How Dirac Made Every Number

A couple weeks back I offered a challenge taken from Graham Farmelo’s biography (The Strangest Man) of the physicist Paul Dirac. The physicist had been invited into a game to create whole numbers by using exactly four 2’s and the normal arithmetic operations, for example:

While four 2’s have to be used, and not any other numerals, it’s permitted to use the 2’s stupidly, as every one of my examples here does. Dirac went off and worked out a scheme for producing any positive integer from them. Now, if all goes well, Dirac’s answer should be behind this cut and it hasn’t been spoiled in the reader or the mails sent out to people reading it.

How Big Is This Number? Answered

My little question about just how big a number was got answered just exactly right by John Friedrich, so if you wondered about how I could say a number took about seven million digits just to write out, there’s your answer. Friedrich gives it as a number with 6,846,169 digits, and I agree. Better, the calculator I found which was able to handle this (MatCalcLite, a free calculator app I have on my iPad) agrees too: it claims that is about which has that magic 6,846,169 digits.

Friedrich uses logarithms to work it out, and this is one of the things logarithms are good for in these days when you don’t generally need them to do multiplications and divisions. You can look at logarithms as letting you evaluate the lengths of numbers — how many digits they need to work out — rather than the numbers themselves, and this brings to the field of accessibility numbers that would otherwise be too big to work with, even on the calculator. (Another thing logarithms are good for is that they’re quite nice to work with if you have to do calculus, so once you’re comfortable with them, you start looking for chances to slip them into analysis.)

One nagging little point about Friedrich’s work, though, is that you need to know the logarithm of 3 to work it out. (Also you need the logarithm of 10, or you could try using the common logarithm — the logarithm base ten — of 3 instead.) For finding the actual number that’s fine; trying to get this answer with any precision without looking up the logarithm of 3 is quirky if not crazy.

But what if you want to do this purely by the joys of mental arithmetic? Could you work out without finding a table of logarithms? Obviously you can’t if you want a really precise answer, and here counts as precise, but could you at least get a good idea of how big a number it is?

The Best Thing About Polynomials

[ Curious: one of the search engine terms which brought people here yesterday was “inner obnoxious”. I can think of when I’d used the words together, eg, in a phrase like “your inner obnoxious twelve-year-old”, the person who makes any kind of attempt at instruction difficult. But who’s searching for that? I find also that “the gil blog by norm feuti” and “heavenly nostrils” brought me visitors so, good for everyone, I think. ]

So polynomials have a number of really nice properties. They’re easy to work with, which is a big one. We might work with difficult mathematical objects, but, rather as with people, we’ll only work with the difficult if they offer something worthwhile in trade, such as solving problems we otherwise can’t hope to tackle. Polynomials are nice and friendly, uncomplaining, and as mathematical objects go, quite un-difficult. Polynomials can be used to approximate any function, which is another big one, as long as we don’t take that “any function” too literally. We still have to think about it some. But here’s an advantage so big it’s almost invisible: to evaluate a polynomial we take some number x and raise it to a variety of powers, which we get by multiplying x by itself over and over again. We take each of those powers and multiply them by a corresponding number, a coefficient. We then add up the products of those coefficients with those powers of x. In all that time we’ve done something great.

Early April’s Math Comics

I had started to think the mathematics references in the comics pages were fading out and I might not have an installment to offer anytime soon. Then, on April 3, Pab Sugenis’s The New Adventures of Queen Victoria — a clip art comic strip which supposes the reader will recognize an illustration of King Edward VI — skipped its planned strip for the day (Sugenis’s choice, he says) and ran a Fuzzy Bunny Time strip calling on pretty much the expected rabbits and mathematics comic strip. (Some people in the Usenet group alt.fan.cecil-adams, which I read reliably and write to occasionally, say Sugenis was briefly a regular; perhaps so, but I don’t remember.) This would start a bumper crop of math strips for the week.

Some Names Which e Doesn’t Have

I’ve outlined now some of the numbers which grew important enough to earn their own names. Most of them are counting numbers; the stragglers are a handful of irrational numbers which proved themselves useful, such as π (pi), or attractive, such as φ (phi), or physically important, such as the fine structure constant. Unnamed except in the list of categories is the number whose explanation I hope to be the first movement of this blog: e.

It’s an important number physically, and a convenient and practical number mathematically. For all that, it defies a simple explanation like π enjoys. The simplest description of which I’m aware is that it is the base of the natural logarithm, which perfectly clarifies things to people who know what logarithms are, know which one is the natural logarithm, and know what the significance of the base is. This I will explain, but not today. For now it’s enough to think of the base as a size of the measurement tool, and to know that switching between one base and another is akin to switching between measuring in centimeters and measuring in inches. What the logarithm is will also wait for explanation; for now, let me hold off on that by saying it’s, in a way, a measure of how many digits it takes to write down a number, so that “81” has a logarithm twice that of “9”, and “49” twice that of “7”, and please don’t take this description so literally as to think the logarithm of “81” is equal to that of “49”.

I agree it’s not clear why we should be interested in the natural logarithm when there are an infinity of possible logarithms, and we can convert a logarithm base e into a logarithm base 10 just by multiplying by the correct number. That, too, will come.

Another common explanation is to say that e describes how fast savings will grow under the influence of compound interest. A dollar invested at one-percent interest, compounded daily, for a year, will grow to just about e dollars. Compounded hourly it grows even closer; compounded by the second it grows closer still; compounded annually, it stays pretty far away. The comparison is probably perfectly clear to those who can invest in anything with interest compounded daily. For my part I note when I finally opened an individual retirement account I put a thousand dollars into an almost thoughtfully selected mutual fund, and within mere weeks had lost $15. That about finishes off compound interest to me.