Napier’s Logarithm

This post is a response to Mr. Cornally’s post on logarithms.  However, since it is intended for his students, and I am not one of his students, I thought I would post my ideas in a separate location to allow their dialog to be untainted by my thoughts.  After all, I am a strong believer that the discussion is much more important than the conclusion. [edit: Doh! I forgot about that whole ping back thing.  Now I just have to pray that his students aren't the type to click on links.]

The Description of the Wonderful Canon of Logarithms,
and the use of which not only in Trigonometry, but also in all
Mathematical Calculations, most fully and easily explained in the most
expeditious manner.

By the author and discoverer
John Napier.
Baron of Merchiston, etc. Scotland.

ON THE AMAZING CANON OF LOGARITHMS.

Preface.
Since nothing is more tedious, fellow mathematicians, in the practice of the mathematical arts, than the great delays suffered in the tedium of lengthy multiplications and divisions, the finding of ratios, and in the extraction of square and cube roots– and in which not only is there the time delay to be considered, but also the annoyance of the many slippery errors that can arise: I had therefore been turning over in my mind, by what sure and expeditious art, I might be able to improve upon these said difficulties.

In the end after much thought, finally I have found an amazing  way of shortening the proceedings, and perhaps the manner in which the method arose will be set out elsewhere: truly, concerning all these matters, there could be nothing more useful than the method that I have found. For all the numbers associated with the multiplications, and divisions of numbers, and with the long arduous tasks of extracting square and cube roots are themselves rejected from the work, and in their place other numbers are substituted, which perform the tasks of these rejected by means of addition, subtraction, and division by two or three only. Since indeed the secret is best made common to all, as all good things are, then it is a pleasant task to set out the method for the public use of mathematicians. Thus, students of mathematics, accept and freely enjoy this work that has been produced by my benevolence   Farewell.

So Napier came up with the concept of the logarithm in the fifteen hundreds.*  Back in the day, things like taking the square root of a number required much more than pressing a few buttons on your TI-83.  Not only did they not have graphing calculators, but they didn’t even have any calculators at all.  What a hard life it must have been.  Sure we can do multiplication and division with relatively small numbers, but what about topics like astronomy where we’re dealing with distances of millions of miles or more.  I would hate to have to perform calculations with those numbers by hand.  I think Napier hated that too.

let us consider a basic algebra problem for a minute.  what is $x^{13} \times x^{16}$ ?  easy, you say… All we have to do is add the $13$ and $16$ to get $x^{29}$.  We use this fact so often, but rarely give it very much thought.  Let us put a number in for $x$, say $2$.  Now we have the equation: $2^{13} \times 2^{16}=2^{29}$.  I know, this doesn’t really seem like a big deal.  However, this tells me something that I would have had a very difficult time figuring out any other way.  Namely, that $8,192 \times 65536=536870912$.  How do I know?  Well if I know that $2^{13}=8192$ and that $2^{16}=65536$, and that $2^{29}=536870912$ then I can just substitute those numbers into my equation and I have the product of two really large numbers (without having to rely on a calculator).  This is an novel idea.  I have just reduced the task of multiplying two numbers in the thousands to the job of adding two 2-digit numbers.

Let’s try another example.  What about computing $\sqrt[3]{2^{33}}$  No problem, I just have to divide $33$ by $3$ to get $11$.  So we have $\sqrt[3]{2^{33}} = 2^{11}$.  Nothing special, right?  Wrong.  I now know that $\sqrt[3]{8589934592} = 2048$.  Imagine trying to calculate that cubed root by hand.

This is the power of a logarithm.  It reduces multiplication and division of giant numbers to addition and subtraction with small numbers.  Exponents and roots become as simple as multiplication and division.

Let’s try an example:

Compute the area of a rectangle with a base of 132874 and a height of 324938 without using a calculator.

Well that problem is easy enough to set up.  we know formula for the area of a rectangle is base times height, or in this case:

$A=132874 \times 324938$.

Here is where things get difficult.  Try computing that by hand and you’ll start to understand the difficulties of being a mathematician in the fourteen hundreds.  However, if we know for a fact that $132874 \approx 10^{5.12344}$ and that $324938 \approx 10^{5.51180}$ then we can rewrite our equation into something easier to calculate.

$A=10^{5.12344} \times 10^{5.51180}$

Well if we just add the exponents, we get $10^{10.63524} = 43175760898$ (the answer is actually $43175811812$ but I was pretty close.  If I used more decimals I would have been much closer).

But wait says the skeptic.  How do you know that $10^{10.63524} = 43175760898$?  Doesn’t that seem much harder than computing the basic multiplication?  Yes, that is a very difficult calculation to make.  This is where common logarithms comes in.  Let’s make a pact to always use a base ten system.  Now if some kind mathematician would spend their lives writing a book of logarithms – a dictionary if you will – translating simple numbers into what they are as a power of ten, then we can be rid of multiplication and powers of large numbers once and for all.  All we will have to do to multiply two numbers is look up what they are as a power of ten, add the exponents, and translate the result back into a regular number with the same book.  For hundreds of years that is exactly what people would do.

*It should be mentioned that Napier’s logarithms were much more cumbersome than the logs we use today.  Of course, the subject of algebra was much more cumbersome than it is today.  His logarithms weren’t based as much off of exponents, but he thought of the topic in a dynamic, geometric-esque way.  Imagine two parallel tracks, each with a train at one end.  both trains start off at the same place, but on one track a train is moving at a constant speed and on the other track, a  train is constantly accelerating. Napier’s version of a logarithm was essentially a way to jump back and forth between those two trains.

On a side note, I have not been writing too much educationally related stuff in my blog recently.  It isn’t that I have nothing to say, but more so that last Friday I turned in a 28 page paper and today I submitted a 40 page paper.  I am pretty burnt out on writing about education right now and decided it would be much more fun to dive into the math world that I love so much.  Maybe this weekend I will get back to writing about education.