What is a slide rule, and why was it’s invention such a big deal?
In: 1999
It lets you quickly multiply or divide numbers.
You learned long multiplication and division at school. It takes a bunch of time. Addition and subtraction is much quicker.
There is a mathematical function called “logarithm”, written “log”. It takes a number and gives another number. Exactly how it works isn’t interesting right now. But the interesting part is that adding two logs together is the same as multiplying the original numbers and then taking the log. I.e. (log X) + (log Y) = log (X * Y).
So lets take a stick and put marks along it, like a ruler. “log 1” is zero, so we’ll choose a starting point towards one end of the stick and mark that “1”. Then for every other number X, we calculate P = (log X), and mark the number on the stick at P millimetres from our starting point. Now we can calculate “log X” by measuring the distance along the stick.
OK, just having ONE of those sticks is not very interesting or useful. But if we have TWO such sticks, that makes things interesting.
First, pick two numbers A and B. Find those numbers on your two sticks. Now, line up the sticks so that A on the first stick is next to “1” on the second stick. Like this:
1=============A=============X=====
1=============B======================
On the first stick, read off the number X.
X is at the position (log A) + (log B) millimetres from “1”. We know this because the distance from 1 to A on the first stick is (log A), and the distance from 1 to B on the second stick is (log B). And by simply lining the sticks up, we have added those distances together, so X is a distance of ((log A) + (log B)) from “1”.
But wait, maths tells us that ((log A) + (log B)) is (log (A * B)). So X is a distance of (log (A * B)) from “1”. And because of the way we marked the sticks, we know that the number X is a distance of (log X) from the “1”, too.
So log (A * B) = log X.
So A * B = X.
OK, there was a lot of explanation there, but if you ignore the “why” then actual usage is really simple:
To multiply two numbers, line up one of the numbers on the first stick with the “1” on the second stick. Look for the other number on the second stick, and read off the result on the first stick.
Division is the opposite of multiplication, so:
To divide two numbers A / B, line up A on the first stick with B on the second stick. Look for the “1” on the second stick, and read off the result on the first stick.
A slide rule is just the two sticks described here, attached to each other so they can slide to carry out these calculations.
Actual slide rules may have a lot more features, but the above describes the basics.
Some caveats:
1. “log” only works with positive numbers. But when multiplying or dividing numbers, if one of the number is negative then you can just change it to positive and do the calculations, then change the result to negative. If both numbers are negative you can just change both of them to positive and do the calculations, and you will get the correct result. This is easy enough to do in your head without slowing down.
2. A slide rule has a limited range. What if you need to multiply huge numbers? Well, you remove zeros from the end of the numbers (or shift the decimal point to the left) until you get them small enough to be sensible. Then you do the calculation, then add all the zeroes back on to the result (or shift the decimal point to the right). Similar rules exist for tiny numbers and for division. Again, easy enough to do in your head.
3. Reading numbers off a scale has limited precision. But you can get accurate enough to do some amazing things with just a good slide rule. A lot of engineering can be done with 3 or 4 digits of precision – they have enough safety margins.
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