However, it's worthwhile mastering all of these approaches since generally you'll want to chain various computations, avoiding the need of jotting down intermediate results or resetting the index.
In other words, depending upon the problem, you may find yourself dancing among the various scales using a combination of these techniques. Know them all ahead of time and you're always ready to switch gears in the heat of battle!
I'll be illustrating these four multiplication methods with a Pickett & Eckel 1006-ES pocket slide rule since it sports all of the scales mentioned. Anyway, it's an extremely cute and silky smooth operating unit. And I love the idea of something so powerful fitting in a shirt pocket! But be sure to follow along with whatever rule you have in hand; the photos here are simply to give you an idea of where the slide and cursor wind up. Note that you can always click a photo to enlarge it for more detail.
We'll begin with the basic approach to multiplying two numbers with a slide rule, which was presumably the start of it all some 400 years ago. Recall the essence is that you're finding a number on the D scale corresponding to the logarithm of the first operand, then moving along the C scale an additional distance corresponding to the logarithm of the second operand.
Example 1.1: Find the product of 3.5 and 2.7:
- Set the cursor at 3.5 on the D scale.
- Align the left index with the hairline.
- Move the cursor to 2.7 on the C scale.
- Read the result under the hairline on the D scale: 9.45.
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| Left index on D:3.5, cursor on C:2.7, result at D:9.45 |
Tip: When trying to make out that last digit, you might be asking yourself, is that a 9.44, 9.45 or 9.46? Well, when multiplying two digit numbers, you can always think about what the product of the concluding digits of the operands are up to. In this case, a 5 (from the 3.5) and a 7 (from the 2.7) imply that the final digit of the result must be a 5 (since 5 times 7 is 35).
Example 1.2: Find the product of 4.2 and 6.5:
First off, try the procedure from Example 1.1.
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| Left index on D:4.2, but C:6.5 is off scale on the right |
Uh-oh! You'll note that 6.5 on the C scale is way to the right, beyond the stator. You might be tempted to rearrange the problem to 6.5 times 4.2 (since multiplication of real numbers is commutative). Give it a whirl. No joy; this time the 4.2 is even further off-scale on the right.
The solution in either case is to use the right index instead. Here's the scoop. A base ten logarithmic scale -- either C or D in this case -- runs from 1 to 10. So, we start with 1.0, then 1.1, 1.2, and so on up to 9.8, 9.9, and then...But instead of 10.0, we'll relabel it 1.0 again, just figured at the next order of magnitude. The scale has simply wrapped around and will continue so in perpetuity.
So the procedure now becomes:
- Set the cursor at 4.2 on the D scale.
- Align the right index with the hairline.
- Move the cursor to 6.5 on the C scale.
- Read the result under the hairline on the D scale: 27.30.
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| Right index on D:4.2, cursor on C:6.5, result at D:27.30 |
Note: The numbers on the C and D scales run from 1 to 10. Because our original attempt put the slide way off to the right and we had to do an index-swap, we know that the product must be the next order of magnitude greater, i.e., 27.30, not 2.730. In any event, estimation will always steer you properly (4 times 6 must be more than 10).
The lesson we draw is that this basic method of multiplication using the C and D scales may force an index-swap on occasion. Efficiency is lost any time you need to reset the slide. However, with experience, you can anticipate when the swap is going to be required and not make the false move first. In particular, if an estimation of the product suggests a mantissa larger than ten will result, then switch to the right index at once. So:
2 times 4: left index as usual
4 times 2: left index as usual, but
3 times 4: use the right index instead
And obviously, the order of magnitude has nothing to do with this, so:
20 times 40: left index as usual
0.4 times 2000: left index as usual, but
3000 times 0.4: use the right index instead
Wrap-Up: So, here are the ups and downs of the basic method of multiplication using the C and D scales:
- Advantages: The C and D scales are large, easy to read and available on all slide rules from the simplest to the most complex.
- Disadvantage: Sometimes you'll need to switch from the left to the right index if you didn't anticipate a bump in order of magnitude coming.