Ken Steiglitz: It’s the Number of Zeroes that Counts

We present the third installment in a series by The Discrete Charm of the Machine author Ken Steiglitz. You can find the first post here and the second, here.

The scales of space and time in our universe; in everyday life we hang out very near the center of this picture: 1 meter and 1 second.

As we’ll see in The Discrete Charm the computer world is full of very big and very small numbers. For example, if your smartphone’s memory has a capacity of 32 GBytes, it means it can hold 32 billion bytes, or 32000000000 bytes. It’s awfully inconvenient and error-prone to count this many zeros, and it can get much worse, so scientists, who are used to dealing with very large and small numbers, just count the number of zeros. In this case the memory capacity is 3.2×1010 bytes. At the other extreme, pulses in an electronic circuit might occur at the rate of a billion per second, so the time between pulses is a billionth of a second, 0.000000001, a nanosecond, 1 × 10−9 seconds. In the time-honored scientific lingo, a factor of 10 is an “order of magnitude,” and back-of-the-envelope estimates often ignore factors of 2 or 3. What’s a factor of 2 or 3 between friends? What matters is the number of zeroes. In the last example, a nanosecond is 9 orders of magnitude smaller than a second.

Such big and small numbers also come up in discussing the size of transistors, the number of them that fit on a chip, the speed of communication on the internet in bits per second, and so on. The figure shows the range of magnitudes we’re ever likely to encounter when we discuss the sizes of things and the time that things take. At the low extremes I indicate the size of an electron and the time between the crests of gamma-ray waves, just about the highest frequency we ever encounter. The electron is about 6 orders of magnitude smaller than a typical virus (and a single transistor on a chip); the frequency of gamma rays is about 10 orders of magnitude faster than a gigahertz computer chip.

To this computer scientist a machine like an automobile is pretty boring. It runs only one program, or maybe two if you count forward and reverse gear. With few exceptions it has four wheels, one engine, one steering wheel—and all cars go about as fast as any other, if they can move in traffic at all. I could take my father’s 1941 Plymouth out for a spin today and hardly anyone would notice. It cost about \$845 in 1941 (for a four-door sedan), or about \$14,000 in today’s dollars. In other words, in our order-of-magnitude world, it is a product that is practically frozen in time. On the other hand, my obsolete and clumsy laptop beats the first computer I ever used by 5 orders of magnitude in speed and memory, and 4 orders of magnitude in weight and volume. If you want to talk money, I remember paying about 50¢ a byte for extra memory for a small laboratory computer in 1971—8 orders of magnitude more expensive than today, or maybe 9 if you take inflation into account.

The number of zeros is roughly the logarithm (base-10), and plots like the figure are said to have logarithmic scales. You can see them in the chapter on Moore’s law in The Discrete Charm, where I need them to get a manageable picture of just how much progress has been made in computer technology over the last few decades. The shrinkage in size and speedup has been, in fact, exponential with the years—which means constant-size hops in the figure, year by year. Anything less than exponential growth would slow to a crawl. This distinction between exponential and slower-than-exponential growth also plays a crucial role in studying the efficiency of computer algorithms, a favorite pursuit of theoretical computer scientists and a subject I take up towards the end of the book.

Counting zeroes lets us to fit the whole universe on a page.

Ken Steiglitz is professor emeritus of computer science and senior scholar at Princeton University. His books include The Discrete Charm of the MachineCombinatorial OptimizationA Digital Signal Processing Primer, and Snipers, Shills, and Sharks. He lives in Princeton, New Jersey.

This post is part of a series, explore additional posts here<< Ken Steiglitz: Garage Rock and the Unknowable