Way Off Base (by David A. Wheeler)
This little essay is an exercise in mathematical recreations;
I hope you find it amusing!
Most people use base 10 for their number system.
Computer people often find base 2, 8, or 16 convenient.
But surely, we're missing out.. why not try some really bizarre bases?!
First, a quick definition: in a base B,
each position to the left has its digit multiplied by one greater power
of B, while each position to the right of the decimal point is multiplied
by one less power of B.
With that in mind, let's look at a few bases:
Base 1 has some interesting properties; each position is raised to the
power of one.
If we only get one symbol, and declare it to be "0" (with the value zero),
the only value you can represent is zero.
We could cheat a little bit, and declare that the one symbol we get is "1"
(with the value of one); this way "1" represents one, "11" represents two,
and so on. It's basically like using scratch marks on a wall, but you
don't get to group them.
There's the problem of being able to represent zero, but by cheating
further we could declare that an empty area (with no marks) represents zero.
Alternatively, you could say that you can't represent zero with this
number system, instead you have to represent zero through arithmetic
Note that the decimal point is irrelevant; "11", "1.1", and ".11" all
have the same value (two) in base 1.
You can still use fractions to represent numbers other than
whole numbers: "1/11" is
one half, "11/111" is two-thirds, and so on.
You could even cheat further by using two symbols ("0" and "1" standing
for zero and one), but this kind of cheating wouldn't help you.
Since one to any power is still only one,
adding zeros as a placeholder won't help you.
For example, "10", "100", and "1000" would all have the same value (one).
xkcd has cartoon about the marvelous
powers of 1, which will only make sense if you've seen
Charles and Ray Eames' "Powers of Ten" documentary
by the Simpsons).
Base pi and e
Since we can use integer digits up to (but not including) the
base, counting starts off easily enough: 0, 1, 2, 3.
However, the value of four is tricky, because "10" in base pi is the
value pi. Since pi is an irrational number,
the value "four" will require an infinite
number of digits to completely represent accurately.
Base e does the same sort of thing.
Base i and multiplicands of it
Using the square root of -1, traditionally called "i" or "j",
has its own oddities.
First, there's a symbol choice to be made: does a "1" represent a one or an i?
Let's assume for the moment it means the traditional value of one
(we'll revisit this assumption in a moment).
Using just i is a lot like base 1, but worse.
If we use our "cheating" trick, there are still few values we can
As you can tell, it's cyclic; only four different values can be
represented in base i as a single number.
You can go further with arithmetic, i.e., four would be written
as "1+1+1+1", but certainly the base isn't helping.
- "1" has the value of 1
- "11" has the value of i^1+(1), which is i+1.
- "111" has the value of i^2+(i+1), which is i.
- "1111" has the value of i^3+(i), which is 0.
- "11111" has the value of i^4+(0), which is 1.
- "111111" has the value of i^5+(1), which is i+1.
Do things get better if the lone symbol represents i?
To keep things clear, let's use the symbol "i".. and it turns
out the answer is that it doesn't help:
- "i" has the value of i
- "ii" has the value of i*i+(i), which is i-1.
- "iii" has the value of i*i^2+(i-1), which is -1.
- "iiii" has the value of i*i^3+(-1), which is 0.
- "iiiii" has the value of i*i^4+(0), which is i.
- "iiiiii" has the value of i*i^5+(i), which is i-1.
One approach to solving this problem is to
use the other cheating approach we mentioned in the discussion about
let's permit two symbols ("0" and "1", with their traditional meaning).
This helps quite a bit; now we can count one, two, three as
"1", "10001", "100010001" (using the "1" means one system).
Now at least we can represent all whole numbers - though it's
One odd thing about this approach is that there are now many ways to
represent a number -
"10001" and "100000001" both represent the value two.
You can even represent a few complex numbers quite easily -
traditional "2+i" becomes "10011".
Another solution, which avoids this kind of cheating,
is to use a larger absolute value, but still a complex number,
for the base.
We can, for example, choose 10i as a base.
Doing this has truly baroque impacts that are hard to characterize,
and you can do even more interesting things by writing "complex" numbers
and adding them with "complex" numbers multiplied by i.
Multiplying such numbers in particular is bizarre.
You could use symbols such as "1", "2", as representing their
traditional value, or use them to represent 1i, 2i, and so on;
either way they're bizarre.
Base 0 (zero)
Here we come to the truly worthless base.
In theory we can have no symbols, but let's stretch
and claim we can use one symbol (0).
Unfortunately, without the decimal point, we can only represent the
value of zero, since "0", "00", "000" and so on all evaluate to zero.
Adding the decimal point makes things worse... we now must evaluate 0/0.
I suppose you could argue that base 0 can "represent" all numbers as 0/0,
but since it can't distinguish between them it's not exactly an
You can have fractional bases, but those are actually studied in
High school student Billy Dorminy has even developed an encryption
algorithm using fractional bases, in his science project
titled "Improper Fractional Base Encryption".
Thus, fractional bases are too useful to be considered further
in this paper :-).
Other Related Works
Other people have also thought about number bases.
You could see
Trinary for information about base 3
(see also this article in American Scientist, Nov-Dec 2001).
There's a special form of base 3 called
``balanced ternary'' (base 3)
which uses symbols for 0, +1, and -1.
``Logical Alternative to the Existing Positional
Number System'' by Robert R. Forslund
discusses using the digits 1 to b instead of 0 to (b-1) for a given base b.
is base -2; it appears that this system was used by the experimental Polish
computers SKRZAT 1 and BINEG in 1950.
In Negabinary, negative and positive numbers can be represented
without a sign bit, and arithmetic operations are more complicated.
Donald Knuth's "The Art of Computer Programming", volume 2,
contains Chapter 4.1, "Arithmetic"; that has more
information than perhaps you wanted to know about implementing arithmetic
I've been told that "Number: From Ahmes to Cantor" by
Midhat Gazale, ISBN 0-691-00515-X, Chapter 2 discusses
positional number systems in great detail.
Everything Gray Code (in gzipped Postscript format)
discusses gray code, a different way to use binary digits to represent
Henry S. Warren, Jr.'s "Hacker's Delight" chapter 12
discusses some unusual bases, including
base -2 (with digits 0 or 1),
bases -1+i and -1-i (again, digits 0 or 1),
and hints at a few others such as base 2i with digits 0, 1, 2, and 3.
the Sora language has a varying base, e.g.,
the units are base 12, but the next higher place is base 20.
I haven't covered negative bases; perhaps I'll add those later.
In short, there's a reason you never saw these before!
Hopefully, you found a little fun in this romp through useless bases.
If you enjoyed this article, you might enjoy my article on the
Four fours problem
My home page at www.dwheeler.com.
David A. Wheeler, 2000-09-22; revised 2012-09-08