Base 3 Multiplication Table

[Ahmend Studioz]

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Five years ago, Tsinghua University in Beijing received a donation of nearly 2,500 bamboo strips. Muddy, smelly and teeming with mould, the strips probably originated from the illegal excavation of a tomb, and the donor had purchased them at a Hong Kong market. Researchers at Tsinghua carbon-dated the materials to around 305 bc, during the Warring States period before the unification of China.

Each strip was about 7 to 12 millimetres wide and up to half a metre long, and had a vertical line of ancient Chinese calligraphy painted on it in black ink. Historians realized that the bamboo pieces constituted 65 ancient texts and recognized them to be among the most important artefacts from the period.

Those 21 strips turned out to be a multiplication table, Feng and his colleagues announced in Beijing today during the presentation of the fourth volume of annotated transcriptions of the Tsinghua collection.

When the strips are arranged properly, says Feng, a matrix structure emerges. The top row and the rightmost column contain, arranged from right to left and from top to bottom respectively, the same 19 numbers: 0.5; the integers from 1 to 9; and multiples of 10 from 10 to 90.

The oldest known multiplication tables were used by the Babylonians about 4000 years ago. However, they used a base of 60. The oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC. This multiplication table shows is another way to represent multiplication.

In mathematics, a base n multiplication table is a mathematical table used to define a multiplication operation for numbers with base n system. Here in this page you can find a base N multiplication tables for numbers with base 1 - 24 as well as individual table for a single base (click on the table header and you will be taken to the individual table. Scroll down to see them all. You can print ( or save as pdf) individual tables by clicking on the printer icon on the top of each table.

for school exercise i need to build an multiplication table 10x10 using element-by-element function, and make it as short as possible.this is the code i wrote (working but too long), please suggest some twicks to this code.thanks in advance (:

In these respects, duodecimal is considered superior to decimal, which has only 2 and 5 as factors, and other proposed bases like octal or hexadecimal. Sexagesimal (base sixty) does even better in this respect (the reciprocals of all 5-smooth numbers terminate), but at the cost of unwieldy multiplication tables and a much larger number of symbols to memorize.

Georges Ifrah speculatively traced the origin of the duodecimal system to a system of finger counting based on the knuckle bones of the four larger fingers. Using the thumb as a pointer, it is possible to count to 12 by touching each finger bone, starting with the farthest bone on the fifth finger, and counting on. In this system, one hand counts repeatedly to 12, while the other displays the number of iterations, until five dozens, i.e. the 60, are full. This system is still in use in many regions of Asia.[2][3]

Languages using duodecimal number systems are uncommon. Languages in the Nigerian Middle Belt such as Janji, Gbiri-Niragu (Gure-Kahugu), Piti, and the Nimbia dialect of Gwandara;[4] and the Chepang language of Nepal[5] are known to use duodecimal numerals.

There are also varying proposals of how to distinguish a duodecimal number from a decimal one. The most common method used in mainstream mathematics sources comparing various number bases uses a subscript "10" or "12", e.g. "5412 = 6410". To avoid ambiguity about the meaning of the subscript 10, the subscripts might be spelled out, "54twelve = 64ten". In 2015 the Dozenal Society of America adopted the more compact single-letter abbreviation "z" for "dozenal" and "d" for "decimal", "54z = 64d".[24]

Other proposed methods include italicizing duodecimal numbers "54 = 64", adding a "Humphrey point" (a semicolon instead of a decimal point) to duodecimal numbers "54;6 = 64.5", prefixing duodecimal numbers by an asterisk "*54 = 64", or some combination of these. The Dozenal Society of Great Britain uses an asterisk prefix for duodecimal whole numbers, and a Humphrey point for other duodecimal numbers.[24]

This system uses "-qua" ending for the positive powers of 12 and "-cia" ending for the negative powers of 12, and an extension of the IUPAC systematic element names (with syllables dec and lev for the two extra digits needed for duodecimal) to express which power is meant.[26][27]

The case for the duodecimal system was put forth at length in Frank Emerson Andrews' 1935 book New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics. Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized either by the adoption of ten-based weights and measure or by the adoption of the duodecimal number system.[11]

Both the Dozenal Society of America and the Dozenal Society of Great Britain promote widespread adoption of the duodecimal system. They use the word "dozenal" instead of "duodecimal" to avoid the more overtly decimal terminology. However, the etymology of "dozenal" itself is also an expression based on decimal terminology since "dozen" is a direct derivation of the French word douzaine, which is a derivative of the French word for twelve, douze, descended from Latin duodecim.

The duodecimal tables are easy to master, easier than the decimal ones; and in elementary teaching they would be so much more interesting, since young children would find more fascinating things to do with twelve rods or blocks than with ten. Anyone having these tables at command will do these calculations more than one-and-a-half times as fast in the duodecimal scale as in the decimal. This is my experience; I am certain that even more so it would be the experience of others.

But the final quantitative advantage, in my own experience, is this: in varied and extensive calculations of an ordinary and not unduly complicated kind, carried out over many years, I come to the conclusion that the efficiency of the decimal system might be rated at about 65 or less, if we assign 100 to the duodecimal.

In "Little Twelvetoes", American television series Schoolhouse Rock! portrayed an alien being with twelve fingers and twelve toes using duodecimal arithmetic, using "dek" and "el" as names for ten and eleven, and Andrews' script-X and script-E for the digit symbols.[31][32]

The Dozenal Society of America argues that if a base is too small, significantly longer expansions are needed for numbers; if a base is too large, one must memorise a large multiplication table to perform arithmetic. Thus, it presumes that "a number base will need to be between about 7 or 8 through about 16, possibly including 18 and 20".[36]

To convert numbers between bases, one can use the general conversion algorithm (see the relevant section under positional notation). Alternatively, one can use digit-conversion tables. The ones provided below can be used to convert any duodecimal number between 0;1 and BB,BBB;B to decimal, or any decimal number between 0.1 and 99,999.9 to duodecimal. To use them, the given number must first be decomposed into a sum of numbers with only one significant digit each. For example:

This decomposition works the same no matter what base the number is expressed in. Just isolate each non-zero digit, padding them with as many zeros as necessary to preserve their respective place values. If the digits in the given number include zeroes (for example, 7,080.9), these are left out in the digit decomposition (7,080.9 = 7,000 + 80 + 0.9). Then, the digit conversion tables can be used to obtain the equivalent value in the target base for each digit. If the given number is in duodecimal and the target base is decimal, we get:

As explained in recurring decimals, whenever an irreducible fraction is written in radix point notation in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base.

The number of denominators that give terminating fractions within a given number of digits, n, in a base b is the number of factors (divisors) of b n \displaystyle b^n , the nth power of the base b (although this includes the divisor 1, which does not produce fractions when used as the denominator). The number of factors of b n \displaystyle b^n is given using its prime factorization.

For decimal, 10 n = 2 n 5 n \displaystyle 10^n=2^n\times 5^n . The number of divisors is found by adding one to each exponent of each prime and multiplying the resulting quantities together, so the number of factors of 10 n \displaystyle 10^n is ( n + 1 ) ( n + 1 ) = ( n + 1 ) 2 \displaystyle (n+1)(n+1)=(n+1)^2 .

For duodecimal, 10 n = 2 2 n 3 n \displaystyle 10^n=2^2n\times 3^n . This has ( 2 n + 1 ) ( n + 1 ) \displaystyle (2n+1)(n+1) divisors. The sample denominator of 8 is a factor of a gross 12 2 = 144 \textstyle 12^2=144 in decimal), so eighths cannot need more than two duodecimal fractional places to terminate. 5 8 = 0.76 12 . \textstyle \frac 58=0.76_12.

Because both ten and twelve have two unique prime factors, the number of divisors of b n \displaystyle b^n for b = 10 or 12 grows quadratically with the exponent n (in other words, of the order of n 2 \displaystyle n^2 ).

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