Inverted Scales
These instructions cover the use of:
- finding reciprocals using CI or DI
- fast, chained multiplication using CI
Terminology
In this tutorial, we always assume the C,D,CI,DI scales run from 1 to 10.
We will refer to numeric positions as decimal numbers between 1 and 10, such is 1.23CI.
Remember that the CI,DI scales run backwards, so 10CI is the left index, not the right.
To find y = 1/x, if x is on CI:
- set cursor at xCI,
- read coefficient of y on C
- Assign the decimal point.
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DI works exactly the same, except with the D scale. Reciprocals are self-inverse, so you can switch C and CI in the algorithm above and get the same results.
Examples
- 1/6
- set cursor at 6CI, (remember scale runs backwards!)
- read coefficient of answer 1.67 on C
- get the exponent: 6 = 6E0, -0-1 = -1
- final answer 1.67E-1 = 0.167
- 1/0.00042
- set cursor at 4.2CI,
- read coefficient of answer 2.38 on C
- get the exponent: .00042 = 4.2E-4, -(-4)-1 = 3
- final answer 2.38E3 = 2380
To find z = w*x*y with one Step:
- set wCI at xD,
- read result on D at yC
- Assign the decimal point after each step.
- To continue, set on final result and start again with more numbers.
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The motions here are exactly the same as in One-step divide and multiply. But instead of using the C scale to divide a number, we use the CI scale to divide by a reciprocal, which is just multiplying. This first step has all the advantages of reciprocal method division, except for multiplication. The second phase is still just multiplying with the C index. Note that you must assign the decimal point with the rules for multiplication in both steps, don't use the rules for division or reciprocals.
Because multiplication commutes, it doesn't matter the order of your numbers, so you have a little more freedom than in division.
Be aware that this method can fail because the last step goes offscale. You can of course continue with some other method of multiplication.
Examples
- find 3*4*5
- set 3CI over 4D.
- read result 1.2 on D, at 1C ( because 10C is offscale ).
- (you don't have to be too accurate reading 1.2, you're only using it for keeping track of decimal points )
- adjust decimal point 1.2 -> 12. ( see decimal point tutorial ).
- read final result 6 on D at 5C
- adjust decimal point again - 12*5 is going to be 60.
- find 2*3*4*5*6*7
- set 2CI over 3D,
- read subresult 6 on D at 10C
- read subresult 2.4->24 on D at 4C
- set cursor at 4C
- set 5CI at cursor
- read subresult 1.2->120 on D at 1C
- read subresult 7.2->720 on D at 6C
- set cursor at 6C
- set 7CI at cursor
- read final result 5.04->5040 on D at 1C
Try the second example multiplying without the CI scale. Notice it takes twice as long, and the final result is farthur off the mark because of accumulated error.
Also try it without taking much care in the settings - you'll notice how drastically the result is wrong. But if you take care to zoom, use the magnifier, and set the points exactly with the subpixel features, You can easily get the exact result 5040.
So your results can be improved by setting the scales carefully, and also improved by noticing ways to reduce the steps in your calculation, like this one here. The plethora of folded and inverse scales aren't just a waste of space, they're for the "power users", who are serious about speed and precision. If you imagine the sheer amount of number-crunching required of a student or technical professional before computers were prevalent, you can understand why these scales, and the tricks and shortcuts they enable, were so valuable.
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