1? When it can be expressed, it ought to exist. But is there somewhere to look for it in our number line?

I'm afraid no. Our number line is linear, and thus has corresponding limitations. It may define the set of real numbers, but when it comes to imaginary numbers, like √-1, it fails. The next question in our list, "Is there a way to define it?"

Inspired by one of my buddies, Akhil, I began thinking about something called a "number plane".
I perceive it different from complex number plane, because it's intended also to include real numbers.

Akhil's number plane was a plot of the super-real infinity versus the sub-real infinity. It was braned by 1; the quadrants above this brane represent numbers till ∞ and -∞, while those below the brane represent numbers till 1/∞ and -1/∞. It was vertically partitioned, to project 0.

My concept of the number plane isn't much different, since it was induced by Akhil's. However in my number plane, imaginary values are also assigned a value, due to the quadratic nature of the plot. Consider the number plane below:

The vertical axis, like in Akhil's number plane, represents 0. However, the brane was eliminated in the system, and replaced by 0. I've termed this point as the "super-point", since it simply can represent the nature of the number line. The quadrants above the super-point is assigned to real numbers; left hand side for the negative ones, right hand side for the positives. On the other end, the imaginary, or the complex numbers are assigned the bottom 2 quadrants. Each of the quadrants are infinitely continual. I shall hereby refer to my number plane as the "number cone".

I've tried developing a way to make it all easier: This system can be compared to 2 distinct number lines, flexed and 2 dimensionally joined. In the below diagram, the system marked red can be compared to a flexed real number line, while the blue system is for the imaginary number system.

One may think, how is a number cone different from 2 distinct number lines? The answer to this question turns out the one beautiful number, 0. 0 is a part of every sub number-systems (except natural and irrational numbers). It's real, as well as imaginary; positive, as well as negative. This makes it a strong glue to adhere real, imaginary, positive and negative, positive imaginary, as well as negative imaginary numbers to a single system. This is the reason why 0 comprises the center of this number plane.

Another concept as I would introduce, the linear number-line projection. Consider the below diagram.

What can be viewed above is a projection of linear number line from the number plane. I've simply implied a simple number plane by horizontally joining any point in the "positive number slope" to its counterpart at the "negative number slope". However, as mentioned afore, there arises a need of scale, when we are to compare a projected linear number line and the flexed number line. Cos (θ/2) will express the required scale, where θ is the angle of flex, in this case being 90 degrees.

This is just one of my ideas, in need of your thoughts and opinions!