## Checking Back in On That 117-Year-Old Roller Coaster

I apologize to people who want to know the most they can about the comic strips of the past week. I’ve not had time to write about them. Part of what has kept me busy is a visit to Lakemont Park, in Altoona, Pennsylvania. The park has had several bad years, including two years in which it did not open at all. But still standing at the park is the oldest-known roller coaster, Leap The Dips.

My first visit to this park, in 2013, among other things gave me a mathematical question to ask. That is, could any of the many pieces of wood in it be original? How many pieces would you expect? One of the dips of Leap The Dips. These hills are not large ones. The biggest drop is about nine feet; the coaster is a total of 41 feet high at its greatest. The track goes back and forth in a figure-eight layout several times, and in the middle of each ‘straightaway’ leg is a dip like this.

Problems of this form happen all the time. They turn up whenever there’s something which has a small chance of happening, but many chances to happen. In this case, there’s a small chance that any particular piece of wood will need replacing. But there are a lot of pieces of wood, and they might need replacement at any ride inspection. So there’s an obvious answer to how likely it is any piece of wood would survive a century-plus. And, from that, how much of that wood should be original.

And, since this is a probability question, I found reasons not to believe in this answer. These reasons amount to my doubting that the reality is much like the mathematical abstraction. I even found evidence that my doubts were correct. The station for the Leap The Dips roller coaster, Lakemont Park, Altoona, Pennsylvania. There are two separate cars visible on the tracks by the station. When I last visited there was only one car on the tracks. The cars have a front and a back seat, and while there is a bar to grab hold of, there are no other restraints, which makes the low-speed ride more exciting.

The sad thing to say about revisiting Lakemont Park — well, one is that the park has lost almost all its amusement park rides. It’s got athletic facilities, and a couple miniature golf courses, but besides two wooden and one kiddie roller coaster, and an antique-cars ride, there’s not much left of its long history as an amusement park. But the other thing is that Leap The Dips was closed when I was able to visit. The ride’s under repairs, and seems to be getting painted too. This is sad, but I hope it implies better things soon.

## How To Find Logarithms Without Using Powerful Computers

I got to remembering an old sequence of mine, and wanted to share it for my current audience. A couple years ago I read a 1949-published book about numerical computing. And it addressed a problem I knew existed but hadn’t put much thought into. That is, how to calculate the logarithm of a number? Logarithms … well, we maybe don’t need them so much now. But they were indispensable for computing for a very long time. They turn the difficult work of multiplication and division into the easier work of addition and subtraction. They turn the really hard work of exponentiation into the easier work of multiplication. So they’re great to have. But how to get them? And, particularly, how to get them if you have a computing device that’s able to do work, but not very much work?

Machines That Think About Logarithms sets out the question, including mentioning Edmund Callis Berkeley’s book that got me started on this. And some talk about the kinds of logarithms and why we use each of them.

Machines That Do Something About Logarithms sets out some principles. These are all things that are generically true about logarithms, including about calculating logarithms. They’re just the principles that were put into clever play by Harvard’s IBM Automatic Sequence-Controlled Calculator in the 1940s.

Machines That Give You Logarithms explains how to use those tools. And lays out how to get the base-ten logarithm for most numbers that you would like with a tiny bit of computing work. I showed off an example of getting the logarithm of 47.2286 using only three divisions, four additions, and a little bit of looking up stuff.

Without Machines That Think About Logarithms closes out the cycle. One catch with the algorithm described is that you need to work out some logarithms ahead of time and have them on hand, ready to look up. They’re not ones that you care about particularly for any problem, but they make it easier to find the logarithm you do want. This essay talks about which logarithms to calculate, in order to get the most accurate results for the logarithm you want, using the least custom work possible.

And there we go. Logarithms are still indispensable for mathematical work, although I realize not so much because we ever care what the logarithm of 47.2286 or any other arbitrary number is. Logarithms have some nice analytic properties, though, and they make other work easier to do. So they’re still in use, but for different problems.

## A Leap Day 2016 Mathematics A To Z: Fractions (Continued)

Another request! I was asked to write about continued fractions for the Leap Day 2016 A To Z. The request came from Keilah, of the Knot Theorist blog. But I’d already had a c-word request in (conjecture). So you see my elegant workaround to talk about continued fractions anyway.

## Fractions (continued).

There are fashions in mathematics. There are fashions in all human endeavors. But mathematics almost begs people to forget that it is a human endeavor. Sometimes a field of mathematics will be popular a while and then fade. Some fade almost to oblivion. Continued fractions are one of them.

A continued fraction comes from a simple enough starting point. Start with a whole number. Add a fraction to it. $1 + \frac{2}{3}$. Everyone knows what that is. But then look at the denominator. In this case, that’s the ‘3’. Why couldn’t that be a sum, instead? No reason. Imagine then the number $1 + \frac{2}{3 + 4}$. Is there a reason that we couldn’t, instead of the ‘4’ there, have a fraction instead? No reason beyond our own timidity. Let’s be courageous. Does $1 + \frac{2}{3 + \frac{4}{5}}$ even mean anything?

Well, sure. It’s getting a little hard to read, but $3 + \frac{4}{5}$ is a fine enough number. It’s 3.8. $\frac{2}{3.8}$ is a less friendly number, but it’s a number anyway. It’s a little over 0.526. (It takes a fair number of digits past the decimal before it ends, but trust me, it does.) And we can add 1 to that easily. So $1 + \frac{2}{3 + \frac{4}{5}}$ means a number a slight bit more than 1.526.

Dare we replace the “5” in that expression with a sum? Better, with the sum of a whole number and a fraction? If we don’t fear being audacious, yes. Could we replace the denominator of that with another sum? Yes. Can we keep doing this forever, creating this never-ending stack of whole numbers plus fractions? … If we want an irrational number, anyway. If we want a rational number, this stack will eventually end. But suppose we feel like creating an infinitely long stack of continued fractions. Can we do it? Why not? Who dares, wins!

OK. Wins what, exactly?

Well … um. Continued fractions certainly had a fashionable time. John Wallis, the 17th century mathematician famous for introducing the ∞ symbol, and for an interminable quarrel with Thomas Hobbes over Hobbes’s attempts to reform mathematics, did much to establish continuous fractions as a field of study. (He’s credited with inventing the field. But all claims to inventing something big are misleading. Real things are complicated and go back farther than people realize, and inventions are more ambiguous than people think.) The astronomer Christiaan Huygens showed how to use continued fractions to design better gear ratios. This may strike you as the dullest application of mathematics ever. Let it. It’s also important stuff. People who need to scale one movement to another need this.

In the 18th and 19th century continued fractions became interesting for higher mathematics. Continued fractions were the approach Leonhard Euler used to prove that e had to be irrational. That’s one of the superstar numbers of mathematics. Johan Heinrich Lambert used this to show that if θ is a rational number (other than zero) then the tangent of θ must be irrational. This is one path to showing that π must be irrational. Many of the astounding theorems of Srinivasa Ramanujan were about continued fractions, or ideas which built on continued fractions.

But since the early 20th century the field’s evaporated. I don’t have a good answer why. The best speculation I’ve heard is that the field seems to fit poorly into any particular topic. Continued fractions get interesting when you have an infinitely long stack of nesting denominators. You don’t want to work with infinitely long strings of things before you’ve studied calculus. You have to be comfortable with these things. But that means students don’t encounter it until college, at least. And at that point fractions seem beneath the grade level. There’s a handful of proofs best done by them. But those proofs can be shown as odd, novel approaches to these particular problems. Studying the whole field is hardly needed.

So, perhaps because it seems like an odd fit, the subject’s dried up and blown away. Even enthusiasts seem to be resigned to its oblivion. Professor Adam Van Tyul, then at Queens University in Kingston, Ontario, composed a nice set of introductory pages about continued fractions. But the page is defunct. Dr Ron Knott has a more thorough page, though, and one with calculators that work well.

Will continued fractions make a comeback? Maybe. It might take the discovery of some interesting new results, or some better visualization tools, to reignite interest. Chaos theory, the study of deterministic yet unpredictable systems, first grew (we now recognize) in the 1890s. But it fell into obscurity. When we got some new theoretical papers and the ability to do computer simulations, it flowered again. For a time it looked ready to take over all mathematics, although we’ve got things under better control now. Could continued fractions do the same? I’m skeptical, but won’t rule it out.

Postscript: something you notice quickly with continued fractions is they’re a pain to typeset. We’re all right with $1 + \frac{2}{3 + \frac{4}{5}}$. But after that the LaTeX engine that WordPress uses to render mathematical symbols is doomed. A real LaTeX engine gets another couple nested denominators in before the situation is hopeless. If you’re writing this out on paper, the way people did in the 19th century, that’s all right. But there’s no typing it out that way.

But notation is made for us, not us for notation. If we want to write a continued fraction in which the numerators are all 1, we have a brackets shorthand available. In this we would write $2 + \frac{1}{3 + \frac{1}{4 + \cdots }}$ as [2; 3, 4, … ]. The numbers are the whole numbers added to the next level of fractions. Another option, and one that lends itself to having numerators which aren’t 1, is to write out a string of fractions. In this we’d write $2 + \frac{1}{3 +} \frac{1}{4 +} \frac{1}{\cdots + }$. We have to trust people notice the + sign is in the denominator there. But if people know we’re doing continued fractions then they know to look for the peculiar notation.

## Echoing “Fourier Echoes Euler”

The above tweet is from the Analysis Fact of The Day feed, which for the 5th had a neat little bit taken from Joseph Fourier’s The Analytic Theory Of Heat, published 1822. Fourier was trying to at least describe the way heat moves through objects, and along the way he developed thing called Fourier series and a field called Fourier Analysis. In this we treat functions — even ones we don’t yet know — as sinusoidal waves, overlapping and interfering with and reinforcing one another.

If we have infinitely many of these waves we can approximate … well, not every function, but surprisingly close to all the functions that might represent real-world affairs, and surprisingly near all the functions we’re interested in anyway. The advantage of representing functions as sums of sinusoidal waves is that sinusoidal waves are very easy to differentiate and integrate, and to add together those differentials and integrals, and that means we can turn problems that are extremely hard into problems that may be longer, but are made up of much easier parts. Since usually it’s better to do something that’s got many easy steps than it is to do something with a few hard ones, Fourier series and Fourier analysis are some of the things you get to know well as you become a mathematician.

The “Fourier Echoes Euler” page linked here shows simply one nice, sweet result that Fourier proved in that major work. It demonstrates what you get if, for absolutely any real number x, you add together $\cos\left(x\right) - \frac12 \cos\left(2x\right) + \frac13 \cos\left(3x\right) - \frac14 \cos\left(4x\right) + \frac15 \cos\left(5x\right) - \cdots$ et cetera. There’s one step in it — “integration by parts” — that you’ll have to remember from freshman calculus, or maybe I’ll get around to explaining that someday, but I would expect most folks reading this far could follow this neat result.

## How Richard Feynman Got From The Square Root of 2 to e

I wanted to bring to greater prominence something that might have got lost in comments. Elke Stangl, author of the Theory And Practice Of Trying To Combine Just Anything blog, noticed that among the Richard Feynman Lectures on Physics, and available online, is his derivation of how to discover e — the base of natural logarithms — from playing around.

e is an important number, certainly, but it’s tricky to explain why it’s important; it hasn’t got a catchy definition like pi has, and even the description that most efficiently says why it’s interesting (“the base of the natural logarithm”) sounds perilously close to technobabble. As an explanation for why e should be interesting Feynman’s text isn’t economical — I make it out as something around two thousand words — but it’s a really good explanation since it starts from a good starting point.

That point is: it’s easy to understand what you mean by raising a number, say 10, to a positive integer: 104, for example, is four tens multiplied together. And it doesn’t take much work to extend that to negative numbers: 10-4 is one divided by the product of four tens multiplied together. Fractions aren’t too bad either: 101/2 would be the number which, multiplied by itself, gives you 10. 103/2 would be 101/2 times 101/2 times 101/2; or if you think this is easier (it might be!), the number which, multiplied by itself, gives you 103. But what about the number $10^{\sqrt{2}}$? And if you can work that out, what about the number $10^{\pi}$?

There’s a pretty good, natural way to go about writing that and as Feynman shows you find there’s something special about some particular number pretty close to 2.71828 by doing so.

## Without Machines That Think About Logarithms

I’ve got a few more thoughts about calculating logarithms, based on how the Harvard IBM Automatic Sequence-Controlled Calculator did things, and wanted to share them. I also have some further thoughts coming up shortly courtesy my first guest blogger, which is exciting to me.

The procedure that was used back then to compute common logarithms — logarithms base ten — was built on several legs: that we can work out some logarithms ahead of time, that we can work out the natural (base e) logarithm of a number using an infinite series, that we can convert the natural logarithm to a common logarithm by a single multiplication, and that the logarithm of the product of two (or more) numbers equals the sum of the logarithm of the separate numbers.

From that we got a pretty nice, fairly slick algorithm for producing logarithms. Ahead of time you have to work out the logarithms for 1, 2, 3, 4, 5, 6, 7, 8, and 9; and then, to make things more efficient, you’ll want the logarithms for 1.1, 1.2, 1.3, 1.4, et cetera up to 1.9; for that matter, you’ll also want 1.01, 1.02, 1.03, 1.04, and so on to 1.09. You can get more accurate numbers quickly by working out the logarithms for three digits past the decimal — 1.001, 1.002, 1.003, 1.004, and so on — and for that matter to four digits (1.0001) and more. You’re buying either speed of calculation or precision of result with memory.

The process as described before worked out common logarithms, although there isn’t much reason that it has to be those. It’s a bit convenient, because if you want the logarithm of 47.2286 you’ll want to shift that to the logarithm of 4.72286 plus the logarithm of 10, and the common logarithm of 10 is a nice, easy 1. The same logic works in natural logarithms: the natural logarithm of 47.2286 is the natural logarithm of 4.72286 plus the natural logarithm of 10, but the natural logarithm of 10 is a not-quite-catchy 2.3026 (approximately). You pretty much have to decide whether you want to deal with factors of 10 being an unpleasant number or do deal with calculating natural logarithms and then multiplying them by the common logarithm of e, about 0.43429.

But the point is if you found yourself with no computational tools, but plenty of paper and time, you could reconstruct logarithms for any number you liked pretty well: decide whether you want natural or common logarithms. I’d probably try working out both, since there’s presumably the time, after all, and who knows what kind of problems I’ll want to work out afterwards. And I can get quite nice accuracy after working out maybe 36 logarithms using the formula: $\log_e\left(1 + h\right) = h - \frac12 h^2 + \frac13 h^3 - \frac14 h^4 + \frac15 h^5 - \cdots$

This will work very well for numbers like 1.1, 1.2, 1.01, 1.02, and so on: for this formula to work, h has to be between -1 and 1, or put another way, we have to be looking for the logarithms of numbers between 0 and 2. And it takes fewer terms to get the result as precise as you want the closer h is to zero, that is, the closer the number whose logarithm we want is to 1.

So most of my reference table is easy enough to make. But there’s a column left out: what is the logarithm of 2? Or 3, or 4, or so on? The infinite-series formula there doesn’t work that far out, and if you give it a try, let’s say with the logarithm of 5, you get a good bit of nonsense, numbers swinging positive and negative and ever-larger.

Of course we’re not limited to formulas; we can think, too. 3, for example, is equal to 1.5 times 2, so the logarithm of 3 is the logarithm of 1.5 2 plus the logarithm of 2, and we have the logarithm of 1.5, and the logarithm of 2 is … OK, that’s a bit of a problem. But if we had the logarithm of 2, we’d be able to work out the logarithm of 4 — it’s just twice that — and we could get to other numbers pretty easily: 5 is, among other things, 2 times 2 times 1.25 so its logarithm is twice the logarithm of 2 plus the logarithm of 1.25. We’d have to work out the logarithm of 1.25, but we can do that by formula. 6 is 2 times 2 times 1.5, and we already had 1.5 worked out. 7 is 2 times 2 times 1.75, and we have a formula for the logarithm of 1.75. 8 is 2 times 2 times 2, so, triple whatever the logarithm of 2 is. 9 is 3 times 3, so, double the logarithm of 3.

We’re not required to do things this way. I just picked some nice, easy ways to factor the whole numbers up to 9, and that didn’t seem to demand doing too much more work. I’d need the logarithms of 1.25 and 1.75, as well as 2, but I can use the formula or, for that matter, work it out using the rest of my table: 1.25 is 1.2 times 1.04 times 1.001 times 1.000602, approximately. But there are infinitely many ways to get 3 by multiplying together numbers between 1 and 2, and we can use any that are convenient.

We do still need the logarithm of 2, but, then, 2 is among other things equal to 1.6 times 1.25, and we’d been planning to work out the logarithm of 1.6 all the time, and 1.25 is useful in getting us to 5 also, so, why not do that?

So in summary we could get logarithms for any numbers we wanted by working out the logarithms for 1.1, 1.2, 1.3, and so on, and 1.01, 1.02, 1.03, et cetera, and 1.001, 1.002, 1.003 and so on, and then 1.25 and 1.75, which lets us work out the logarithms of 2, 3, 4, and so on up to 9.

I haven’t yet worked out, but I am curious about, what the fewest number of “extra” numbers I’d have to calculate are. That is, granted that I have to figure out the logarithms of 1.1, 1.01, 1.001, et cetera anyway. The way I outlined things I have to also work out the logarithms of 1.25 and 1.75 to get all the numbers I need. Is it possible to figure out a cleverer bit of factorization that requires only one extra number be worked out? For that matter, is it possible to need no extra numbers? My instinctive response is to say no, but that’s hardly a proof. I’d be interested to know better.

## Machines That Give You Logarithms

As I’ve laid out the tools that the Harvard IBM Automatic Sequence-Controlled Calculator would use to work out a common logarithm, now I can show how this computer of the 1940s and 1950s would do it. The goal, remember, is to compute logarithms to a desired accuracy, using computers that haven’t got abundant memory, and as quickly as possible. As quickly as possible means, roughly, avoiding multiplication (which takes time) and doing as few divisions as can possibly be done (divisions take forever).

As a reminder, the tools we have are:

1. We can work out at least some logarithms ahead of time and look them up as needed.
2. The natural logarithm of a number close to 1 is $log_e\left(1 + h\right) = h - \frac12h^2 + \frac13h^3 - \frac14h^4 + \frac15h^5 - \cdots$.
3. If we know a number’s natural logarithm (base e), then we can get its common logarithm (base 10): multiply the natural logarithm by the common logarithm of e, which is about 0.43429.
4. Whether the natural or the common logarithm (or any other logarithm you might like) $\log\left(a\cdot b\cdot c \cdot d \cdots \right) = \log(a) + \log(b) + \log(c) + \log(d) + \cdots$

Now we’ll put this to work. The first step is which logarithms to work out ahead of time. Since we’re dealing with common logarithms, we only need to be able to work out the logarithms for numbers between 1 and 10: the common logarithm of, say, 47.2286 is one plus the logarithm of 4.72286, and the common logarithm of 0.472286 is minus two plus the logarithm of 4.72286. So we’ll start by working out the logarithms of 1, 2, 3, 4, 5, 6, 7, 8, and 9, and storing them in what, in 1944, was still a pretty tiny block of memory. The original computer using this could store 72 numbers at a time, remember, though to 23 decimal digits.

So let’s say we want to know the logarithm of 47.2286. We have to divide this by 10 in order to get the number 4.72286, which is between 1 and 10, so we’ll need to add one to whatever we get for the logarithm of 4.72286 is. (And, yes, we want to avoid doing divisions, but dividing by 10 is a special case. The Automatic Sequence-Controlled Calculator stored numbers, if I am not grossly misunderstanding things, in base ten, and so dividing or multiplying by ten was as fast for it as moving the decimal point is for us. Modern computers, using binary arithmetic, find it as fast to divide or multiply by powers of two, even though division in general is a relatively sluggish thing.)

We haven’t worked out what the logarithm of 4.72286 is. And we don’t have a formula that’s good for that. But: 4.72286 is equal to 4 times 1.1807, and therefore the logarithm of 4.72286 is going to be the logarithm of 4 plus the logarithm of 1.1807. We worked out the logarithm of 4 ahead of time (it’s about 0.60206, if you’re curious).

We can use the infinite series formula to get the natural logarithm of 1.1807 to as many digits as we like. The natural logarithm of 1.1807 will be about $0.1807 - \frac12 0.1807^2 + \frac13 0.1807^3 - \frac14 0.1807^4 + \frac15 0.1807^5 - \cdots$ or 0.16613. Multiply this by the logarithm of e (about 0.43429) and we have a common logarithm of about 0.07214. (We have an error estimate, too: we’ve got the natural logarithm of 1.1807 within a margin of error of $\frac16 0.1807^6$, or about 0.000 0058, which, multiplied by the logarithm of e, corresponds to a margin of error for the common logarithm of about 0.000 0025.

Therefore: the logarithm of 47.2286 is about 1 plus 0.60206 plus 0.07214, which is 1.6742. And it is, too; we’ve done very well at getting the number just right considering how little work we really did.

Although … that infinite series formula. That requires a fair number of multiplications, at least eight as I figure it, however you look at it, and those are sluggish. It also properly speaking requires divisions, although you could easily write your code so that instead of dividing by 4 (say) you multiply by 0.25 instead. For this particular example number of 47.2286 we didn’t need very many terms in the series to get four decimal digits of accuracy, but maybe we got lucky and some other number would have required dozens of multiplications. Can we make this process, on average, faster?

And here’s one way to do it. Besides working out the common logarithms for the whole numbers 1 through 9, also work out the common logarithms for 1.1, 1.2, 1.3, 1.4, et cetera up to 1.9. And then …

We started with 47.2286. Divide by 10 (a free bit of work) and we have 4.72286. Divide 4.72286 is 4 times 1.180715. And 1.180715 is equal to 1.1 — the whole number and the first digit past the decimal — times 1.07337. That is, 47.2286 is 10 times 4 times 1.1 times 1.07337. And so the logarithm of 47.2286 is the logarithm of 10 plus the logarithm of 4 plus the logarithm of 1.1 plus the logarithm of 1.07337. We are almost certainly going to need fewer terms in the infinite series to get the logarithm of 1.07337 than we need for 1.180715 and so, at the cost of one more division, we probably save a good number of multiplications.

The common logarithm of 1.1 is about 0.041393. So the logarithm of 10 (1) plus the logarithm of 4 (0.60206) plus the logarithm of 1.1 (0.041393) is 1.6435, which falls a little short of the actual logarithm we’d wanted, about 1.6742, but two or three terms in the infinite series should be enough to make that up.

Or we could work out a few more common logarithms ahead of time: those for 1.01, 1.02, 1.03, and so on up to Our original 47.2286 divided by 10 is 4.72286. Divide that by the first number, 4, and you get 1.180715. Divide 1.180715 by 1.1, the first two digits, and you get 1.07337. Divide 1.07337 by 1.07, the first three digits, and you get 1.003156. So 47.2286 is 10 times 4 times 1.1 times 1.07 times 1.003156. So the common logarithm of 47.2286 is the logarithm of 10 (1) plus the logarithm of 4 (0.60206) plus the logarithm of 1.1 (0.041393) plus the logarithm of 1.07 (about 0.02938) plus the logarithm of 1.003156 (to be determined). Even ignoring the to-be-determined part that adds up to 1.6728, which is a little short of the 1.6742 we want but is doing pretty good considering we’ve reduced the whole problem to three divisions, looking stuff up, and four additions.

If we go a tiny bit farther, and also have worked out ahead of time the logarithms for 1.001, 1.002, 1.003, and so on out to 1.009, and do the same process all over again, then we get some better accuracy and quite cheaply yet: 47.2286 divided by 10 is 4.72286. 4.72286 divided by 4 is 1.180715. 1.180715 divided by 1.1 is 1.07337. 1.07337 divided by 1.07 is 1.003156. 1.003156 divided by 1.003 is 1.0001558.

So the logarithm of 47.2286 is the logarithm of 10 (1) plus the logarithm of 4 (0.60206) plus the logarithm of 1.1 (0.041393) plus the logarithm of 1.07 (0.029383) plus the logarithm of 1.003 (0.001301) plus the logarithm of 1.001558 (to be determined). Leaving aside the to-be-determined part, that adds up to 1.6741.

And the to-be-determined part is great: if we used just a single term in this series, the margin for error would be, at most, 0.000 000 0052, which is probably small enough for practical purposes. The first term in the to-be-determined part is awfully easy to calculate, too: it’s just 1.0001558 – 1, that is, 0.0001558. Add that and we have an approximate logarithm of 1.6742, which is dead on.

And I do mean dead on: work out more decimal places of the logarithm based on this summation and you get 1.674 205 077 226 78. That’s no more than five billionths away from the correct logarithm for the original 47.2286. And it required doing four divisions, one multiplication, and five additions. It’s difficult to picture getting such good precision with less work.

Of course, that’s done in part by having stockpiled a lot of hard work ahead of time: we need to know the logarithms of 1, 1.1, 1.01, 1.001, and then 2, 1.2, 1.02, 1.002, and so on. That’s 36 numbers altogether and there are many ways to work out logarithms. But people have already done that work, and we can use that work to make the problems we want to do considerably easier.

But there’s the process. Work out ahead of time logarithms for 1, 1.1, 1.01, 1.001, and so on, to whatever the limits of your patience. Then take the number whose logarithm you want and divide (or multiply) by ten until you get your working number into the range of 1 through 10. Divide out the first digit, which will be a whole number from 1 through 9. Divide out the first two digits, which will be something from 1.1 to 1.9. Divide out the first three digits, something from 1.01 to 1.09. Divide out the first four digits, something from 1.001 to 1.009. And so on. Then add up the logarithms of the power of ten you divided or multiplied by with the logarithm of the first divisor and the second divisor and third divisor and fourth divisor, until you run out of divisors. And then — if you haven’t already got the answer as accurately as you need — work out as many terms in the infinite series as you need; probably, it won’t be very many. Add that to your total. And you are, amazingly, done.

## Machines That Do Something About Logarithms

I’m going to assume everyone reading this accepts that logarithms are worth computing, and try to describe how Harvard’s IBM Automatic Sequence-Controlled Calculator would work them out.

The first part of this is kind of an observation: the quickest way to give the logarithm of a number is to already know it. Looking it up in a table is way faster than evaluating it, and that’s as true for the computer as for you. Obviously we can’t work out logarithms for every number, what with there being so many of them, but we could work out the logarithms for a reasonable range and to a certain precision and trust that the logarithm of (say) 4.42286 is going to be tolerably close to the logarithm of 4.423 that we worked out ahead of time. Working out a range of, say, 1 to 10 for logarithms base ten is plenty, because that’s all the range we need: the logarithm base ten of 44.2286 is the logarithm base ten of 4.42286 plus one. The logarithm base ten of 0.442286 is the logarithm base ten of 4.42286 minus one. You can guess from that what the logarithm of 4,422.86 is, compared to that of 4.42286.

This is trading computer memory for computational speed, which is often worth doing. But the old Automatic Sequence-Controlled Calculator can’t do that, at least not as easily as we’d like: it had the ability to store 72 numbers, albeit to 23 decimal digits. We can’t just use “worked it out ahead of time”, although we’re not going to abandon that idea either.

The next piece we have is something useful if we want to work out the natural logarithm — the logarithm base e — of a number that’s close to 1. We have a formula that will let us work out this natural logarithm to whatever accuracy we want: $\log_{e}\left(1 + h\right) = h - \frac12 h^2 + \frac13 h^3 - \frac14 h^4 + \frac15 h^5 - \cdots \mbox{ if } |h| < 1$

In principle, we have to add up infinitely many terms to get the answer right. In practice, we only add up terms until the error — the difference between our sum and the correct answer — is smaller than some acceptable margin. This seems to beg the question, because how can we know how big that error is without knowing what the correct answer is? In fact we don’t know just what the error is, but we do know that the error can’t be any larger than the absolute value of the first term we neglect.

Let me give an example. Suppose we want the natural logarithm of 1.5, which the alert have noticed is equal to 1 + 0.5. Then h is 0.5. If we add together the first five terms of the natural logarithm series, then we have $0.5 - \frac12 0.5^2 + \frac13 0.5^3 - \frac14 0.5^4 + \frac15 0.5^5$ which is approximately 0.40729. If we were to work out the next term in the series, that would be $-\frac16 0.5^6$, which has an absolute value of about 0.0026. So the natural logarithm of 1.5 is 0.40729, plus or minus 0.0026. If we only need the natural logarithm to within 0.0026, that’s good: we’re done.

In fact, the natural logarithm of 1.5 is approximately 0.40547, so our error is closer to 0.00183, but that’s all right. Few people complain that our error is smaller than what we estimated it to be.

If we know what margin of error we’ll tolerate, by the way, then we know how many terms we have to calculate. Suppose we want the natural logarithm of 1.5 accurate to 0.001. Then we have to find the first number n so that $\frac1n 0.5^n < 0.001$; if I'm not mistaken, that's eight. Just how many terms we have to calculate will depend on what h is; the bigger it is — the farther the number is from 1 — the more terms we'll need.

The trouble with this is that it’s only good for working out the natural logarithms of numbers between 0 and 2. (And it’s better the closer the number is to 1.) If you want the natural logarithm of 44.2286, you have to divide out the highest power of e that’s less than it — well, you can fake that by dividing by e repeatedly — and what you get is that it’s e times e times e times 2.202 and we’re stuck there. Not hopelessly, mind you: we could find the logarithm of 1/2.202, which will be minus the logarithm of 2.202, at least, and we can work back to the original number from there. Still, this is a bit of a mess. We can do better.

The third piece we can use is one of the fundamental properties of logarithms. This is true for any base, as long as we use the same base for each logarithm in the equation here, and I’ve mentioned it in passing before: $\log\left(a\cdot b\cdot c\cdot d \cdots\right) = \log\left(a\right) + \log\left(b\right) + \log\left(c\right) + \log\left(d\right) + \cdots$

That is, if we could factor a number whose logarithm we want into components which we can either look up or we can calculate very quickly, then we know its logarithm is the sum of the logarithms of those components. And this, finally, is how we can work out logarithms quickly and without too much hard work.

## Machines That Think About Logarithms

I confess that I picked up Edmund Callis Berkeley’s Giant Brains: Or Machines That Think, originally published 1949, from the library shelf as a source of cheap ironic giggles. After all, what is funnier than an attempt to explain to a popular audience that, wild as it may be to contemplate, electrically-driven machines could “remember” information and follow “programs” of instructions based on different conditions satisfied by that information? There’s a certain amount of that, though not as much as I imagined, and a good amount of descriptions of how the hardware of different partly or fully electrical computing machines of the 1940s worked.

But a good part, and the most interesting part, of the book is about algorithms, the ways to solve complicated problems without demanding too much computing power. This is fun to read because it showcases the ingenuity and creativity required to do useful work. The need for ingenuity will never leave us — we will always want to compute things that are a little beyond our ability — but to see how it’s done for a simple problem is instructive, if for nothing else to learn the kinds of tricks you can do to get the most of your computing resources.

The example that most struck me and which I want to share is from the chapter on the IBM Automatic Sequence-Controlled Calculator, built at Harvard at a cost of “somewhere near 3 or 4 hundred thousand dollars, if we leave out some of the cost of research and development, which would have been done whether or not this particular machine had ever been built”. It started working in April 1944, and wasn’t officially retired until 1959. It could store 72 numbers, each with 23 decimal digits. Like most computers (then and now) it could do addition and subtraction very quickly, in the then-blazing speed of about a third of a second; it could do multiplication tolerably quickly, in about six seconds; and division, rather slowly, in about fifteen seconds.

The process I want to describe is the taking of logarithms, and why logarithms should be interesting to compute takes a little bit of justification, although it’s implicitly there just in how fast calculations get done. Logarithms let one replace the multiplication of numbers with their addition, for a considerable savings in time; better, they let you replace the division of numbers with subtraction. They further let you turn exponentiation and roots into multiplication and division, which is almost always faster to do. Many human senses seem to work on a logarithmic scale, as well: we can tell that one weight is twice as heavy as the other much more reliably than we can tell that one weight is four pounds heavier than the other, or that one light is twice as bright as the other rather than is ten lumens brighter.

What the logarithm of a number is depends on some other, fixed, quantity, known as the base. In principle any positive number will do as base; in practice, these days people mostly only care about base e (which is a little over 2.718), the “natural” logarithm, because it has some nice analytic properties. Back in the day, which includes when this book was written, we also cared about base 10, the “common” logarithm, because we mostly work in base ten. I have heard of people who use base 2, but haven’t seen them myself and must regard them as an urban legend. The other bases are mostly used by people who are writing homework problems for the part of the class dealing with logarithms. To some extent it doesn’t matter what base you use. If you work out the logarithm in one base, you can convert that to the logarithm in another base by a multiplication.

The logarithm of some number in your base is the exponent you have to raise the base to to get your desired number. For example, the logarithm of 100, in base 10, is going to be 2 because 102 is 100, and the logarithm of e1/3 (a touch greater than 1.3956), in base e, is going to be 1/3. To dig deeper in my reserve of in-jokes, the logarithm of 2038, in base 10, is approximately 3.3092, because 103.3092 is just about 2038. The logarithm of e, in base 10, is about 0.4343, and the logarithm of 10, in base e, is about 2.303. Your calculator will verify all that.

All that talk about “approximately” should have given you some hint of the trouble with logarithms. They’re only really easy to compute if you’re looking for whole powers of whatever your base is, and that if your base is 10 or 2 or something else simple like that. If you’re clever and determined you can work out, say, that the logarithm of 2, base 10, has to be close to 0.3. It’s fun to do that, but it’ll involve such reasoning as “two to the tenth power is 1,024, which is very close to ten to the third power, which is 1,000, so therefore the logarithm of two to the tenth power must be about the same as the logarithm of ten to the third power”. That’s clever and fun, but it’s hardly systematic, and it doesn’t get you many digits of accuracy.

So when I pick up this thread I hope to explain one way to produce as many decimal digits of a logarithm as you could want, without asking for too much from your poor Automatic Sequence-Controlled Calculator.

## Writing About E (Not By Me)

It’s tricky to write about $e$. That is, it’s not a difficult thing to write about, but it’s hard to find the audience for this number. It’s quite important, mathematically, but it hasn’t got an easy-to-understand definition like pi’s “the circumference of a circle divided by its diameter”. E’s most concise definition, I guess, is “the base of the natural logarithm”, which as an explanation to someone who hasn’t done much mathematics is only marginally more enlightening than slapping him with a slice of cold pizza. And it hasn’t got the sort of renown of something like the golden ratio which makes the number sound familiar and even welcoming.

Still, the Mean Green Math blog (“Explaining the whys of mathematics”) has been running a series of essays explaining $e$, by looking at different definitions of the number. The most recent of this has been the twelfth in the series, and they seem to be arranged in chronological order under the category of Algebra II topics, and under the tag of “E” essays, although I can’t promise how long it’ll be before you have to flip through so many “older” page links on the category and tag pages that it’s harder to find that way. If I see a master page collecting all the Definitions Of E essays into one guide I’ll post that.

## Why I Don’t Believe It’s 1/e

The above picture, showing the Leap-the-Dips roller coaster at Lakemont Park before its renovation, kind of answers why despite my neat reasoning and mental calculations I don’t really believe that there’s a chance of something like one in three that any particular board from the roller coaster’s original, 1902, construction is still in place. The picture — from the end of the track, if I’m not mistaken — dates to shortly before the renovation of the roller coaster began in the late 90s. Leap-the-Dips had stood without operating, and almost certainly without maintenance, from 1986 (coincidental to the park’s acquisition by the Boyer Candy company and its temporary renaming as Boyertown USA, in miniature imitation of Hershey Park) to 1998.

The result of this period seems almost to demand replacing every board in the thing. But we don’t know that happened, and after all, surely some boards took it better than others, didn’t they? Not every board was equally exposed to the elements, or to vandalism, or to whatever does smash up wood. And there’s a lot of pieces of wood that go into a wooden roller coaster. Surely some were lucky by virtue of being in the right spot? So in my head I worked out an estimate of about one in three that any particular board would have remained from the Leap-The-Dips’ original, 1902, configuration, even though I didn’t really believe it. Here’s how I got that figure.

First, you have to take a guess as to how likely it is that any board is going to be replaced in any particular stretch of time. Guessing that one percent of boards need replacing per year sounded plausible, what with how neatly a chance of one-in-a-hundred fits with our base ten numbering system, and how it’s been about a hundred years in operation. So any particular board would have about a 99 percent chance of making it through any particular year. If we suppose that the chance of a board making it through the year is independent — it doesn’t change with the board’s age, or the condition of neighboring boards, or anything but the fact that a year has passed — then the chance of any particular board lasting a hundred years is going to be $0.99^{100}$. That takes a little thought to work out if you haven’t got a calculator on hand.

## Quick Little Calculus Puzzle

fluffy, one of my friends and regular readers, got to discussing with me a couple of limit problems, particularly, ones that seemed to be solved through L’Hopital’s Rule and then ran across some that don’t call for that tool of Freshman Calculus which you maybe remember. It’s the thing about limits of zero divided by zero, or infinity divided by infinity. (It can also be applied to a couple of other “indeterminate forms”; I remember when I took this level calculus the teacher explaining there were seven such forms. Without looking them up, I think they’re $\frac00, \frac{\infty}{\infty}, 0^0, \infty^{0}, 0^{\infty}, 1^{\infty}, \mbox{ and } \infty - \infty$ but I would not recommend trusting my memory in favor of actually studying for your test.)

Anyway, fluffy put forth two cute little puzzles that I had immediate responses for, and then started getting plagued by doubts about, so I thought I’d put them out here for people who want the recreation. They’re both about taking the limit at zero of fractions, specifically: $\lim_{x \rightarrow 0} \frac{e^x}{x^e}$ $\lim_{x \rightarrow 0} \frac{x^e}{e^x}$

where e here is the base of the natural logarithm, that is, that number just a little high of 2.71828 that mathematicians find so interesting even though it isn’t pi.

The limit is, if you want to be exact, a subtly and carefully defined idea that took centuries of really bright work to explain. But the first really good feeling that I really got for it is to imagine a function evaluated at the points near but not exactly at the target point — in the limits here, where x equals zero — and to see, if you keep evaluating x very near zero, are the values of your expression very near something? If it does, that thing the expression gets near is probably the limit at that point.

So, yes, you can plug in values of x like 0.1 and 0.01 and 0.0001 and so on into $\frac{e^x}{x^e}$ and $\frac{x^e}{e^x}$ and get a feeling for what the limit probably is. Saying what it definitely is takes a little more work.

## Hopefully, Saying Something True

I wanted to talk about drawing graphs that represent something, and to get there have to say what kinds of things I mean to represent. The quick and expected answer is that I mean to represent some kind of equation, such as “y = 3*x – 2” or “x2 + y2 = 4”, and that probably does come up the most often. We might also be interested in representing an inequality, something like “x2 – 2 y2 ≤ 1”. On occasion we’re interested just in the region where something is not true, saying something like “y ≠ 3 – x”. (I’ve used nice small counting numbers here not out of any interest in these numbers, or because larger ones or non-whole numbers or even irrational numbers don’t work, but because there is something pleasantly reassuring about seeing a “1” or a “2” in an equation. We strongly believe we know what we mean by “1”.)

Anyway, what we’ve written down is something describing a relationship which we are willing to suppose is true. We might not know what x or y are, and we might not care, but at least for the length of the problem we will suppose that the number represented by y must be equal to three times whatever number is represented by x and minus two. There might be only a single value of x we find interesting; there might be several; there might be infinitely many such values. There’ll be a corresponding number of y’s, at least, so long as the equation is true.

Sometimes we’ll turn the description in terms of an equation into a description in terms of a graph right away. Some of these descriptions are like as those of a line — the “y = 3*x – 2” equation — or a simple shape — “x2 + y2 = 4” is a circle — in that we can turn them into graphs right away without having to process them, at least not once we’re familiar and comfortable with the idea of graphing. Some of these descriptions are going to be in awkward forms. “x + 2 = – y2 / x + 2 y /x” is really just an awkward way to describe a circle (more or less), but that shape is hidden in the writing.

## Before Drawing a Graph

I want to talk about drawing graphs, specifically, drawing curves on graphs. We know roughly what’s meant by that: it’s about wiggly shapes with a faint rectangular grid, usually in grey or maybe drawn in dotted lines, behind them. Sometimes the wiggly shapes will be in bright colors, to clarify a complicated figure or to justify printing the textbook in color. Those graphs.

I clarify because there is a type of math called graph theory in which, yes, you might draw graphs, but there what’s meant by a graph is just any sort of group of points, called vertices, connected by lines or curves. It makes great sense as a name, but it’s not what what someone who talks about drawing a graph means, up until graph theory gets into consideration. Those graphs are fun, particularly because they’re insensitive to exactly where the vertices are, so you get to exercise some artistic talent instead of figuring out whatever you were trying to prove in the problem.

The ordinary kind of graphs offer some wonderful advantages. The obvious one is that they’re pictures. People can very often understand a picture of something much faster than they can understand other sorts of descriptions. This probably doesn’t need any demonstration; if it does, try looking at a map of the boundaries of South Carolina versus reading a description of its boundaries. Some problems are much easier to work out if we can approach it as a geometric problem. (And I admit feeling a particular delight when I can prove a problem geometrically; it feels cleverer.)

## Descartes and the Terror of the Negative

When René Descartes first described the system we’ve turned into Cartesian coordinates he didn’t put it forth in quite the way we build them these days. This shouldn’t be too surprising; he lived about four centuries ago, and we have experience with the idea of matching every point on the plane to some ordered pair of numbers that he couldn’t have. The idea has been expanded on, and improved, and logical rigor I only pretend to understand laid underneath the concept. But the core remains: we put somewhere on our surface an origin point — usually this gets labelled O, mnemonic for “origin” and also suggesting the zeroes which fill its coordinates — and we pick some direction to be the x-coordinate and some direction to be the y-coordinate, and the ordered pair for a point are how far in the x-direction and how far in the y-direction one must go from the origin to get there.

The most obvious difference between Cartesian coordinates as Descartes set them up and Cartesian coordinates as we use them is that Descartes would fill a plane with four chips, one quadrant each in the plane. The first quadrant is the points to the right of and above the origin. The second quadrant is to the left of and still above the origin. The third quadrant is to the left of and below the origin, and the fourth is to the right of the origin but below it. This division of the plane into quadrants, and even their identification as quadrants I, II, III, and IV respectively, still exists, one of those minor points on which prealgebra and algebra students briefly trip on their way to tripping over the trigonometric identities.

Descartes had, from his perspective, excellent reason to divide the plane up this way. It’s a reason difficult to imagine today. By separating the plane like this he avoided dealing with something mathematicians of the day were still uncomfortable with. It’s easy enough to describe a point in the first quadrant as being so far to the right and so far above the origin. But a point in the second quadrant is … not any distance to the right. It’s to the left. How far to the right is something that’s to the left?

## Descartes’ Flies

There are a healthy number of legends about René Descartes. Some of them may be true. I know the one I like is the story that this superlative mathematician, philosopher, and theologian (fields not so sharply differentiated in his time as they are today; for that matter, fields still not perfectly sharply differentiated) was so insistent on sleeping late and sufficiently ingenious in forming arguments that while a student at the Jesuit Collè Royal Henry-Le-Grand he convinced his schoolmasters to let him sleep until 11 am. Supposedly he kept to this rather civilized rising hour until he last months of his life, when he needed to tutor Queen Christina of Sweden in the earliest hours of the winter morning.

I suppose this may be true; it’s certainly repeated often enough, and comes to mind often when I do have to wake to the alarm clock. I haven’t studied Descartes’ biography well enough to know whether to believe it, although as it makes for a charming and humanizing touch probably the whole idea is bunk and we’re fools to believe it. I’m comfortable being a little foolish. (I’ve read just the one book which might be described as even loosely biographic of Descartes — Russell Shorto’s Descartes’ Bones — and so, though I have no particular reason to doubt Shorto’s research and no question with his narrative style, suppose I am marginally worse-informed than if I were completely ignorant. It takes a cluster of books on a subject to know it.)

Place the name “Descartes” into the conversation and a few things pop immediately into mind. Those things are mostly “I think, therefore I am”, and some attempts to compose a joke about being “before the horse”. Running up sometime after that is something called “Cartesian coordinates”, which are about the most famous kind of coordinates and the easiest way to get into the problem of describing just where something is in two- or three-dimensional space.

## Can A Ball Of Yarn Threaten Three-Dimensional Space?

All that talk about numbering spots on the New York Thruway had a goal, that of establishing how we could set up a coordinate system for the points on a line. It turns out just as easy to do this for a curve, even one a little bit complicated like a branch of the Thruway. About the only constraints we said anything about were that we shouldn’t have branches. Lurking unstated was the idea that we didn’t have loops. For the Thruway that’s nothing exceptional; if we had a traffic circle in the middle of a high-speed limited-access highway we wouldn’t very long have a high-speed highway. Worse, we’d have some point — where the loop crosses itself — that would have two numbers describing its position. We don’t want to face that. But we’ve got this satisfying little system where we can assign unique numbers to all the points on a single line, or even a curve.

The natural follow-up idea is whether we can set up a system where we can describe a point on a surface or even in all of space using the same sort of coordinates scheme. And there’s the obvious answer of how to do it, using two numbers to describe where something is on a surface, since that’s a two-dimensional thing; or three numbers to describe where it is in space, since that’s a three-dimensional thing. So I’m not going to talk about that just now. I want to do something more fun, the kind of thing that could do nicely in late-night conversations in the dorm lounge if undergraduates still have late-night conversations in the dorm lounge.

If we have a long enough thread, or a strand of yarn, or whatever the quite correct term is, we know this can be set up with a coordinate system by marking off distance along that thread. We imagined doing that, more or less, with the numbering system on the Thruway and imagining the straightening out and curving and other moving around of the highway’s center line. As long as we didn’t stretch or compress the strand any, we could spread it out in any shape we liked, and have coordinates for whatever path the strand traces out.

## Searching For Infinity On The New York Thruway

So with several examples I’ve managed to prove what nobody really questioned, that it’s possible to imagine a complicated curve like the route of the New York Thruway and assign to all the points on it, or at least to the center line of the road, a unique number that no other point on the road has. And, more, it’s possible to assign these unique numbers in many different ways, from any lower bound we like to any upper bound we like. It’s a nice system, particularly if we’re short on numbers to tell us when we approach Loudonville.

But I’m feeling ambitious right now and want to see how ridiculously huge, positive or negative, a number I can assign to some point on the road. Since we’d measured distances from a reference point by miles before and got a range of about 500, or by millimeters and got a range of about 800,000,000, obviously we could get to any number, however big or small, just by measuring distance using the appropriate unit: lay megaparsecs or angstroms down on the Thruway, or even use some awkward or contrived units. I want to shoot for infinitely big numbers. I’ll start by dividing the road in two.

After all, there are two halves to the Thruway, a northern and a southern end, both arranged like upside-down u’s across the state. Instead of thinking of the center line of the whole Thruway, then, think of the center lines of the northern road and of the southern. They’re both about the same 496-mile length, but, it’d be remarkable if they were exactly the same length. Let’s suppose the northern belt is 497 miles, and the southern 495. Pretty naturally the northern belt we can give numbers from 0 to 497, based on how far they are from the south-eastern end of the road; similarly, the southern belt gets numbers from 0 to 495, from the same reference point.

## Searching For 800,000,000 On The New York Thruway

So we’ve introduced, with maybe more words than strictly necessary, the idea that we can set up a match between the numbers from 0 to 496 and particular locations on the New York Thruway. There are a number of practical quibbles that can be brought against this scheme. For example: could we say for certain that the “outer” edge of this road, which has roughly the shape of an upside-down u, isn’t loger than the “inner” edge? We may need more numbers for the one side than the other. And the mile markers, which seemed like an acceptable scheme for noting where one was, are almost certainly only approximately located.

But these aren’t very important. We can imagine the existence of the “ideal” Thruway, some line which runs along the median of the whole extent of the highway, so there’s no difference in length running either direction, and we can imagine measuring it to arbitrarily great precision. The actual road approximates that idealized road. And this gives what I had really wanted, a kind of number line. All the numbers from zero to 496 (or so) match a point on this ideal Thruway line, and all the points on this Thruway match some number between zero and 496. That the line wriggles all over the place and changes direction over and over, well, do we really insist that a line has to be straight?

Well, we can at least imagine taking this “ideal” Thruway, lifting it off the globe and straightening it out, if we really want to. Here we invoke a host of assumptions even past the idea that we can move this curvy idealized road around. We assume that we can straighten it out without changing its length, for example. This isn’t too unreasonable if we imagine this curve as being something like a tangled bit of string and that we straighten it out without putting any particular tension on it; but if we imagined the idealized road as being a rubber band, held taut at the New York City and Ripley, New York, ends and pinned in place at the major turns we notice that isn’t actually guaranteed. Let’s assume we can do this straightening-out without distorting the lengths, though.