Slide Rule Basics
Whether you are multiplying 1200 by 4, or 120 by 4, it is all represented as 1.2 x 4 on a slide rule. As the result is read out to be 4.8, where to place the decimal point?
Except for simple cases that you can estimate the results, it’s often difficult, or counter-intuitive to place the decimal point correctly for slide rule calculations.
In a hand calculation, n x m = 3.141 x 5.926 = zzzzzzzz, we start with a 4-digit number multiplying another 4-digit number. Then at the end of the multiplication, we try to place the decimal point.
It’s done by counting the number of digits after the decimal point, n has 3 digits, m has 3 digits, so the result should have 6 digits after the decimal point. That’s probably the rule we all learned in school.
In slide rule calculation, significant digits after the decimal point are dropped to two. In this case, 3.141 turns into 3.14, and 5.926 becomes 5.93. Similarly the result will only have two digits after the decimal point, so it is obvious that counting digits after decimal points is not going to work for slide rule.
Instead, we have to count the digits before the decimal point.
1. Multiplication
Let’s start with a concrete example, m = 3.145, n = 2.4.
To calculate it by hand, you start with 3145 x 24, which = 75480. Then since there are 3 + 1 = 4, 4 digits after the decimal point, the result becomes 7.5480
To make that same calculation on a slide rule, you first “walk” 3.14 on D scale, then 2.4 on C scale. The result reads out as 7.55
This time, we are lucky that the results match hand calculation without adjustment of decimal points.
But what if m = 31.45, n = 24.01, the slide rule will still read 7.55 and we certainly don’t have 4 digits after the decimal point as we would have hand calculation.
So let’s instead look at the digits before the decimal point.
In this case, m has 2 digits before decimal point, n has 2 digits as well. The result should have 2 + 2 — 1 = 3 digits before decimal point. So 7.55 should become 755.0 to be the correct answer.
Why subtract 1, you ask.
Here comes the rule:
When you walk the second multiplicand on C scale, if it goes to the right, subtract one; otherwise no subtraction is needed.
For example, 2 x 6 → the walk of 6 on C is outside the range of D, so we have to “extend” the rule by sliding the other “1” to align with 2, this time 6 is one the left of “1” when 1.2 is read on D. Since 1 + 1 = 2, so 1.2 should be 12, that is two digits before decimal point.
On the other hand, 2 x 4, isn’t going to result in 80 (2 digits) because the walk of 4 on C is on the right side of “1”. So 1 + 1 — 1 = 1, only one digit before decimal point, 8 should be the right answer.
2. Division
In division with slide rule, we also keep track of digits before the decimal point.
To divide m / n, we put m on D and it is aligned with n on C. Then we “walk” backwards on C to “1” in order to read the result on D.
For example 8 / 4 = 2. This time the “walk” on C is towards left, so the result should have 1–1 + 1 = 1, 1 digit before decimal point. 2 has one digit before the decimal point. It is the correct result.
Here comes the rule:
In division, when you walk on C scale to the left, add one;
Another example, 4 / 8, after align 4 and 8, the left side “1” on C is outside, so we have to use the right side “1” to read D scale, we get 5. But this time, the result should have 1–1 = 0 digit ahead of decimal point, so the result isn’t going to 5, but 0.5
Summary
Here is the decimal point rule for the slide rule
< : + 1 | x — 1 >
Reference
https://web.archive.org/web/20180924160456/http://www.sawicki.cc/how_to_use.htm
