Just three more ways today, although with all the ones that have been suggested I think we’ll get to the 25!
(15) Multiplication with Log Tables. People had already been using trig tables to multiply, but when logarithms were discovered they became THE way to multiply numbers. The idea behind log tables is that logarithms turn multiplication (nasty) into addition (fun!) without having to derive a bunch off trig formulas. In particular, we’re going to use the fact that log(x·y)=log(x)+log(y).
Here’s how we can find 875×978 with logarithm tables. We’ll start by writing the two numbers in Scientific Notation: 875 would become 8.75×102 and 978 would become 9.78×102. We’re going to multiply 8.75 and 9.78, and then adjust that product by the appropriate power of 10 (in this case, 104).
The next step for the multiplication is to look in your Table of Common [Base 10] Logarithms. If you don’t have a copy handy, you can look here. It turns out that log(8.75) is 0.9420081 [that’s our log(x)] and log(9.78) is 0.9903389 [that’s our log(y)].
Now we’ll add those together to get 1.9323470. This must be the log of our product! So we work backwards with the table, looking for the number whose log is 1.932347. Unfortunately, the table only gives results between 0 and 1, so initially it seems like we’re stymied, but we can be sneaky and subtract 1. We’ll come back to that in a moment.
So now we’re looking for a number whose log is 0.932347.
Looking back at our table, we see that log(8.55)=0.9319661 and log(8.56)=0.9324738, so the number we’re looking for must be between 8.55 and 8.56. We can pick 8.56, which is the closer number, or do a little interpolation. If we round, we’re looking for 0.9323 and instead we got 0.9320 and 0.9325. The number we wanted was about 60% of the way from the smaller to the larger, and so instead of choosing 8.55 or 8.56 we could go about 60% of the distance between them, and guess that the product was 8.556. Let’s do that. [We could be even more accurate if we used a calculator to figure the exact percentage, but that seems to defeat the purpose of using the log table to multiply.]
So now we have a product of 8.556, but clearly that’s not exactly right. We first have to account for the fact that we subtracted 1. Notice that 1+log(blah) is the same as log(10)+log(blah) [because we’re using the common logarithm], and THAT is the same as log(10·blah). Here log(blah) is 0.9323470, and we just saw that blah was approximately 8.556, so the product that we want — the product of 8.75 and 9.78 — is approximately 85.56.
We’re almost there! Remember how originally we wanted the product of 875 and 978 but we first wrote those in Scientific Notation? We need to adjust our answer of 85.56 by 104, leading us to the conclusion that 875×978 is approximately (drum roll please) 855600. The correct answer is 855750, so we’re certainly in the right ballpark and more accurate than we were with the trig tables, but, well, it’s a wonder to me that this method was so popular for so long when so many other ways are even more accurate. Maybe I’m missing something.
(I’ve multiplied with log tables before, but refreshed my memory with this site on the Obsolete Skills Wiki, which also explains how to Get off the couch to Change Channels on the TV set and how to Make Change in [old] Shilling and Pence.)
(16) Slide Rule. This takes the ideas of the log table, but bypasses the actual looking up. Instead, the numbers are scaled on the slide rule in such a way that you don’t have to do much at all. William Oughtred is credited in many places (online) with the slide rule, and the date 1622 shows up, so this happened just a few years after the invention of logarithms. Pretty quick thinking!
Let’s do a simple example first: 2.5×3. You can use an actual slide rule, or use the java version here. The slide rule looks complicated because it can do a lot of things, but we’re mostly going to be looking at the bottom of the slidey part in the middle (C) and the fixed part at the bottom (D).
To multiply 2.5 by 3, you start with2.5 and align the 1 on C with the 2.5 on D. Then look for 3 on C, and right below it will be the product (in this case, 7.5):
See how once you’ve aligned the 1 with the first factor 2.5, you can move the slider doohickey to the 3 on C so that you can see what’s right below it?
Now let’s look at a more complicated example: back to 875×978. As with log tables, we’ll start by writing the numbers in scientific notation (8.75×102 and 9.78×102) and then we’ll multiply 8.75 and 9.78 and adjust our final answer.
Normally we’d align the 1 of C with 8.75 beneath it on D, and then start looking to the right. But 8.75 is so large, we quickly run out of room:
So we’re sneaky, and instead of aligning the 1 (on C) with the 8.75, we align 10 (on C) with 8.75. We’ll have to adjust by multiplying by an extra power of 10 at the end, kind of like we did with the log tables before (and, really, it’s not a coincidence that this happened in both calculations).
So we’ve aligned the 10 on C with the 8.75 on the bottom. Then we move the slider over to 9.78 on C, and look below to see the product! The product looks like, ummm, it seems to be a bit over 8.55 but not yet at 8.56, so we’ll say 8.555.
As our final step, we need to multiply by 10 because we aligned with the 10 instead of with the 1 (as discussed above), and also by 104 because we’d had to write the numbers in scientific notation. This means our final answer is 8.555×105, or 855500. As we’ve seen before, it’s not exact but it is pretty close and it’s a lot faster than looking up in tables!
(17) The Gunter scale. This is the precursor to the slide rule, invented by Edmund Gunter, and it didn’t slide at all but it was BIG: two feet long was standard. It was a big ole piece of wood with a logarithm scale on it, and if you wanted to measure a product like 2.5×3 you’d measure the physical distances to 2.5 and to 3, add them together, and see where you ended up on the scale. Essentially it was a Slide Rule that didn’t slide: you had to do that part by hand.
(There’s a picture here and more information about Gunter here, including the fact that he coined the terms for Cosine and Cotangent, though Cosecant is a bit older. I hadn’t known about this method until Pat Ballew brought it up in a recent comment: thanks Pat!)