A Digression

• Did it bother you that I used the approximate value 6.3 inches instead of the exact value 160 mm? It should have! Amazingly, it turns out that the approximation is an unusually good one, so good that the number of pixels is the same either way. To a few more decimal places, the exact value is 6.2992126.

Let's look at that calculation in detail. We have two physical sizes: a = 6.3, the approximate value, and b = 6.2992126, the better approximation. To get the number of pixels, we just multiply by d = 600, the number of dots per inch. On the one hand we have da = 3780; on the other hand we have db = 3779.52756, which rounds up to 3780. So, like I said, it's the same either way.

However, it's almost not the same. If d were just a little bit larger, the difference da − db = d(a − b) would increase from 0.47244 to 0.5 and affect the rounding. How much larger?

d	= 0.5 / (a − b)
	= 0.5 / 0.0007874
	= 635.00127 000254 000508 001016 002032 004064 008128 …

Strange! The decimal expansion looks like a geometric series with ratio 2 and terms that are nearly powers of 2 … but let's come back to that later. For now, we can check our work by setting d = 635 and seeing if the difference da − db is close to 0.5.

da = 635 × 6.3	= 4000.5
db = 635 × 6.2992126	= 4000.000001 (exactly)

Very strange! If that's still not enough to pique your interest, here are a few more strange things. In the calculation of d above, the number 7874 has 127 as a factor—yes, 127 is prime—so it seems that we're dividing by 127 and somehow getting a result that contains 127. It works the other way 'round, too.

1/127 = 0.007874 015748 031496 062992 125984 251968 503937

That group of digits in the middle also looks strangely familiar. What's going on here??

Now we've reached the part where I encourage you to stop and investigate these mysteries. They're not too mysterious, especially if you're aware of certain facts. While you're doing that, I'll just pad the essay with some remarks about rounding so that it will be harder for you to peek ahead by accident.