Special Quotients

In this installment we'll investigate how to handle quotients in which squares, square roots, cubes, cube roots and π figure in the denominator. Of course, we've already learned how to deal with them in the numerator in the fairly extensive Special Products installments. So, if you need some review on the scales involved, head back there before proceeding.

Division by Square Roots


As you'd expect, we'll be using the A and B scales for this sort of problem. There are three methods (which I've encountered, anyway). While it may seem redundant to learn three ways to accomplish the same thing, in fact it's worth doing. We'll find the alternatives useful to know when dealing with the problematic division by cube roots, coming up shortly. Try them all out on the following problem.

Exercise 1.1: Compute 2/sqrt(3).

Method #1: This is the direct approach which would occur first to most people. We'll simply carry out a canonical division, but let the B scale provide the square root.
  1. Set the cursor at 2 on the D scale.
  2. Align 3 on the B scale with the hairline.
  3. Move the cursor to the left index.
  4. Read the result under the hairline on D: 1.15.

Cursor at D:2, B:3 to hairline, cursor to left index, result at D:1.15

Should I hasten to remind you once again that you can click the picture to enlarge it for details?

Recall that the B scale is half as long as the D scale, and so must represent square roots. Further recall that the first half of B is devoted to arguments whose whole number part has an odd number of digits, while the second portion is for the even type. So, in Exercise 1.1, you used the leftmost 3.

Method #2: In this approach we begin by computing the reciprocal of the desired fraction first, then undo it at the end by taking the reciprocal again. That may sound a bit Byzantine, but in fact ends up being the only way to handle division by cube roots with simpler slide rules. (You'll see that in action in just a moment). So, let's just go ahead and learn it with square roots now as an investment for the future. We'll attack the same problem as before, but begin by calculating sqrt(3)/2.
  1. Set the cursor at the leftmost 3 on the A scale since it has one digit only.
  2. Align 2 of the C scale with the hairline.
  3. Move the cursor to the right index.
  4. Read the reciprocal under the hairline on the DI scale: 1.15.
It looks like this:

Cursor at A:3, align C:2 with hairline, cursor to right index, result at DI:1.15

Did you notice in Method #1 we used the B and D scales, while in Method #2 it was the A, C and DI scales? Having a choice like that will prove useful with chained operations later.

Now the Pickett 1006-ES pocket slide rule is pretty versatile in that it sports all of the scales just mentioned and quite a few others. But what if you're using a simpler unit, say with the Rietz layout. There is no DI scale, so how do you undo that reciprocal at the end?

Not to worry; it can still be done, although it takes one more movement. Let's try it out on an Aristo 803.
  1. Set the cursor at the leftmost 3 on the A scale.
  2. Align 2 of the C scale with the hairline.
  3. Move the cursor to the right index.
  4. Without touching the cursor, close the rule, aligning C and D.
  5. Read the result on CI under the hairline: 1.15.
I'll summarize the steps with two pics:

Cursor at A:3, C:2 to hairline, cursor to right index
Close rule, result at CI:1.15

In effect, we're doing a scale-transfer here, handing off the non-existent DI scale to CI which does exist. Keep this technique in mind; it'll prove even more useful in a few moments.

Division by Squares


This is pretty uneventful! The rationale hinges on the fact that--thinking in reverse now--the C scale is twice as long as the two portions of the A scale. Thus, if we start on A, then anything measured along C must represent a square. For something this direct, a single method gets the job done. Let's try it out on the following problem:

Exercise 2.1: Compute 15/(3.2^2).
  1. Set the cursor at the second 15 (1.5) on the A scale. (15 has two digits).
  2. Align 3.2 of the C scale with the hairline. Remember this is seen as a square now from A's point of view.
  3. Move the cursor to the left index.
  4. Read the result under the hairline on the A scale: 1.46.
In summary:

Cursor to A:15, C:3.2 to hairline, cursor to left index, result at A:1.46

If the fine divisions of the A scale tax your eyes, in fact this problem could be worked on C, CI and D only, using a chained operation: divide 15 by 3.2 twice in a row. The first time just employ canonical division (C and D scales), and for the second division, multiply by the reciprocal of 3.2 on the CI scale, reading the result on D. Not too shabby, but it does require two movements of the cursor.

Division by Cube Roots


Consider the following problem.

Exercise 3.1: Compute 12/(15^(1/3)).

In words, find 12 divided by the cube root of 15. This ought to be just as easy as square roots, right? Well, the trouble is, while virtually all slide rules sport A and B scales, very few feature a K-equivalent of the B scale. In other words, you simply don't see a K scale on the slide all that often. However, in my collection, one does have it: the Faber-Castell 2/82. The manufacturer labels it K'. As long as I've got it out, let's see how to divide by cube roots using it.
  1. Set the cursor at 12 (1.2) on the D scale.
  2. Align the middle 15 (1.5) of the K' scale with the hairline.
  3. Move the cursor to the right index.
  4. Read the result under the hairline at D: 4.86.
On the Faber-Castell 2/82, a duplex-flip will be required since the K and K' scales appear on the back side of C and D. Hence the three pictures here (which you can click to enlarge for details):

Cursor at D:1.2, duplex-flip
K':15 to hairline, duplex-flip
Cursor to right index, result at D:4.86

Did you notice in step 2 that we needed the middle of the three 15s which appear on the K' scale? That's because it's a two-digit number.

As mentioned, the K' scale isn't much in vogue. So what's a person to do otherwise? Simple; we'll use the analogue of Method #2 from above (which applied to square roots). The idea is to work from the reciprocal, then at the end invert again to right things. Here goes, with the same problem, this time on the simpler Pickett 1006-ES.
  1. Set the cursor at the middle 15 (1.5) on the K scale.
  2. Align 12 (1.2) on the C scale with the hairline.
  3. Move the cursor to the left index.
  4. Read the result under the hairline on the DI scale.

Cursor at K:15, C:12 to hairline, cursor to left index, result at DI:4.86

I'll remind you from the discussion above, that if you're absent DI, you can still carry on with the CI scale after closing the rule.

Division by Cubes


Once more, this is pretty pedestrian, easily accommodated by ordinary division, simply recognizing that if K is our vantage point, then distances on C must represent cubes. (They're three times longer than their compatriots on K). This problem is a snap, then.

Exercise 4.1: Compute 750/(4.5^3).

Here we go:
  1. Set the cursor at the third 7.5 (750, three digits) on the K scale.
  2. Align 4.5 of the C scale with the cursor.
  3. Move the cursor to the left index.
  4. Read the result under the hairline on K: 8.2.
Pictorially:

Cursor at K:7.5, C:4.5 to hairline, cursor to left index, result at K: 8.2

Here's a brief recap of what we've seen with squares, square roots, cubes and cube roots. As a rule, dividing by squares and cubes is the least troublesome. You can pretty much carry on with canonical division, remembering that you're starting and ending on the A or K scale, respectively. But don't forget to set the cursor on the numerator within the correct range (two choices for A, three choices for K).

Square roots are also fairly straightforward. And the only reason cube roots may be a botheration is if your slide rule is lacking a desired scale. But as we've seen, there are always workarounds. In particular, keep that business of making up for an absent DI scale by closing the rule and using CI as a surrogate.

Division by π


This'll be pretty easy to nail down, if you're mastered the techniques in Multiplication by π and π/4. There are two approaches. The first is available on any slide rule, while the second is a shortcut for those rules sporting a DF scale. Here's the problem we'll lick with both methods.

Exercise 5.1: Compute 17/π.

Recalling that most slide rules feature a π gauge mark, let's use it! This first approach is just canonical division.
  1. Set the cursor at 17 on the D scale.
  2. Align π on the C scale with the hairline.
  3. Move the cursor to the right index.
  4. Read the result under the hairline on the D scale: 5.41.
It looks something like this

Cursor to D:17, align C:π with hairline, cursor to right index, result at D:5.41

Then again, if you've got the folded scale DF at your beck, our second method which requires no slide movement at all is particularly sweet. Recall that the DF scale is shifted along by π units, with respect to D, hence represents multiplication by that ubiquitous constant.
  1. Set the cursor at 17 (1.7) on the DF scale.
  2. Read the result under the hairline on the D scale: 5.41.
Graphically:

Cursor to DF:1.7, result at D:5.41
Of course, this is just the reverse of how we learned to multiply by π earlier.

And there we have it, a handful of techniques for dividing by quantities which pop up commonly. By this point, you should be a real expert on the A, B and K scales!

Next installment: Chained Operations: Basic Methods