Remember that the same was done in the english language.
Remember Robin Hood and the seven score yeomen in his merry band of outlaws?
La franchise ne consiste pas à dire tout ce que l’on pense, mais à penser tout ce que l’on dit.
H. de Livry
… or Abe Lincoln and his fourscore and seven years?
[quote]
…for multiplication they introduced a method of successive doubling. For example, to multiply 28 by 11, one constructs a table of multiples of 28 like the following:
1 28
2 56
4 112
8 224
16 448]
I’ve seen gifted 8 year olds {third graders} to this - they hadn’t yet learned the multiplication tables but got correct answers as quickly as their classmates. I didn’t know it was Greek math, neither did they. It was just an end run around the job of learning the tables.
That’s also how I used to convert my number system problems…from octal to decimal…
An old math pub. says, " In the writings of Horace and Cicero, reference is made to the three forms of counting boards used by the Romans: a grooved table with beads, a marked table for counters, and the primitive dust board."
The grooved table isn’t tough to imagine. The marked table is easy to see. The primitive dust board is explained as a board covered with a layer of fine chalk, talc or sand used bofore the development of papyrus, parchment, and paper.
Perhaps the Romans picked one or more of these techinques up from the Greeks - while the dust table might not help us much (too much like paper) the other counting/calculating methods might have helped them over come the clumsiness of the Greek letters and might make more sense to us, too.
If these things were anywhere near as functional as the abacus a good Greek or Roman mathmatician might not have been all that slow.
Jois:
That’s similar to an optimization used (until fairly recently) in programming.
Multiplication used to be slow, but shifting a number of adding were very quick instructions, so it was worth it to take multiplication by a constant where you have to multiply by that constant frequently, and speed it up.
To multiply by 320, you would take the number, shift it left six places, tally that, and shift it another two places, adding the result to the tally. This accomplishes a multiplication by 64 + a multiplication by 256, which, when added, equal a multiplication by 320 (a common screen width for old graphics cards, back when this method was worth using.)
But, it’s interesting that you’ve seen kids using it. It’s a good optimization, because it only relies on multiplication by two, which is easy, and addition, which is easy. No special rules to learn, like carrying, which can trip up a math newbie.
Not terribly relevant, but it’s always neat to hear about some clever shortcut someone finds.
It’s my guess that Greeks and Romans had these horribly bright people sprinkled here and there in their populations, too.
A person like this would be able to do the math regardless of how clumsy the system was, probably resolving the ackward parts mentally, in their own personal math systems.
Another gifted boy “invented” base 5 over the summer and put it to work in his third grade math class - failing math totally until he switched to base ten for a few days, and then again as he moved to base eleven.
We say “carry” and “borrow” because of the line “abacus” used in the 15th and 16th centuries - it was lined paper like graph paper, when a certain number of counters were placed in one line or square, a counter had to be “carried” to the next line or square.
So maybe if you did your math on a counting table or abacus the words or names or even symbols of the numbers wouldn’t be very important???
> the decimal point may seem a misnomer
> when used with other bases, but you
> can use that notation with any base.
Just for the sake of superfluous pedantry: you can indeed use that notation with any base but “decimal point” means base 10. Binary point, ternary point, … . The general term is “radix point”.
I wish I had some cites for this, but its from a History of Math class I had about 2 years ago…
In many aincent civilizations (Sumerian, Babylonian, Egyptian, and maybe Greek anyway), there was a difference between the math that accountants used and the math that mathemeticians used. The math that accountants used was actual quite sophisticated, but was looked down upon by the mathemeticians and astronomers.
In particular, the accountants used abacus like devices, and so they had a place value system, a concept of zero, and possibly negative numbers.
So for day to day math, computation was quite fast, and the numbering system was easy to use. Its only when you had to translate your number into something the non-accountants could understand (i.e. somthing without place values or zeros) that math got hard.
Civiliation Before Greece and Rome by H.W.F. Saggs (A b-day present just rec’d today!)
While there was a lot of borrowing…“Mathematics was an area where the peoples of Mesopotamia and Egypt differed considerably in their achievements. In Mesopotamia there was considerable achievement, extending to the use of algebracic processes. (Please allow me to claim credit for typos.) But Egyptians never went beyond elementary arithmetic…their fractions were clumsy…” And they multiplied the way we described earlier in this thread, did division and fractions the same way.
And you have a good memeory!
Sumerians had two different systems - a decimal like on the powers of ten and nother on the powers of 60 - which we still use for clocks and degrees in a circle.
Decimal was for business and sexagesimal for math and astronomical calculations…
And then the text goes into detail.
From skimming it seems no one caught up to the Babylonians quickly - they were familiar with the fact behind Pythagoras 1200 years before he was born. (Egypt used ropes to do this calculation after flooding destroyed boundries every year with the flood season.)
“But it may be said that the Babylonians concerned were employing a sophisticated mathematical astronomy relating to planetary and lunar motion which was not surpassed until the coming of Copernicus.”
So we are saying with the possible exception of the Babylonians, math was pretty unsophisticated, used tables, charts and abacus-like aides to do math.
Must have been pretty patient people.
Oh, I’m gonna keep using these #%@&* codes 'til I get 'em right.
My understanding is that the abacus was common in Rome and less so in Greece for everyday computations. Bronze versions with little “counters” that fit in grooves have been found. As someone above points out, once some one is comfortable with an abacus, they are in fact using place holders to express numbers. The Arabic system is a way to use written places as holders of value the same way an abacus does. I wonder if the Arabic system may be indebted to the Chinese?
I am still amazed at the level of mathmatical and astronomical competence of the Romans, and especially Babylonians and Sumerians.
Keep in mind that the innate brain capacity of your genius Greek was no more or less than a PhD today (probably on average more) - they just had less to work with. Also a complicated series of computations performed over several years, could, once completed be the basis of a lifetime’s work.
Newton’s genius stands out here, he had to create calculus (or steal it from Liebniz) to generate the large volume and complexity of calculatios he performed in his lifetime.