How Many Different Sequences of Eight Bases Can You Make
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Base of operations 8 - the Best Number System!
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 of operations 10 kind of has a toehold. Yet, that shouldn't concur us back from the fun of thinking and comparing and analyzing. Nothing's incommunicable, and the adoption of Base 8 some twenty-four hour period past some civilization is quite a few steps up from unthinkable. Suppose, for example, some iv-fingered aliens or, more horrifying even so, another French Revolution, came along and forced information technology on us, along with ten-day weeks and m-day years, shudder, shudder. (Well, at least the former scenario can be proven to be impossible - see my "Human being Race is Special" page.)
We humans are "binary" creatures. By that I hateful that nosotros can only reliably double and halve quantities using our senses. To be honest, I'k actually only referring to lengths hither, merely that'south one of our most mutual and fundamental measurements. If y'all 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 lot'll know you have information technology right because if you didn't there would be a visible imbalance between the two sides.
Then far, that doesn't audio similar a big deal; you tin hitting the midpoint whether you're living in a Base of operations 10 globe, where the halfway marker is called .5, or a Base 8 earth, in which information technology's called .4 (because four is half of viii.)
But in Base 8 you can continue halving halves to zoom in on any number that tin be written. For example, suppose y'all needed to mark the .31 spot on this unit length:
________________________________________________________________
We concentrate first on locating .3 . The halfway bespeak is .iv :
_______________________________|________________________________ .4
Nosotros observe .2 halfway between 0 and .4 :
_______________|_______________.________________________________ .two
And .three is halfway between .ii and .4 :
_______________._______|_______.________________________________ .3
Now nosotros 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, nosotros nail .31 halfway between .3 and .32 :
_______________._______.|._.___.________________________________ .31
Imagine that, two-place precision without whatsoever sort of measuring device! And with the bespeak of your pen, you could take it to another place, such as .316. If y'all 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 subsequently the decimal point (I know I'm using a lot of Base 10 terminology here) nosotros would very quickly zoom in on the spot where our fingertip doesn't move anymore.
On the other hand, what we just did cannot by and large be done in Base 10. That requires guesswork. You can simply take a stab at the .1, .two, .3, .4, .half-dozen, .7, .8, or .9 spots. Yes, yous can nail the numbers that occur naturally in Base of operations 8, such as one/2 (.5), and multiples of i/iv (.25), 1/8 (.125), etc., but just those few. Think near the times when you've needed to find the .seven spot betwixt the smallest divisions of some calibration. I'll bet what you do is two quick halvings to get to .75 and so say, "Ok, information technology's a piffling less than that." If you needed .viii, you would move upwards "a piddling" from .75 .
(By a similar process of doubling we could speedily construct any desired length greater than 1 using the unit length. Just 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, e'er existence mindful of the powers of 2. For case, if you lot needed to construct a length of xix from a unit, you would do four doublings to get to 16, add together on a unmarried doubling to go xviii, and and so add on one unit of measurement. In Base 8, the representation for xix is 23, which shows straight that we demand 2 x eight, plus 3 units.)
In third form the instructor had us folding paper into fractions. Getting exact halves was a cinch, but when Mrs. Koehler challenged u.s. to fold a piece of paper in thirds, I got very frustrated. In Base 8, 1/3 is .252525..., and by continual halvings y'all can zero in on that. (I just did an experiment - with excellent results. So there.)
The signal of this is, and I wish I could conjure upwards the most compelling words, is that in Base 8 we can connect every number to the real world merely using our guts. This is only truthful for number systems based on powers of 2, and so it would besides be true for number systems based on ii, 4, and 16. I don't think I'd get much of an argument by maxim that Base viii represents the all-time compromise betwixt the purity of Base of operations 2 and the compactness and convenience of Base ten. In fact, moving from Base of operations 10 to Base of operations 8 would inappreciably give rise to the need for more than digits. Two octal digits are almost as precise every bit two decimal digits.
I suppose someone could put my feet to the fire and contend that our senses are capable of dividing into thirds too. But that would take both hands to marking the third-style points, crave much more time and effort to gauge the equalness of the iii segments, and yield correspondingly less reliable results. (I but tried it on a piece of paper and missed by a mile. So at that place.)
Notice how builders naturally went for the binary organization 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 of operations 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 ten multiplication table. The same is true of the add-on 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 yous have a dozen apples, y'all can divide them equally into 2 groups of 6 each, 3 groups of 4 each, iv groups of iii each, half dozen groups of 2 each, or 12 groups of 1 each. The of import matter is that not merely are 2 and 4 [every bit in number systems based on 16] factors of 12, but 3 is likewise. . . .
The convenience of such factoring in applied arithmetics is bully, and there are people who wish that we had used 12 equally the base of our number organisation instead of 10. The number 10 has only two factors, 2 and 5, and cannot be divided evenly by either 3 or 4. The simply reason 10 won out over 12 is probably the anatomical accident that nosotros have v fingers on each paw. Now if we had had 6 on each mitt --
I say the importance of a factor of three is grossly trumped up; that it is not something we have a natural feel for, and division past 4, which is very natural in Base 8, gives something very similar anyhow. If a factor of 3 is then important, why not five and 7, etc? Moreover, Base of operations 12 all of a sudden gives united states multiplication and addition tables 44% larger than in Base of operations 10 - yeowww!
I say, at present if nosotros had only ignored our thumbs --
Equally admitted at the beginning, nosotros're not likely to convert over tomorrow with a large smile on our face up. And it certainly wouldn't exist worth the endeavour if nosotros're going to wipe ourselves out or revert to cavemen in the adjacent few decades. Information technology'south difficult to envision any given generation proverb, "Sure, nosotros'll take a swoop for hereafter generations. Information technology'due south for all the kids, human being." I'm guessing information technology'll be more than only 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 gear up of numerals for Base of operations 8. In that way, Base of operations 10 and Base eight could coexist until the fourth dimension comes when Base 10 has faded silently away forever (or until 3-fingered aliens descend upon united states.) Afterward all, we seem to survive with lots of different units of measure going on. Do nosotros ever. (Encounter 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 seven are non symmetric near a vertical axis, and viii, which is symmetric, is not used in Base eight (which volition be chosen Base 10, heehee.) 0 and 1, in their bones forms are symmetric, but they need new characters, anyhow, every bit they practise in Base of operations 10, to finally eliminate the mess acquired past 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 opposite 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 nosotros won't become into that here. Ok, we volition, since I brought it up. It'south 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 event 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 concluded up with for our Base 8 numerals:
Just an thought; this doesn't take to be the last word.
I suppose at present y'all want me to tell y'all 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 folio. But oot answers come right to listen, the nuwhth being always so slightly tongue in cheek - we tin 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 respond is to leave this to the people putting together our universal second language (run into my page on that), since Base 8 will be the official number system in that linguistic communication, right? And then the trouble is solved one time and for all for everybody, every bit opposed to new and different words being added to each of the thousands of languages out there. If the console wants a bit of advice, I would advise that there be only pronunciations for numbers, no associated written words. In the universal second language numbers volition only be written as numerals. Numerals will be considered elemental, as letters are for words, and representing numerals with alphabetic character combinations will exist viewed as nonsensical.
So you can hit the ground running when Base 8 comes forth, hither is the addition table:
Base 8 Improver Table 0 ane 2 three four 5 vi vii ten 1 2 three 4 v vi 7 ten 11 2 three 4 5 6 7 10 11 12 3 iv 5 vi 7 10 eleven 12 13 4 five 6 seven 10 xi 12 13 14 5 6 7 10 xi 12 13 14 15 vi 7 x xi 12 13 14 xv 16 7 ten 11 12 thirteen fourteen 15 16 17 x 11 12 13 14 15 16 17 20
Got it? 7 + 5 = xiv.
Now here'southward the multiplication table.
Warning: Figuring that everybody would requite this i glance and surf off into the wild blueish yonder, I planted an error or so to snag a few visitors into taking a closer look.
Base viii Multiplication Table 1 2 iii four 5 six 7 10 2 iv half dozen eight 12 14 16 20 3 six eleven 14 17 22 25 30 four 10 14 20 24 xxx 34 xl v 12 17 24 31 36 43 50 six 14 22 30 36 44 52 60 7 xvi 25 34 43 52 63 70 x 20 30 forty 50 60 seventy 100
v x five = 31 . . . what a gas! Observe information technology's multiplication past 4 that gives the prissy sequence of answers alternately ending in 0 and 4.
DON'T GET LEFT Backside! You can get started in Base of operations 8 by leaving out all the 8'southward and 9's while trying to autumn to sleep this evening: i sheep . . . 2 sheep . . . 3 sheep . . . four sheep . . . v sheep . . . 6 sheep . . . 7 sheep . . . 10 sheep . . . 11 sheep . . . 12 sheep . . . thirteen sheep . . . fourteen sheep . . . 15 sheep . . . 16 sheep . . . 17 sheep . . . xx sheep . . . 21 sheep . . . 22 sheep . . . 23 sheep . . . 24 sheep . . . 25 sheep . . . 26 sheep . . . 27 sheep . . . xxx 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 . . .
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