A resource for students age 11-14
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Mathematics Concepts

## The Positional Principle

The decimal notation, or notation in any other base, is one of the milestones of human thought. All these notations are based on the "positional principle."

"The principle is so simple," wrote the 18th-19th-century mathematician Pierre-Simon, marquis de Laplace, "that this very simplicity is the reason for our not being sufficiently aware how much admiration it deserves." The numeral 0 plays an important role in this principle. It employs the convention that any numeral other than 0 when raised to the power 0 has the value 1.

Representation in larger bases

The largest digit used in the decimal system is 9. When the base b is larger than 10, we would need additional symbols to denote the digits higher than 9 and lower than b. For instance, when b = 11, the symbol A is used for the digit after 9.
Thus (113)10 = A × 111 + 3 = (A3)11.
For b = 16 (Hexadecimal system), the following is the notation for digits:
0123456789ABCDEF
where A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15, which in binary notation is 1010, 1011, 1100, 1101, 1110, and 1111, respectively.

Example:
6B7F16 = 6 × 163 + 11 × 162 + 7 × 16 + 15 = 24576+2816+112+15 = 2751910.

The notation using letters A, B,…, Z will work for bases up to 36 because 10 + 26 (English alphabet letters) = 36. When b > 36, we will need to use other symbols. Fortunately we have an infinite number of such symbols available. For instance, for 37 we can use 100101.

Corresponding to base (radix) values of 2,3,4,5,6,7,8,9,10,11,12,16,20, and 60, the numeral notation systems are called binary, ternary, quaternary, quinary, senary, septenary, octenary (or octal), nonary, denary (decimal), undenary, duodenary, hexadecimal, vigesimal, and sexagesimal, respectively. It is clear that for computers, the binary sytem is best. It is also clear that the decimal notation is more convenient for humans in paper and oral communication because we are used to it and it takes less number of digits than in binary. For large numbers like a googol, which is 10100, their number of digits in binary will be approximately 3.3 times their number of digits in decimal representation - see Number of digits in base b representation. We save a few trees every day by having all humans use the decimal notation rather than the binary when communicating on paper. However, this question remains: Is the decimal notation the best system for humans? It appears that the duodecimal notation has some advantages over the decimal system. These advantanges have to do with the nature of the divisors of 12 as compared to those of 10. However, it will be clearly very difficult to change from the present system.

It is interesting to think about the advantages of using a particular base. For instance, consider the Abacus. How many beads are needed for an Abacus that can handle binary notation for numbers as large as can be handled on a standard Abacus for decimal representation? It turns out that you will need less beads. In fact an Abacus for base 3 will need the least number of beads. An exploration of this would make a useful school project for gifted students.

(G.R.T.)

>> Conversion between bases

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