Addition and Multiplication Tables in Various Bases

Many children grow superstitious, and think that you cannot carry except in tens; or that it is wrong to carry in anything but tens. The use of algebra is to free them from bondage to all this superstitious nonsense, and help them to see that the numbers would come just as right if we carried in eights or twelves or twenties. It is a little difficult to do this at first, because we are not accustomed to it; but algebra helps to get over our stiffness and set habits and to do numeration on any basis that suits the matter we are dealing with.

Mary Everest Boole
Philosophy And Fun Of Algebra,
London: C. W. Daniel, LTD, 1909

The question of conversion between number systems with various bases has been addressed on one of the very first pages at this site. Later I added a page that describes the conversion procedure algorithmically. There is also a page with an intriguing subject of appearance of primes in base 36.

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.) Base 36 uses up all decimal digits and all the letters of the English alphabet.


This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

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-west to south-east 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 Calendar tables.
  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 the four entries in any such square is a square number.
    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 cubes and tetrahedral numbers.

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. Weighing 12 coins
  2. Selecting counterbalances

Similar weighing problems appear in several books:

  1. W.W. Rouse Ball and H.S.M. Coxeter, Mathematical Recreations and Essays, Dover, 1987
  2. A. Beck, M.N. Bleicher, D. W. Crowe, Excursions into Mathematics, A K Peters, 2000
  3. D. Fomin,S. Genkin,I. Itenberg, Mathematical Circles (Russian Experience), AMS, 1996
  4. Ya.I. Perelman, Fun with Maths and Physics, Mir Publishers, Moscow, 1988
  5. 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 le; 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. Napier bones is another great tool to study multiplication (in various systems.) Addition is better handled with Abacus, or its Chinese (Suan pan) or Japanese (Soroban) variants.
  2. Conversion of fractions 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 number guessing game helps internalize the binary system.
  4. Binary system is indispensible for grasping the right strategy in the game of Nim.
  5. Binary system also proves useful for designing infinite Latin squares.
  6. Binary and ternary systems underlie the constructions of the Cantor set and the Sierpinski Gasket.

What Can Be Multiplied?


Related material
Read more...

  • Expansion of Integers in an Integer Base
  • Base (Binary, Decimal, etc.) Converter
  • Base Converter
  • Implementation of Base Conversion Algorithms
  • Conversion of Fractions in Various Bases
  • Scoring: the simplest of the impartial games
  • History of the Binary System
  • Calculation of the Digits of pi by the Spigot Algorithm of Rabinowitz and Wagon
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