# Are <1 base systems possible?

I stuck a postin where it didn’t belong with a wiseassed comment, and accidentally got to thinking.

Is it possible to have a number system with a base less than one? If so, are there any useful or interesting properties? What would the digits be? Could a base 0.5 number system use 0.1, 0.2, 0.3, and 0.4? The value of ‘higher’ places diminishes, but the additive quantity still increases. What about a base system using negative numbers? I played around a bit with Excel, and my desk didn’t burn down, but I don’t know if there was any meaning to the results I saw.

Thanks

“Base 0.5” wouldn’t be much different from base 2; you just swap the stuff to the left of the units position with the stuff to the right. For example, thirteen in base 0.5 would be 1.011, instead of 1101 in binary, and four and one-eighth would be 1000.01 in base 0.5 instead of 100.001 in binary.

In general you can map “base x” back and forth with “base 1/x” this way; so base 0.25 is just base 4 backwards, base 0.2 is base 5 backwards, and so on. Other fractional bases are more problematic, since it’s not obvious what the digits should be.

Negative bases are also quite possible. For example in base -2, 101 is (-2)[sup]2[/sup]+1=5, 11010 is six, and 1010 is negative ten. In general you can write any positive or negative integer uniquely in base -n using the digits 0, 1, …, n-1.

And if you really want to abuse your brain, you can also use certain complex numbers for bases. For example, base -1+i, with digits 0 and 1, can be used to represent any Gaussian integer (that is, any complex number whose real and imaginary parts are both integers). One is 1, two is 1100, three is 1101, four is 111010000, negative 4 is 10000, i is 11, and so on. You can even do arithmetic in this base; just remember that 1+1=1100, and 111+11=0, and everything will work out.

Not every Gaussian integer works for a base; for example, base +1+i doesn’t work, at least not with the digits 0 and 1 (not every Gaussian integer can be expressed in this “base”). Another example which does work is base -3+i, if you use the digits 0, 1, …, 9.

You could invent a system where the base of the system was some fractional number. I suspect it would be cumbersome. But fundamentally all the number system is is a way of naming integers.

Number system bases are typically examples of “counting numbers”. These are a special subset of integers, starting from 1. They are used in (duh!) counting, which is the ultimate source of number bases.

But imagine you have a base 0.5 system. You begin to count in your new system. The first item is numbered 0.5, the next is 1.0, the third is 1.5. You see what is happening here? You have given the items different names but you are still counting integral numbers of them.

This isn’t two much different, however, from, say, base 2 numbers. The first is 1, the second 10, the third 11. Different names, same objects.

Your fractional number base system is possible. Whether it is very useful is a question for the mathematicians. Who knows, maybe it will be the basis for a whole new mathematical field. We’ll call it, um, Plutonian-based arithmetic, ok? Oh all right–Rhythmdvl math, then.

You can’t have a base system of less than 1. The base means the number of SYMBOLS used, and it is just by convention that for base 2 for instance we use 0 and 1. Any other two DIFFERENT symbols would do, such as A and B. Then A would be the first number, followed by B, and next would be AB. A would mean 0, B would mean 1, so AB wpi;d be 01…wait now I’m all mixed up! Why in base two do we have 0 meaning 0, then 1 meaning 1, but 01 doesn’t mean 2? Oh I remember, it’s the PLACES that have to be considered. I think the first place is for how many units, the second place is for how many tens, then hundreds, etc., so 253 means 2 units…now wait, it’s the other way around. The last place is the units, then we go left, thus making 253 mean three units, five tens, and two hundreds. Multiplying in each case we have 3 times unit, 5 times 10, and 2 times 100, which gives 3, 50, and 200. then you add and get 253.
Now in binary, the first on the RIGHT means how many units, then moving left how many tens, I think. Thus 10 means no units plus one ten, no that won’t work because 10 is supposed to mean 2. How about zero units, then moving left we get one 2, thus 10 means 2. But I’m not sure that’s the way it works. Now let’s do a number systiem what has onlyl ONE symbol, say 0. Then 0 would be 0 and 00 would have to be 1, then 000 would be 2, etc. lThis is much easier to understand. But I know you can’t have a number system with less than one symbol in it because less than one symbol means either no symbol or else fractions of the symbol.
It does no good to say that you could have a system of base 5 by using .1, …2, etc because now you have six different symbols and you’re just into hexagonals with the added clumsiness of having to put in the . here and there.
Remember, the number system just means SYMBOLS and is called by how many different symbols you have, but always counting zero as a symbol, which is not just nothing but also a place holder. And then there are those columns, which are confusing…

This reminds me of a question I’ve had for a long time: Is there a society that has used a counting system other than base ten? I can imagine doing it, but has a whole society managed to come to a consensus on anything else? It’s so pervasive now, and of course so fundamental in human anatomy, that it’s hard to imagine doing something else.

The Mayans used base 20.

In addition to the Mayan’s sort-of base 20 system (after you get to 360, Mayan numbers start to count by 18’s as well), the Babylonians used a base 60 system (although they had separate words for multiples of ten as well, e.g. there was a word for 10 times 60 as well as one for 60 times 60).

What Math Geek said. You’d run into particular trouble with fractions that aren’t reciprocals or roots of an integer, but you could come up with something.

For instance, let’s say you wanted to have base sqrt(2). Every integer can be represented this way, as you need just use the even powers of sqrt(2) (those which are also powers of 2).

Math Geek: Where can I find more info on the use of Gaussian integers as bases?

The specific question asked in the OP was tangentially discussed (without apparent resolution) in this thread about a year ago. That thread wound up being dedicated mostly to debate over whether a “base 1” system had any meaning…

I agree with Math Geek on the fractional base idea - I’d never thought about it that way, but it certainly seems to be valid. The only quibble I have (and it’s minor) is that negative bases do run into uniqueness problems:

For example, 1[sub]10[/sub] can be represented by either 1[sub]-10[/sub]or -19[sub]-10[/sub]:
1*(-10)[sup]0[/sup] = 1;
(-1) * {1*(-10)[sup]1[/sup] + 9*(-10)[sup]0[/sup]} = (-1)*(-10 + 9) = 1.

I don’t think that negative base number systems use negative signs. This is because every integer can be uniquely expressed without a sign. However, I don’t have a cite for this–can anyone back it up?

Correction: I meant to say that negative base systems don’t need to use negative signs. Since I don’t know of any place where they are used, I can’t really talk about how they’re used, now can I?

I heard about this from one of my undergrad professors. You can try his home page, which has references to his published papers as well as some pretty pictures. You’ll probably find the most information in his papers; the ones in Mathematics Magazine and the Mathematics Intelligencer are likely the most introductory. (At least one of those papers also talks about negative bases.)

That’s an excellent point. I think you’re probably right.

Martin Gardner discussed this subject in a Scientific American column that was reprinted in his book Knotted Doughnuts and Other Mathematical Entertainments. Gardner uses the word “negabinary” to describe the notation based on the powers of -2, and states, “The main virtue of negabinary is that every positive and every negative integer can now be uniquely represented in binary notation without the use of signs.” Gardner provides a chart listing the negabinary notations of the integers from 1 to 20, and from -1 to -20. For example, 19 is expressed as 10111 (16+4-2+1), and -19 is 111101 (-32+16-8+4+1). Every positive number has an odd number of digits, and every negative number has an even number of digits.