The things you learn in ECE

Today in my Networks & Systems class we learned something I thought was actually really interesting. I’ll try to break it down in plain english right here. In a perfect world I would use MathML for this but I have to be realistic.

An optical image on a computer screen can be represented as a function with a domain corresponding to the pixel resolution (2 variables) and a codomain (or range) corresponding to the the RGB value of each pixel (3 variables). For example, my proffessor’s display is 1680 by 1050 and uses 32 bit color, which is really almost always 24 bit color, 8 bits for each color (red, green, and blue). Therefore the domain has a total number of pixels equal to the height times the width, which is 1680 times 1050, or 1.764 million. The range has a total number of possible colors equal to 2 raised to the power of 24, because in binary representation the total number of possible values is 2 raised to the number of digits.

Now, the number of possible colors for any pixel “x” on the screen is equal to the total number of possible values for the codomain, in this case 2^²⁴. The number of possible colors for two pixels on the screen is the number of possible values for the first pixel times the number for the second pixel, or 2^²⁴ squared. For three pixels it is 2^²⁴ cubed, and so on.

In general, the total number of possible different functions, that is, possible images that can be displayed on this screen in 24 bit color, is the codomain raised to the power of the domain. Yes, (2^²⁴)^^1,764,000. This is equivalent to 2 raised to the power 42,336,000, which my calculator can’t seem to handle. I can estimate that 2^^42336000 is approximately equal to a lot.

So there you have it: mankind’s modern mysteries solved by simple mathematics. You can apply this analysis to your screen, too. Just take 2 and raise it to the power (24 times the width times the height) of your screen. Then punch that into your calculator and you should get something along the lines of “Error.”

Math is power.