Slide Rules and Decimals
I’ve been doing some posts on slide rules, mostly because they’re fun! One thing that seems to be a bit thought-provoking is locating the decimal point in the result. There are a couple of tried-and-true methods for doing that, but occasionally one method may be preferred, or easier, than another, depending on the configuration of the formula.
Digits or Span
One method is the digits (or span) method noted in the Bishop (1955) book. This chart shows that method.
| Number | Span or Digit | Number | Span or Digit |
|---|---|---|---|
| 3520 | 4 | 0.352 | 0 |
| 352.0 | 3 | 0.0352 | -1 |
| 35.2 | 2 | 0.00352 | -2 |
| 3.52 | 1 | 0.000352 | -3 |
| 0.352 | 0 | 0.0000352 | -4 |
How this translates to the slide rule for adjusting those numbers using the C and D scales is shown below.
| Left | Projection | [] | Right | Projection |
|---|---|---|---|---|
| \(\times\) | SUM | [] | \(\times\) | SUM -1 |
| \(\div\) | DIFF | [] | \(\div\) | DIFF +1 |
Reverse the above if using with CI, CIF or DIF scales as shown in this next chart.
| Left | Projection | [] | Right | Projection |
|---|---|---|---|---|
| \(\times\) | SUM -1 | [] | \(\times\) | SUM |
| \(\div\) | DIFF +1 | [] | \(\div\) | DIFF |
If we have a calculation requiring combined operations such as this next example, using this method is straightforward.
\[A = \frac{25 \times 0.00052 \times 125}{0.00000125 \times 11 \times 0.075}\]
This equates to 2 - (-5) + (-3) - 2 + 3 - (-1) = 6. Added to this sum would be any others, depending on the slide direction during calculations. For this combined operation using only the C and D scales, we have only the 1st division (slide to R) adding +1 for seven total answer digits, giving 1,580,000.
Occasionally this method is not very intuitive, or can be overly complicated. Using this method, there are some configurations where a large difference between the number of factors makes the determination very cumbersome.
\[\frac{24}{0.007 \times 35 \times 1250} \quad and \quad \frac{21 \times 0.000057 \times 350}{44}\]
The above shows two calculations that are not as easy to determine as ones that have the same configuration top and bottom. These examples do not lend themselves to combined operations, and are much easier by performing separate operations, following the multiplication, division, addition, subtraction approach, commonly taught as, “My Dear Aunt Sally.” The first example we do the divisor total, then divide into the dividend (i.e., 24 / 306). The last bottom multiplication gives a -1 (slide R) for 2 - ((-2) + 2 + 4 -1) = -1, giving the result of 0.0784. One way to do the final division is leave the multiplication result (306) on the D scale, slide 24 (C scale) over that, and read the answer on the C scale at the D index.
The second example we do the top multiplication first, for 0.419. Using the CI scale to minimize slide movement, the first multiplication adds nothing (slide R), the second on the C scale (slide R) subtracts one. So we have (2 + (-4) + 3) -2 -1 = -2. This gives a final answer of 0.0095.
Scientific Notation
Now we will depict the same two calculations using scientific notation.
\[\frac{2.4e^{1}}{7e^{-3} \times 3.5e^{1} \times 1.25e^{3}} \quad and \quad \frac{2.1e^{1} \times 5.7e^{-5} \times 3.5e^{2}}{4.4e^{1}}\]
Dealing with the exponents, the first example is 1 - (-3 + 1 + 3) = 0. Here however, we are dealing with powers of 10, (i.e., ratio ~ 10 and result < 1) so move the decimal left one additional place for 0.0784.
The exponents of the second example are (1 + (-5) + 2) - 1 = -3. So the answer 9.5e-3 becomes 0.0095.
Notice the difference between the first and second examples, where the first took some mental gymnastics and the second more straightforward. The first ratio is about 10:1, where the second ratio approaches 100:1. This illustrates where an estimate of what the final answer should be is necessary to realize what slide rule number range makes sense.
And therein lies the main reason electronic calculators took over so quickly from slide rules: time savings and answer magnitude interpretation. And sometimes greater precision was necessary for critical calculations. Of course, like most advances, there were the tradeoffs. We now educate folks in reading digital outputs, without necessarily understanding what is being shown.
In any case, That’s progress. Have a great day, and Happy New Year!