Primer: Positional (Place-Value) and Integer Base Notation

Positional notation is so intuitive that we grasp a fairly complex arithmetic equation with just a glance. We immediately understand 195.02 without having to add 5 to 90 to 100, and finally include the fraction ²/₁₀₀. Positional sexagesimal is just as intuitive, with a little practice.

As you have probably guessed, “positional notation” means the position of each digit relative to the radix marker determines the magnitude of its value. The first place to the left of the radix marker represents the digit times the numerical base taken to the zeroth power (N⁰ always equals one), the second the digit times the numerical base to the first power, the third the digit times the numerical base squared, and so on. Similarly, the first position to the right of the decimal represents the digit times the numerical base taken to the negative one power, the second position the digit times the numerical base taken to the negative two power, and so on. The results are all added together, giving us the total value of the number. This is true of base 10, base 60, or any other numerical base you care to use.

Stated most generally, any number D can be represented in any base N (N>0) by a series of digits d as follows:

D = d_jN^j + d_j-1N^j-1 + … + d₁N¹+d₀N⁰ . d_-1N^-1 + d_-2N^-2 + … + d_-kN^-k

Which, in standard positional notation, is written:

D = (d_jd_j-1 … d₁d₀.d_-1d_-2 …)N,

where all digits d satisfy the constraint 0≤d_i<N.

Take, for example, the decimal (base 10) number “103.95″:

1 x 10² + 0 x 10¹ + 3 x 10⁰ + 9 x 10^-1 + 5 x 10^-2 = 100 + 0 + 3 + ⁹/₁₀ + ⁵/₁₀₀

Which of course, gives us one hundred three and ninety-five hundredths, written in the familiar form of “103.95″ in positional base 10.

Similarly, in sexagesimal, each position to the left of the radix marker represents increasing powers of 60, while those to the right of the radix marker represent increasing negative powers of 60. Consider:

m01.Fd

Each symbol represents a number between zero and fifty nine. Just like in decimal notation, the position of the symbol defines the magnitude of its value. In the number above, m represents “48″, 0 is “0″, 1 is “1″, “.” is the sexagesimal (radix) marker, F is “15″, and d is “39″.

Sexagesimal positional notation yields:

m x 60² + 0 x 60¹ + 1 x 60⁰ + F x 60^-1 + d x 60^-2,

or, using numerals we are more familiar with:

48 x 60² + 0 x 60¹ + 1 x 60⁰ + 15 x 60^-1 + 39 x 60^-2, which gives us:

172,800 + 0 + 1 + ¹⁵/₆₀ + ³⁹/₃₆₀₀ = 3648 ⁹³⁹/₃₆₀₀, or the decimal value:

172,801.2608333… (3 recurring).

Notice how a three digit numeral in base 60 can represent a six digit base 10 number? The difference between decimal and sexagesimal notation grows dramatically with each digit:

100,000 represents 777,600,000, while
xxx,xxx equals 46,655,999,999!

Which would you find easier to remember, a six digit number or an eleven digit number?

© 2007 Jean-Michel Smith. The contents of this page may be used or redistributed under the terms of the Creative Commons Attribution License, Version 2.5 or later.

Autonomy positional notation sexagesimal