The Last Number
The Journal of Cool Ideas, V.1, pp.1
The last number is defined as the largest natural number. It is provisionally assigned the symbol Omega, Ω. There is no number larger in magnitude than Omega. If you add one to Omega, you get a number smaller than Omega. If you multiply Omega by a number, you get Omega. If you divide a number by Omega, you get Omega.
Using this concept of the last number allows the possibly of providing a definition to division by zero, which has heretofore been considered an undefined operation. We can define Omega as the inverse of zero:
Ω = 1/ 0
Using this definition, it can be seen that Omega is the only number that when multiplied by zero, does not equal zero, but instead equals one. Thus, by using this definition of Omega, division by zero is no longer “undefined”.
Also, by definition, Omega does not have a sign; it is neither positive or negative, as is the case for zero.
Using the definition of Omega may provide some useful results.
II. Number Lines
A way to think about Omega, is to extend the concept of the number line. Instead of envisioning the number line as being a line of infinite length, it can be conceived as a circle of infinite radius. On this circle of infinite radius there exist two “anti-podes”: zero and omega. At the zero side of the number circle, it looks like the standard number line:
zero can be conceived as one “anti-pode”, and omega can be conceived as the “anti-pode” of zero on the opposite side of the infinite number circle. Typically, it is assumed that the natural numbers continue to increase unbounded, in both the positive and negative directions on the number line.
Using the concept of the “number circle”, we can conceptualize the omega side of the number circle, the anti-pode of zero, as sort of looking like the standard number line, but with some unsettling differences. It could be conceived thusly:
As we add positive numbers to Ω – n, we approach and reach omega. As we continue to add positive numbers to omega, we pass through the anti-pode of omega, and now the numbers become negative. At the omega anti-pode, we have the counter-intuitive case that adding positive integers to a negative number increases the magnitude of the negative number.
The number circle, in its entirely, could thus be configured as:
Using the above mentioned definition:
Ω = 1/ 0
We can develop other properties of omega.
n/0 = n x Ω
And, just as n x 0 = 0 so also does n x Ω = Ω
Ω + 1 < Ω – 1
abs(Ω + 1) = abs(Ω – 1)
-1 x Ω = 1 x Ω
n Ω = Ω for n >1
n Ω = 0 for n <1
n Ω = 1 for n =1
n 1/ Ω = 1
Also, we now have a consistent definition for zero raised to the zero:
0 0 = 1
IV. Recommendations for Future Work
These above posited relationships are meant to provoke the exploration of other useful mathematical relationships, given the definition of Omega.
1. Calculus and Analytic Geometry; G.B. Thomas, R.L. Finney; Addison-Wesley Publishing Company; 1979