Back to index of science pages by Donald Sauter.

Right off the bat let me pull the rug out from under you and freely admit myself that this idea probably won't be implemented tomorrow. Looks like Base 10 kind of has a toehold. Still, that shouldn't hold us back from the fun of thinking and comparing and analyzing. Nothing's impossible, and the adoption of Base 8 some day by some civilization is quite a few steps up from unthinkable. Suppose, for example, some four-fingered aliens or, more horrifying yet, another French Revolution, came along and forced it on us, along with 10-day weeks and 1000-day years, shudder, shudder. (Well, at least the former scenario can be proven to be impossible - see my "Human Race is Special" page.)

We humans are "binary" creatures. By that I mean that we can only reliably double and halve quantities using our senses. To be honest, I'm really only referring to lengths here, but that's one of our most common and fundamental measurements. If you have a stick, or a board, or the edge of a table, or a wall, or a column of water, etc., you can reliably put your finger on the midpoint. You'll know you have it right because if you didn't there would be a visible imbalance between the two sides.

So far, that doesn't sound like a big deal; you can hit the midpoint whether you're living in a Base 10 world, where the halfway mark is called .5, or a Base 8 world, in which it's called .4 (because 4 is half of 8.)

But in Base 8 you can continue halving halves to zoom in on any number that can be written. For example, suppose you needed to mark the .31 spot on this unit length:

________________________________________________________________

We concentrate first on locating .3 . The halfway point is .4 :

_______________________________|________________________________ .4

We find .2 halfway between 0 and .4 :

_______________|_______________.________________________________ .2

And .3 is halfway between .2 and .4 :

_______________._______|_______.________________________________ .3

Now we go for the second digit. We split .3 and .4 to get .34 :

_______________._______.___|___.________________________________ .34

We find .32 halfway between .3 and .34 :

_______________._______._|_.___.________________________________ .32

And, finally, we nail .31 halfway between .3 and .32 :

_______________._______.|._.___.________________________________ .31

Imagine that, two-place precision without any sort of measuring
device! And with the point of your pen, you could take it
to *another* place, such as .316. If you haven't thought about
this before, you should be pretty impressed. And remember,
what we just did could be done for *any* number expressed
octally. Even if we were given a number with *many* digits after
the decimal point (I know I'm using a lot of Base 10 terminology here)
we would very quickly zoom in on the spot where our fingertip doesn't
move anymore.

On the other hand, what we just did *cannot* generally be done in
Base 10. That requires guesswork. You can only take a stab at the
.1, .2, .3, .4, .6, .7, .8, or .9 spots. Yes, you can nail the numbers that
occur naturally in Base 8, such as 1/2 (.5), and multiples of
1/4 (.25), 1/8 (.125), etc., but just those few. Think about the
times when you've needed to find the .7 spot between the smallest
divisions of some scale. I'll bet what you do is two quick
halvings to get to .75 and then say, "Ok, it's a little less than that."
If you needed .8, you would move up "a little" from .75 .

(By a similar process of doubling we could quickly construct any desired length greater than 1 using the unit length. But I don't think this has as much practical significance, and notice that what we would do in Base 10 borrows directly from the binary method, anyhow, always being mindful of the powers of 2. For instance, if you needed to construct a length of 19 from a unit, you would do four doublings to get to 16, add on a single doubling to get 18, and then add on one unit. In Base 8, the representation for 19 is 23, which shows directly that we need 2 x 8, plus 3 units.)

In third grade the teacher had us folding paper into fractions.
Getting exact halves
was a cinch, but when Mrs. Koehler challenged us to fold a
piece of paper in thirds, I got *very* frustrated. In Base 8,
1/3 is .252525..., and by continual halvings you can zero in on that.
(I just did an experiment - with excellent results. So there.)

The point of this is, and I wish I could conjure up the most
compelling words, is that *in Base 8 we can connect every number
to the real world* just using our guts. This is *only*
true for number systems based on powers of 2, so it would also
be true for number systems based on 2, 4, and 16. I don't
think I'd get much of an argument by saying that Base 8 represents
the best compromise between the purity of Base 2 and the
compactness and convenience of Base 10. In fact, moving from
Base 10 to Base 8 would hardly give rise to the need for more
digits. Two octal digits are almost as precise as two decimal digits.

I suppose someone could put my feet to the fire and argue that our senses are capable of dividing into thirds as well. But that would take both hands to mark the third-way points, require much more time and effort to gauge the equalness of the three segments, and yield correspondingly less reliable results. (I just tried it on a piece of paper and missed by a mile. So there.)

Notice how builders naturally went for the binary system of halving halves. An inch is divided into halves, quarters, eighths, sixteenths, thirty-secondths, and sixty-fourths. A superb idea but, with all due respect, the fractional notation is strictly for the birds. All those numbers would look great in Base 8. A really messy one like 37/64, which is .578125 in decimal notation, works out to a slim and trim .45 in octal notation.

A nice side benefit of Base 8 is that the multiplication table is 64% as big as the Base 10 multiplication table. The same is true of the addition table. So arithmetic would be that much easier for everybody.

In his book "The Realm of Numbers" Isaac Asimov spends some time extolling the virtues of Base 12:

Observe how useful the dozen is. If you have a dozen apples, you can divide them equally into 2 groups of 6 each, 3 groups of 4 each, 4 groups of 3 each, 6 groups of 2 each, or 12 groups of 1 each. The important thing is that not only are 2 and 4 [as in number systems based on 16] factors of 12, but 3 is also. . . .

The convenience of such factoring in practical arithmetic is great, and there are people who wish that we had used 12 as the base of our number system instead of 10. The number 10 has only two factors, 2 and 5, and cannot be divided evenly by either 3 or 4. The only reason 10 won out over 12 is probably the anatomical accident that we have 5 fingers on each hand. Now if we had had 6 on each hand --

I say the importance of a factor of 3 is grossly trumped up; that
it is not something we have a natural feel for, and division by 4, which is
very natural in Base 8, gives something very similar anyhow. If a factor
of 3 is so important, why not 5 and 7, etc? Moreover, Base 12 all of
a sudden gives us multiplication and addition tables 44% *larger*
than in Base 10 - yeowww!

*I* say, now if we had only ignored our thumbs --

As admitted at the beginning, we're not likely to convert over tomorrow with a big smile on our face. And it certainly wouldn't be worth the effort if we're going to wipe ourselves out or revert to cavemen in the next few decades. It's hard to envision any given generation saying, "Sure, we'll take a dive for future generations. It's for all the kids, man." I'm guessing it'll be more than just the grannies wondering what happened to all the numbers between 77 and 100.

But to pull this out of the depths of complete impossibility, I offer a plan. The plan is simply a new set of numerals for Base 8. In that way, Base 10 and Base 8 could coexist until the time comes when Base 10 has faded silently away forever (or until 3-fingered aliens descend upon us.) After all, we seem to survive with lots of different units of measure going on. Do we ever. (See my proposal for fixing the units mess.)

What occurred to me was to flip our current numerals left-for-right. It works out nice that 2 through 7 are not symmetric about a vertical axis, and 8, which is symmetric, is not used in Base 8 (which will be called Base 10, heehee.) 0 and 1, in their basic forms are symmetric, but they need new characters, anyhow, as they do in Base 10, to finally eliminate the mess caused by the identity with alphabetic characters - ohs and zeros and ells and ones and eyes, oh my.

To refresh your memory, here are the numerals from 0 to 7 (in reverse order - why did I do that, I wonder?):

Here they are flipped left-for-right. (I didn't say mirror image.
Mirrors don't flip left and right, but we won't get into that
here. Ok, we will, since I brought it up. It's *us* that flip
left and right when we turn something toward a mirror. A mirrors pulls
the object through itself, back to front, which has the effect of making
a right-handed object look left-handed.)

And tweaking them a bit for the sake of writeability and uniqueness, this is what I ended up with for our Base 8 numerals:

Just an idea; this doesn't have to be the final word.

I suppose *now* you want me to tell you how to say the
Base 8 numbers, huh? To be honest, I hadn't given it any thought
for all these years until I got to this point in this web page.
But oot answers come right to mind, the nuwhth being ever so
slightly tongue in cheek - we can pronounce the numbers
backwards, too. That's not hard: oh-reez', nuwh, oot, eethr, orf,
vee-ahf', skiss, and nev-ess'.

How's that for English chauvinism?

The ootth answer is to leave this to the people putting together our
universal second language (see my page on that), since Base 8 will be
the official number system in that language, right? Then the
problem is solved once and for all for everybody, as opposed to
new and different words being added to each of the thousands of
languages out there. If the panel wants a bit of advice, I would
suggest that there be only *pronunciations* for numbers,
no associated written words. In the universal second language numbers
will only be written as numerals. Numerals will be considered
elemental, as letters are for words, and representing numerals
with letter combinations will be viewed as nonsensical.

So you can hit the ground running when Base 8 comes along, here is the addition table:

Base 8 Addition Table 0 1 2 3 4 5 6 7 10 1 2 3 4 5 6 7 10 11 2 3 4 5 6 7 10 11 12 3 4 5 6 7 10 11 12 13 4 5 6 7 10 11 12 13 14 5 6 7 10 11 12 13 14 15 6 7 10 11 12 13 14 15 16 7 10 11 12 13 14 15 16 17 10 11 12 13 14 15 16 17 20

Got it? 7 + 5 = 14.

Now here's the multiplication table.

WARNING: Figuring that everybody would give this one glance and surf off into the wild blue yonder, I planted an error or so to snag a few visitors into taking a closer look.

Base 8 Multiplication Table 1 2 3 4 5 6 7 10 2 4 6 8 12 14 16 20 3 6 11 14 17 22 25 30 4 10 14 20 24 30 34 40 5 12 17 24 31 36 43 50 6 14 22 30 36 44 52 60 7 16 25 34 43 52 63 70 10 20 30 40 50 60 70 100

5 x 5 = 31 . . . what a gas! Notice it's multiplication by 4 that gives the nice sequence of answers alternately ending in 0 and 4.

DON'T GET LEFT BEHIND! You can get started in Base 8 by leaving out all the 8's and 9's while trying to fall to sleep tonight: 1 sheep . . . 2 sheep . . . 3 sheep . . . 4 sheep . . . 5 sheep . . . 6 sheep . . . 7 sheep . . . 10 sheep . . . 11 sheep . . . 12 sheep . . . 13 sheep . . . 14 sheep . . . 15 sheep . . . 16 sheep . . . 17 sheep . . . 20 sheep . . . 21 sheep . . . 22 sheep . . . 23 sheep . . . 24 sheep . . . 25 sheep . . . 26 sheep . . . 27 sheep . . . 30 sheep . . . 31 sheep . . . 32 sheep . . . 33 sheep . . . 34 sheep . . . 35 sheep . . . 36 sheep . . . 37 sheep . . . 40 sheep . . . 41 sheep . . . 42 sheep . . . 43 sheep . . . 44 sheep . . . 45 sheep . . . 46 sheep . . . 47 sheep . . . 50 sheep . . . 51 sheep . . . 52 sheep . . . 53 sheep . . . 54 sheep . . . 55 sheep . . . 56 sheep . . . 57 sjeep . . . 60 sheep . . . 61 sheep . . . 62 sheep . . . 63 skeep . . . 64 sheep . . . 65 sheep . . . 66 sheep . . . 67 sseep . . . 70 sheep . . . 71 sweep . . . 72 sneep . . . 73 sheep . . . 74 smeep . . . 75 sheep . . . 76 sheep . . . 77 sbeep . . . 100 sheep . . . 101 szeep . . . szzzzzzz . . .

Contact Donald Sauter: send an email; view guestbook; sign guestbook.

Back to Donald Sauter's main page.

Rather shop than think? Please visit My Little Shop of Rare and Precious Commodities.

Back to the top of this page.

Keywords to assist web searches: base eight base ten base twelve base 16 base sixteen octal hexadecimal.

Parents, if you're considering tutoring or supplemental education for your child, you may be interested in my observations on Kumon.