# Perspective

It's all in how you look at it

## Fractal numbers

WARNING: if you get a headache when reading discussions of mathematics, you may want to skip this one.

No, that's not their real name, but it is a good way to describe them ...

Everyone is familiar with "real numbers", yes? Well, just in case ... in Mathematics, we talk about various classes of numbers. There are "natural" numbers (counting numbers: 1, 2, 3, ...), whole numbers (natural numbers along with 0), integers (whole numbers and their corresponding negative values), rational numbers (numbers that can be written as fractions, which includes all integers), and then real numbers ... Real numbers includes "irrational numbers" (numbers that can't be written as fractions) such as pi or the square root of 2 as well as rational numbers. Oh, and there are complex numbers, things like the square root of -1.

For a long time, complex numbers were considered as something of a novelty - sort of a mathematical plaything with no real application. Okay, they could make some problems easier, but you don't see anything like a square root of a negative number in reality. That's why multiples of the square root of -1 are called "imaginary numbers" (and square root of -1 is written as i) ... you don't see a board which is i meters long. Though technically you don't have a distance of -2.4 either, distances are always positive though they can be fractions - is there some reason why i is any less real than -2.4?

There are problems in electrical circuitry where we use complex numbers, I won't go into it here but "reactance" involves complex numbers. And there are problems in physics where you have to allow complex numbers in order to get sensible answers (specifically in quantum mechanics), so I think any physicist would agree that i isn't any less real than -2.4 or pi or other "real" numbers. However, there are also problems in quantum mechanics where complex numbers aren't enough ...

The real numbers are what is called a completion of the rational numbers, while pi is not a rational number there are fractions that can get pretty close - as close as you want. In the old days, people would use 22/7 as an approximation for pi, it's accurate two 2 decimal places (22/7 = 3.1428571... while pi = 3.14159...), of course with calculators becoming common when I was in school it was easier to use 3.14 (which technically is 314/100 or 157/50 - also a fraction). Modern "scientific" calculators will do fractions (unlike the ones I had in school), then again any scientific calculator will have a dedicated pi key somewhere on it. Or key combination, I guess technically "Shift+EXP" isn't one key.

However ... there is more than one way to "complete" the rational numbers ... this doesn't actually work too well in base ten, but let me describe the situation in base two (binary - the way computers represent numbers). If I have a problem that says a/b = c where a, b and c are all natural numbers (meaning none of them are 0, and there will not be a remainder) I can actually divide either left-to-right (the way you learned in school when you learned to divide longhand) or right-to-left. This doesn't work in base ten because there's more than one way to get a last digit of 6 if you multiplied by 2 (both 2x3 and 2x8 will end in 6), but in base two the digits are either 0 or 1 and there will be only one way to get either as an answer.

That's not completely clear, so let's get a little more concrete here. If I look at 8-bit numbers (which we'd write in base 10 as 0 to 255), technically 255 and -1 are the same thing if I ignore overflows - 256 and 0 are the same so 1 + 255 = 0 when we limit our answers to 8 bits. But then again, technically 3x171 = 1 (3x171 is 513, subtracting 256 until I get a number between 0 and 255 I end up with 1) so I could say that 1/3 = 171. In fact, I can divide any two odd numbers and get some other odd number as an answer. (I can't divide by 2 because I'm using base two, I'd have to allow a decimal point in base two in order to write a representation for 1/2).

Obviously 1/3 = 171 is only going to be "correct" if we limit ourselves to 8 bits, but there will be some other number if we change that to 16 bits, or 32, or even 64 ... and I can get that number by merely dividing using a right-to-left algorithm instead of the usual left-to-right division. Using similar rules, I could also define a right-to-left version of a square root, though there are some limits on that ... just as I can't get a square root of -1 in real numbers, I can't have a square root of 2 in base two - and there are other numbers as well. But for example, I could find a number which when multiplied by itself gave 17 as a result - or -7 for that matter.

I've said that base two was nice because the only digits are 0 and 1, but two isn't the only "nice" base. In fact, we could use any prime number - mathematicians tend to use p when referring to an unspecified prime number - as a base and do much the same (except that we won't be able to divide by p rather than by 2, etc.). The technical name for the classes of numbers obtained in this fashion is p-adic numbers. If we do allow a decimal point we get what are called p-adic rationals.

Of course the geometry of real numbers is the same as the geometry of a line, the geometry of p-adic rationals is the same as a type of fractal called a generalized Cantor set, hence my term above of "fractal numbers". As I mentioned in passing, there are some problems in quantum mechanics that don't have answer when we use complex numbers - some of these problems can be answered if you use p-adic rationals instead. But do they have any application to the real world?

Well ... maybe. I read an article the other day about the unexplained fractal nature of high temperature superconductors. Perhaps it's not so unexplained after all ...

http://www.sciencenews.org/view/generic/id/62006/title/Superconductors_go_fractal

Felixclaudeb Tuesday, August 24, 2010 1:36:10 PM

Fascinating! I'm not terribly good at math, but this is pretty clear. Not surprised to find out about yet another sighting of fractals in the natural world, either. (I've touched on this subject -- clumsily -- in a blog post years ago.) Thanks for sharing.

Stevesgunhouse Wednesday, August 25, 2010 12:43:00 AM

Not much mention, and really not the same ...

The term fractal has a fairly broad definition and is frequently abused ...

The original definition of fractal was an object with a fractional dimensionality, like ... well, like the Cantor set, the Koch "snowflake" curve and so on. It turns out that these particular fractals are extremely regular (as you'd expect from something purely mathematical), are self-similar (small parts look the same as larger parts), and have a certain abstract "beauty" to them.

The term fractal has since come to be applied to Chaos, though obviously Chaos doesn't fit the "extremely regular" part there. The Mandelbrot set is commonly considered as a fractal, but doesn't fit the first definition as it doesn't have one specific fractional dimension.

I guess in one sense p-adic numbers aren't quite fractal - they are one dimensional, just like a line would be. And 1 isn't a particularly interesting fraction ... well, not like some infinite decimal would be.

Stevesgunhouse Monday, October 18, 2010 2:48:03 PM

jehovajah Saturday, March 19, 2011 7:31:33 AM

Steve, got lost with your left to right right to left division, then switching between bases etc.
171
-----
3)513

is that what you mean by left to right division as taught in school?

In which case is this right to left division?

171
----
513(3

Now your modular arithmetic mod(256) makes sense but your switching between 8 bit 16 bit, etc confuses.

Using mod(p) where p is a prime is the basis of p-adic numbers is what you are saying?

How are these numbers structured? Is this what you mean by geometry?

Any way, What i wanted to ask is how do you or anyone else do latex in ones blog?

Stevesgunhouse Saturday, March 19, 2011 7:59:35 AM

When written in a number system that uses a prime number as base, there is only one number that gives the correct last digit.

Here's a multiplication table for base 7:

``` x | 1  2  3  4  5  6
---+------------------
1 | 1  2  3  4  5  6
2 | 2  4  6 11 13 15
3 | 3  6 12 15 21 24
4 | 4 11 15 22 26 33
5 | 5 13 21 26 34 42
6 | 6 15 24 33 42 51```

The operative thing here is the last digit ... if you know that 24 (base 7) x (something) = 453 (base 7), you can look through the row for "4" (the last digit of 24) above and see the last digit of "something" must be 6, since 4x6 is the only combination in that row that ends in 3. Knowing what the last digit is, you could subtract to figure out what the next-to-last digit must be, and continue until you finally get 0.

LaTeX? I don't know that one. Many web browsers support MathML, if you can convert it to MathML then that should work. Otherwise you'd have to take a picture of the output and use that.

jehovajah Monday, March 21, 2011 2:26:33 AM

Thanks Steve. I will look into mathML.

Also thanks for the table.

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