Napier Bones in Various Bases

[turkmmo]

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Napier Bones in Various Bases​

John Napier (1550-1617), a Scottish mathematician, is mostly known for his invention of logarithms - a device that revolutionized calculations by reducing difficult and tedious multiplication to addition of table entries. In 1617, three years after appearance of Mirifici logarithmorum canonis descriptio (A Description of the Wonderful Law of Logarithms), he published Rabdologiae which was recently reproduced as Rabdology by the Charles Babbage Institute in the Reprint Series for the History of Computing. The Elementary Latin Dictionary offered two entries:​

1.Rab- - raving, mad, rage, be mad, ...
2.Dolo - pike, pointed stuff, sword-stick, ...
(my son David who takes Latin in high school, claims that the last part logo has to do with the word "study".) The difficulty of putting the three together would explain why the Institute decided to anglicize the title instead of translating it. Trusting several accounts, it appears that in their day the sticks described in the book and later known as Napier's rods or Napier's bones, were indeed a rave among merchants who carried them along and used them to speed up calculations.
[Richard Persky from University of Texas diverges: Actually, it looks like Latinized Greek to me - rhabdos ("staff, stick") plus logos ("speech," "reason," "knowledge," "study" - it's a slippery little word with a broad range of meanings). So "Rabdologiae" would be the science of sticks, which seems a bit more reasonable than "the science of going berserk with a pointed object".]
Each bone is a multiplication table for a single digit. The digit appears at the top of its bone. Below one carves consecutive products of this digit by all non-zero digits in the system (decimal in Rabdology). Each product occupies a single cell. Digits in a 2-digit number are separated, the first is written above while the second below the bottom-left top-right diagonal. To multiply 187 by 3, put three bones corresponding to digits 1,8, and 7 alongside each other. The third row looks like

​

The product is evaluated diagonally, 5 (=3+2) 6 (=4+2) 1, 187 * 3 = 561. That simple. (Of course, from time to time you will have to carry 1.)
Among other wonderful things John Napier was also the discoverer of the binary system. So it's appropriate that in the applet below the base may change from 3 through 20.

References​

1.H.Eves, Great Moments in Mathematics Before 1650, MAA, 1983
2.M.Gardner, Knotted Doughnuts, W.H.Freeman and Co, 1986

Addition and Multiplication Tables in Various Bases​

The question of conversion between number systems with various bases has been addressed on one of the

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at this site. Later I added a page that describes the

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. There is also a page with an intriguing subject of appearance of primes in

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. ​

Below I wish to discuss the manner in which arithmetic operations (addition and multiplications) are carried out in various bases. The number systems we are looking into are known as positional: each uses a fixed number of digits whose meaning depends on its position in a number representation. The decimal system has been introduced in Europe less than a thousand years ago. Given its appeal and convenience, it's astonishing that it was not invented by the Ancient mathematicians. Even in an unfamiliar base, like 7 or 22, carrying arithmetic operations is incomparably easier than handling Roman numerals.
The applet combines addition and multiplication tables (check a radio button) for bases from 2 through 36. In every base N, there are N digits. In the decimal system, for example, we have 10 of them: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. In base 7, there are seven digits: 0, 1, 2, 3, 4, 5, 6. When N exceeds 10 we start adding English letters as needed. (No distinction is made between capital and lower case letters.)

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uses up all decimal digits and all the letters of the English alphabet.

Practice is all it takes to master various bases; for the rules are the same as in the decimal system. The sum or product of two digits may only produce one or two digit numbers. In the latter case, if necessary, the first digit is carried over to the next operation (on the left.) For example, in base 7, 36 + 144 = 213. Indeed, from right to left, 6 + 4 = 13. Then 3 + 4 + 1 = 11, and finally 1 + 1 = 2.​

Also, 144 × 36 = 6243. Indeed,


144
36
----
1263
465
----
6243

Still toying with the table we may learn a few interesting things. As everyone knows, 2 + 2 = 4. This is true in all base systems. That is, except bases 2, 3, and 4. In base 4, we have 2 + 2 = 10. In base 3, 2 + 2 = 11. However, recollect that (4)10 = (10)4 = (11)3, and everything falls into its right place again. Numbers equal in one base are equal in any other base. Conversion between bases does not violate arithmetic identities. In base 2, 2 + 2 = 4 appears as 10 + 10 = 100 - looking differently but having exactly the same meaning.
The same, of course, is true of 2 × 2 = 4 which is true in all bases starting with 5. In bases 4,3, and 2 it appears as

2 × 2 = 10
2 × 2 = 11
10 × 10 = 100, ​

respectively.
A few more things you may want to verify and, perhaps, justify more rigorously:
1.For every N>1, (N)10 = (10)N.
2.For addition, in the lower right corner, there always appears a 2-digit number whose first digit is 1 while the second digit is the penultimate digit of the system.
3.All tables are symmetric with respect to the diagonal from the upper left to the lower right corner.
4.Consecutive numbers on any north-east to south-west diagonal, in all addition tables, differ by 2. Why?
5.For multiplication, the number in the lower right corner is always obtained from its "addition" counterpart by swapping the two digits.
6.In the last row of multiplication tables, last digits grow by 1 if followed right to left. At the same time, the first digits decrease by 1.
7.The numbers in the last row before the last are also related to each other: if for addition we have a number 1a, then for multiplication its counterpart will be a2.
8.One can use addition tables to play the same game as with the

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.
9.For multiplication tables this is also true provided selected entries are multiplied instead of being added up.
10.As a particular case, in multiplication tables determinants of any 2x2 square are 0.
11.In addition tables, determinants of any 2x2 square equal -1. Determinants of order 3 or higher are all 0.
1.In multiplication tables, consider 2×2 diagonal squares, i.e., squares whose diagonal lies on the main diagonal of the table. The sum of 4 entries in any such square is a

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.
2.The same is true for the sum of entries in a 3×3 diagonal square.
3.What about a more general nxn diagonal square?
12.In multiplication tables, consider the diagonal 0, 3, 8, 15, ... Each entry on this diagonal just touches a diagonal entry which is always 1 more: 0-1, 3-4, 8-9, 15-16, and so on. Is it always the case?
13.This is a simple one: in addtion tables, south-west-to-north-east diagonals each consists of a single number.
14.In multiplication tables, the last row (starting with the third entry) consists of numbers that are pairwise mirror images of each other.
15.Multiplication tables conceal

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.
As we see, number systems with different bases have many features in common. Some features are, of course, unique. For example, in base 6 all prime numbers end with either 1 or 5 (why?). Base 3 has been used in some early computers. In base 3, numbers are represented with digits 0,1, and 2. It's also possible to make go with digits -1,0,1 so that any number will be representable as an algebraic sum (i.e., allowing for both plus and minus) of distinct powers of 3. For example, 27 = 33, 28 = 33 + 30, 29 = 33 + 31 - 30, 30 = 33 + 31, 31 = 33 + 31 + 30, 32 = 33 + 32 - 31 - 30, 33 = 33 + 32 - 31, 34 = 33 + 32 - 31 + 30, 35 = 33 + 32 - 30, 36 = 33 + 32, 37 = 33 + 32 + 30, 38 = 33 + 32 + 31 - 30, 39 = 33 + 32 + 31, 40 = 33 + 32 + 31 + 30, 41 = 34 - 33 - 32 - 31 - 30, etc.
An Aside
While researching for his laws, Kepler (1571-1630) used an ingenious number notation based on the Roman system, where subtraction as well as addition was involved. Kepler used symbols I, V, X, L, but instead of the numbers 1, 5, 10, 50 he selected 1, 3, 9, 27, and so on. (J.R.Newman, The World of Mathematics, v1.)
Number systems serve the purpose of representing numbers in different ways. As the examples below demonstrate, the binary and ternary systems have their niche in this field with very practical applications:

1.

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2.

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Similar weighing problems appear in several books:
1.W.W.Rouse Ball and H.S.M.Coxeter, Mathematical Recreations and Essays, Dover, 1987
2.D.Fomin,S.Genkin,I.Itenberg, Mathematical Circles (Russian Experience), AMS, 1996
3.Ya.I. Perelman, Fun with Maths and Physics, Mir Publishers, Moscow, 1988
4.D.Wells, The Penguin Book of Curious and Interesting Puzzles, Penguin Books, 1992

​

Claude Shannon, one of the fathers of Cybernetics and Information Theory, suggested (well might be in jest) in 1950 (Am Math Monthly) that other represenations may have computational advantages. For an odd base r, he considered r digits -(r-1)/2
ai
(r-1)/2. For r = 3 this leads to the Kepler's approach. For r = 7, there would be 7 digits -3,-2,-1,0,1,2,3 which Shannon denoted 3',2',1',0,1,2,3. The upside of such a representation is that the sign of a number is built in. Positive numbers start with one of 1,2,3, while negative numbers start with one of their counterparts 1',2',3'.
Even stranger was a possibility of including even bases into this framework. The digits would represent midpoints between integers! For r = 10, we would use 9'/2, 7'/2, 5'/2, 3'/2, 1'/2, 1/2, 3/2, 5/2, 7/2, 9/2 as digits. One annoying inconvenience was that integers would have multiple representations:

0 = .(1/2)(9'/2)(9'/2)... = (1/2).(9'/2)(9'/2)... = (1'/2).(9/2)(9/2)... ​

where I kept individual digits in parentheses for simplicity sake.
The upside of the approach was that addition and multiplication tables (being symmetric with respect to primed and not primed digits) would require less effort to memorize.
Remark
1.

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is another great tool to study multiplication (in various systems.) Addition is better handled with

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.
2.

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between various bases, although a consequence of essentially the same representation, is still different from conversion of integers and deserves a special page.
3.A small

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game helps internalize the binary system.
4.Binary system is indispensible for grasping the right strategy in the

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.
5.Binary system also proves useful for designing

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.
6.Binary and ternary systems underlie the constructions of the

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and the

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.

​

In the early 1600s, a Scottish mathematician called John Napier invented a tool called Napier's Bones, which were multiplication tables inscribed on strips of wood or bone.
Napier, who was the Laird of Merchiston, also invented logarithms, which greatly assisted in arithmetic calculations.

In 1621, an English mathematician and clergyman called William Oughtred used Napier's logarithms as the basis for the slide rule (Oughtred invented both the standard rectilinear slide rule and the less commonly used circular slide rule). However, although the slide rule was an exceptionally effective tool that remained in common use for over three hundred years, like the abacus it also does not qualify as a mechanical calculator.