We all know the four basic operations of arithmetic: addition, subtraction, multiplication, and division. In school, we were all forced to trivially add and subtract numbers again and again, keep track of remainders, etc. until it was printed in our brains how these operations work.

It is only much later we find out that there are really only two fundamental operations on the real numbers, namely addition and multiplication. Subtraction is just addition by a negative number and division is just multiplication by a fraction.

In this article, we will explore a fifth operation of arithmetic that, in my humble opinion, is almost just as natural as the others. Moreover, we will show how to use this operation to build a completely new kind of calculus that I call parallel calculus. Just as we have derivatives and integrals, we have parallel derivatives and parallel integrals.

In this alien calculus, the parallel derivative of x² is x/2, which we will show, and the parallel derivative of ln(x) = ln(x)²/x.

We even have parallel Taylor series and parallel Fourier series, parallel Laplace transforms, etc! But let’s start with the beginning before someone gets choked in their coffee before getting to the good part.

Before we begin, I need to inform the reader that the operation that I am about to inflict upon you is well-known - however, I have not seen the corresponding “parallel differential and integral operators” anywhere in the literature. If you can find a source exploring that, I would be very interested in receiving a link from you.

As far as I know, this is unchartered territory and we are about to go on a journey into this unexplored mathematical world.

Let’s start by defining parallel addition and parallel subtraction and discuss where the word parallel in the name of the operation comes from.

In the literature, there are several different notations for this operation. It is always a tough choice for an author to choose a notation that makes…