Sunday, April 06, 2008

Things They Forgot To Teach In History Class

Yesterday, a co-worker and I got into a discussion regarding the origin of the word "grocery", whereupon, I opined that originally a "grocer" was one who received goods in batches of 144 -- 144 of anything being a "gross".

This led to Co-worker wondering how the hell anyone would arbitrarily pick 144 as a standard amount of anything.

I offered my opinion that it was for the same reason that many cultures use a base twelve system -- 12 signs in the Zodiac, 12 inches to the foot, two x 12 hours in a day, five x 12 seconds in a minute, same number of minutes in an hour, so on and so forth -- twelve is the highest number that can be counted to using one hand, and 144 is the highest number that can be counted to using both hands.

The blank look on his face was complete, total and final.

*sigh*

Take your right hand and hold it in front of your face, palm in and fingers spread. Take your right thumb and touch the tip of your right little finger and say "One". Touch the thumb to the tip of the ring finger and say, "Two". Tip of middle finger is "Three", tip of index is "Four".

Now, take your thumb back to your little finger and touch the middle section of the finger -- the part between the two joints -- and say, "Five". Mid-length of the ring finger is "Six"; middle finger is "Seven; and mid-point of your index finger is "Eight".

Everyone knows what to do next -- base section of the little finger is "Nine" all the way to base of your index finger, which is "Twelve".

If you grow up using this method to count things, twelve is a logical, elegant number to base your math on.

Now, let us pretend that you are a Babylonian merchant -- or an Egyptian scribe, or even a London greengrocer -- and you are present someplace where a large amount of thingummies that you own, or are in the process of owning, are being offloaded.

As each thingummy passes you, count on your finger joints as shown above -- and when you get to "Twelve" on your right hand, touch the thumb of your left hand to the tip of your little finger. Start over with your right hand -- and when you again get to "Twelve" on your right hand, your left thumb moves to the tip of your left ring finger.

If you continue this, the highest number of thingummies you'll be able to count before running out of finger joints is going to be twelve twelves. Or one hundred and forty-four.

A "gross" -- one hundred and forty-four -- isn't arbitrary at all. It's simply the highest number you can count to on your fingers.

LawDog

36 comments:

lee n. field said...

You can count to 32 on one hand if you use binary.

Andrew said...

...and 1024 if you use both. :)

Peter said...

All of which proves that the digit-al age is older than we think!

:-)

wolfwalker said...

It's a highly creative explanation, Lawdog. It's also one I never heard of before. Mind if I ask where you learned this?

Anyway, I would've given a different origin for base-twelve counting, on common sense alone: five digits for 1-5, and then the clenched fist means "six."

Old NFO said...

I heard that years ago and I agree- Like Peter said, the digital age is a LOT older!

markm said...

So why'd we end up with base 10? Because that counting algorithm is too complex for the average dolt - or the average Roman Senator, I suspect.

There's a more basic reason for favoring base 12, and especially for ordering and storing items in groups of 12: divisibility. You can only split up 10 donuts evenly three ways: 2x5, 5x2, 10x1. You can split up 12 donuts five ways: 2x6, 3x4, 4x3, 6x2, 12x1. Although there are a few numbers that don't go into 12 evenly, no other number under 20 factors so many ways, so your chances of getting an even division no matter how many people show up are better when you buy things by the dozen.

There are also tricks for easier multiplication and division when the base will factor the right way, so base 12 had advantages for mathematicians (before calculators and computers made mental and paper-and-pencil methods irrelevant to the hard calculations). The ancient Babylonian astronomers, who were the first to accurately measure orbital periods and do the multiplication and division to determine when things would return to their original position in the sky, they went one better: they used base 60, which is divisible by 2,3,4,5,6,10,12,15,20, and 30. But I'd hate to have to memorize their addition tables!

Jay said...

Lee, you try doing the number that involves holding down only your third and fifth fingers...I tried that, it's mechanically difficult (at least for me, I can't bend my pinkie without also bending my ring finger). Also it's slower.

My question: is there any reason you can't use all three finger joints and get to sixteen?

Medic3 said...

Using binary (and assuming all 10 digits being present) you can only reach 1023, not 1024. It's that pesky concept of zero.

512+256+128+64+32+16+8+4+2+1=1023

But points to Lee N.Field and Andrew for mentioning binary first!

Oh, and us double jointed folks have no problem with the more acrobatic counting methods. =)

Don Gwinn said...

I'll let Lawdog tell where he got that, but I have my suspicions.

Anyway, base-12 is easier than base-10 in mental math and on paper? Mental math I'll buy, but with pencil and paper and zero for a placeholder, I would think base-10 would work better.

Pete S. said...

Interesting indeed!

Not a day seems to go by where I don't learn something new.

Kristopher said...

Don: That doesn't follow.

If you go completely to base 12, 10 and 11 would have their own symbols ( say "A" and "B" ), 12 would be written at "10", and normal Zero based mathematical figuring would work as usual on paper.

Hoem/private schooled kids would have to memorize 144 position addition/subtraction and 144 position multiplication/division tables ( instead of the normal 100 position tables ), but everything else would work the same.

Anonymous said...

Arabs. That's where the Dog learned his math. His brother has a very ancient Arab sky marker and its system is as elegant as the 12 count is. He can figure anywhere on that thing, and I still don't know how he does it. But, I didn't go over to the mosque across the street and learn wise things from the magus like my boys did.
LawMom

fuzzys dad said...

I have learned some thing new.
Thank-You Dog

Anonymous said...

There are only 10 kinds of people:
Those who understand binary, and those who don't.

Rebekah said...

Hmmm, I learned something new today. Can I go home now? (Im at work and would like to go back to sleep)

Angela said...

Huh. Thankee Mr. Lawdog.

doc Russia said...

hmmm....

And I thought that twelves was used so often because it is the first number that can be halved, divided by a third, and a fourth.

HKpistole said...

Man, I hate Mafematticks.

Grandpa-Old Soldier said...

Everyone knows that if you take the squar root of pi based on the x factory that the only logical calculation will be base x+pi/the sum of what the hell are you talking about. Give me my calculator.

Nelson said...

@ Kristopher: Extend the system to use the fingertips and the knuckles instead of the finger segments, and you can count to 16. That's called hexadecimal, and it's used all the time in the computer biz. It uses 0-9, and adds A-F for values from 10 to 15. 16 decimal = 10 hex.

It can be a bit difficult to wrap one's head around (it took me years to figure it out, and I'm not sure I could do any arithmetic in it), but you can count to 255 with both hands. Not as high as binary will allow, but still more than most folks need. The written representation is nice, too; 255 can be written as FF.

Anonymous said...

De toe bone connected to de foot bone, de foot bone connected to de ankle bone......
LawMom.

Anonymous said...

gosh Lawdog, dat wuz just downright handy!

(sorry, couldn't hep myself)

;-)

Rogue Medic said...

Thank you for the lesson on history and another on math.

These help give a different perspective on many things.

Brian <> said...

It is even mentioned in the Bible where a certain group of individuals were numbered because they knew the difference between their left and right hands.
Jonah 4:11
But Nineveh has more than a hundred and twenty thousand people who cannot tell their right hand from their left, and many cattle as well. Should I not be concerned about that great city?"

In other words there were 120,000 lower class people, Many were the soldiers of the day, not counting the merchants who did know this method of counting

Jenna said...

A quick thanks before I stumble off to my well earned bed (Spring, well... springing is great. But the 100+ square feet of dirt moved topsy turvy is making itself known to my back. Ahh well. Back aches bring summer bounty.) but I did want to make sure to drop this thank you.

First time in a long while the way my da taught my brother and I to count - and KEEP count - in a hurry. Did me good to know I wasn't the only one walking around with this habit.

Christina LMT said...

Thank you! I love learning new (to me) stuff.

Ted said...

After reading this I pulled out the Dictionary (got an OED for a wedding present--it rocks)to check some etymology. All it has to say is that "gross" comes from the romance languages, but it does add that a "great gross" is 12 gross or 1748.

Rogue Medic said...

Here are the listings for gross and grocer from The Online Etymology Dictionary.

Bradley said...

sadly counting 0 you can only get to 31 on one hand, 1023 using both. remember it is the 6th finger that gives 32, and the 11th that gives 1024.

sadly medic 3 beatme but still posting this. :D

Ahab said...

That is cool.

Anonymous said...

Intriguing. Will have to practice
with that.
And am happy no one mentioned the
Viking who couldn't count to 21
without lifting his kilt.
Anon, Don

Weetabix said...

My fingers aren't limber enough to count that way. I have to use the fingers on the mojo hand I keep in my pocket.

Anonymous said...

I hadn't thought of your explanation for 12-12's. I have never counted that way. However, one thing I thought of is that 12 is the highest consecutive number that has its own name. One through Twelve have unique names. At thirteen, you start recycling the smaller numbers in the teens, twenties, and thirties. Spanish numbers are the same I believe. Might be a similar explanation in that.

MechAg94

perlhaqr said...

I like your method for duodecimal counting on one hands. Hadn't ever seen that before. I did, however, independently invent binary hand counting round about the 2nd grade.

Weird how different paths come up in different people's minds, innit?

AlisonH said...

Cool! Thanks!

Wyera said...

You can also do your 9 times tables on two hands. Hold your hands out, palm down. Fold your left pinky down. Every finger to the right of a folded finger counts as a "one", every finger to the left of the folded finger counts as a "ten". So folding your left pinky down, you're left with 9 ones (9x1). Left ring finger folded? 18 (9x2) and so on.