Addition and Multiplication Tables in Various Bases

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.

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 x 36 = 6243. Indeed,

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.

Now try to discover your own patterns:

archive="JCTKUtility.zip,CTKAlgebra.zip,SysTable.zip">

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Some patterns

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