Binary Math

Most of you know that I work with computers. As a consequence, I am very familiar with powers of 2; that is the number 2 multiplied by itself zero or more times. Mathematicians use the notation x^y as a shorthand to represent a particular number x multiplied by itself y times. It’s read as ‘x [raised] to the y power.’ Thus, as shown below, 2⁵ means 2 × 2 × 2 × 2 × 2 or 32.

2 (2 or 2¹)
4 (2 × 2 or (2²)
8 (2 × 2 × 2 or (2³)
16 (2 × 2 × 2 × 2 or 2⁴)
32 (2 × 2 × 2 × 2 × 2 or 2⁵)
64 (2 × 2 × 2 × 2 × 2 × 2 or 2⁶)
128 (2 × 2 × 2 × 2 × 2 × 2 × 2 or 2⁷)
256 (2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 or 2⁸)
512 (2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 or 2⁹)

Each successive number is double the previous. And they get big very quickly.Very quickly. Three more doublings (2¹²) and we reach 4,096. Four more beyond that (twelve more in all: 2⁹ × 2³ × 2⁴ or 2¹⁶) and we get 65,536. Here’s a table of all powers of two through 2^1,058. Numbers that are powers of 2 are called binary numbers (bi is a Greek prefix meaning two).

When you plot (draw) these numbers on a regular graph, the result is an exponential curve resembling this one, which happens to be a curve describing population growth over time. As we progress forward in time, the steepness (slope) of the graph increases, approaching but never reaching true vertical. At no time does it trend back towards the horizontal. (This is discounting limiting factors such as nutrients, water, land, housing or senescence, which vary by the type of population. Nor does it recognize external events such as the death of the host in the case of viral colonies.)

Okay, so what does all this have to do with my father? Well, if you haven’t picked up the hints I’ve dropped so far (they were relatively sophisticated), replace the word doubling with generation, as in cell generation, and you can begin to understand how the cancer progressed and spread as quickly as it did.

Assuming that there was just one mutant cell on August 15th and a conservative eighteen hour cell division time, that means over 125 generations by today. 2¹²⁵ is a very large number: 42,535,295,865,117,307,932,921,825,928,971,026,432. Even if we start accounting for senescence and other factors, we still wind up with incredibly unnerving values. And that’s how my father could appear to be fine in August and in such dire straits just three months later.

The clock is still ticking. Even as we wait ten to twelve days for the result of the first of several expected biopsies. I wish I couldn’t do the math.

You can learn more about cancer cell population growth here. Admittedly, it is a site aimed at those with a scientific or mathemtical background. However, the Model of Exponential Growth movie found below right on the page should be accessible to almost anybody. Click on the word Play to view it.