3D Quick roots and quick reciprocals

Chapter 3 · pages 88–98 · 1742 words

Square roots can be pulled from a break table much as logs were in section 3B, but the geometry is less forgiving. A log table can be reused every time x changes by a factor of 10; a root table only repeats itself when x changes by a factor of 100, since sqrt(100·x) = 10·sqrt(x). Worse, a plain two-decimal root table has wildly uneven step sizes: going from sqrt(100)=10 to sqrt(121)=11 is a jump of 21 in the argument for one unit of root, while going from sqrt(9604)=98 to sqrt(9801)=99 needs a jump of about 197 — roughly ten times as many raw numbers — to move the root by the same one unit. A well-built break table compensates by using closer-together breaks near the low end (near 11) and more widely spaced breaks near the high end (near 98), so that each row of the table advances the root by a controlled amount; that is exactly the design behind the root break table (its "exhibit 7").

Quick reciprocals raise a different problem: ordinary reciprocals reverse order. Since 3 > 2 but 1/3 < 1/2, working directly with 1/x scrambles a batch's ranking. The fix is to use negative reciprocals: -1/3 is indeed more positive than -1/2, so -1/x preserves the original ordering. Because raw values like 57 are inconvenient to invert directly, it's handy to work with -1000/number instead, and a break table (called "exhibit 8" in the book) is built for exactly that quantity — you read off the nonzero digits from the main table and fix the decimal point from a companion panel, the same two-step lookup style used for logs and roots.

A worked example: to find -1000/43.7, look at the digits 437 in the break table; it falls between two adjacent break entries (4347 and 4425), giving nonzero digits -228, and the decimal-point panel converts that to -22.8.

The volcano heights example shows why you'd bother with roots at all. The raw heights of 219 volcanoes (in hundreds of feet) trail off strongly toward large values:

STEM-and-LEAF, unit = 100 feet (rounded), heights of 219 volcanoes
 0 | many low leaves (short stem, few counts)
 ...
 5*| many leaves (heavily populated middle)
 ...
19 | 39730        <- long climb up to the tallest volcanoes

Taking logs (as in section 3A/3B) overcorrects and now trails the other way, toward low values, so logs are too strong a pull. Taking square roots instead lands in between and produces a nicely symmetric stem-and-leaf, roughly:

STEM-and-LEAF, units 0.1 of sqrt(height in hundreds of feet)
 1*| 4
 2 | 65525
12 | 44488
13*| 2266
14 | 0000

with a median near 6500 feet and extremes near 200 and 19,900 feet — a batch that finally looks even-handed rather than skewed. Comparing the three schematic plots (raw, log, root) side by side makes the improvement even more obvious than reading the three stem-and-leaf displays does, because a schematic plot shows symmetry or skew at a glance through the position of the median and the lengths of the whisker-like fences.

The last topic is reciprocal times. When timing an event — a rat running a maze, a fish stopping swimming, a house collapsing in an earthquake — some trials never finish. A rat that never finishes is really just an extreme case of slowness, much closer to a rat that takes 1000 seconds than to one that finishes in 4 seconds; yet on the raw time scale "never" sits infinitely far from everything. Working with 1000/time (or its negative, -1000/time, to preserve order as "more positive means slower") turns "never" into a perfectly ordinary value near zero, and turns very long times into unremarkable small speeds. Data from a rat-running study illustrate this: raw times for four rats show erratic, seemingly very different amounts of spread, with some rats looking wildly variable and one looking oddly compressed; converting to reciprocal times (1000/seconds) produces a much more consistent pattern of roughly symmetric, similarly sized spreads across the rats, with only one rat genuinely standing out as unusually spread out. That changed impression — driven purely by choice of scale — is the section's point: how we look at numbers can matter enormously. When data include both "never" and literal "zero" values, the recommended fix for the zero is to replace it with half the smallest nonzero value that could have been recorded, while also complaining about how the data were collected.

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