For any positive whole number b, one can create a
place-value notational system of that base as follows:
Write a given number nas a sum of powers of b:
n= akbk+ ak–1bk–1+ … + a2b2+ a1b+ a0
with each number aisatisfying 0≤ai< b. Then
the base brepresentation of nis the k-digit
quantity akak–1…a2a1a0. Such a representation
uses only the symbols 0, 1, 2,…, b– 1.
For example, to write the number 18 in base four—
using the symbols 0, 1, 2, and 3—observe that 18 = 1 ×
42+ 0 ×4 + 2 ×1, yielding the base-4 representation:
102. In the reverse direction, if 5,142 is the base-6 rep-
resentation of a number n, then nis the number 5 ×63
+ 1 ×62+ 4 ×6 + 2 ×1 = 1,142.
One may also make use of negative powers of the
base quantity b. For example, using a decimal point to
separate positive and negative powers of ten, the num-
ber 312.407, for instance, represents the fractional
quantity: 3 ×102+ 1 ×10 + 2 ×1 + 4 ×+ 0 ×
+7 ×. In base 4, the number 33.22 is the quantity
3 ×4 + 3 ×1 + 2 ×+ 2 ×= 15 + + , which
is 15.625 in base 10.
The following table gives the names of the place-
value number systems that use different base values b.
The Babylonians of ancient times used a sexagesimal
system, and the Mayas of the first millennium used a
system close to being purely vigesimal.
The representation of numbers can be well-
represented with the aid of a simple
AUTOMATON
called a number-base machine. Beginning with a row of
boxes extending infinitely to the left, one places in the
rightmost box a finite number of pennies. The automa-
ton then redistributes the pennies according to a preset
rule. A “1 ←2” machine, for example, replaces a pair
of pennies in one box with a single penny in the box
one place to the left. Thus, for instance, six pennies
placed into the 1 ←2 machine “fire” four times to
yield a final distribution that can be read as “1 1 0.”
This result is the number six written as a
BINARY NUM
-
BER
and this machine converts all numbers to their
base-two representations. (The diagram in the entry for
automaton illustrates this.) A 1 ←3 machine yields
base-three representations, and a 1 ←10 machine
yields the ordinary base-ten representations.
Long Division
The process of long division in
ARITHMETIC
can be
explained with the aid of a number-base machine. As
an example, let us use the 1 ←10 machine to divide
the number 276 by 12. Noting that 276 pennies placed
in the 1 ←10 machine yields a diagram with two pen-
nies in the 100s position, seven pennies in the 10s posi-
tion, and six pennies in the units position, and that 12
pennies appears as one penny in a box with two pen-
nies in the box to its right, to divide 276 by 12, one
must simply look for “groups of 12” within the dia-
gram of 276 pennies and keep count of the number of
groups one finds.
2
16
2
4
1
42
1
4
1
103
1
102
1
10
base of a number system 37
base bnumber system
2 Binary
3 Ternary
4 Quaternary
5 Quinary
6 Senary
7 Septenary
8 Octal
9 Nonary
10 Decimal
11 Undenary
12 Duodecimal
16 Hexadecimal
20 Vigesimal
60 Sexagesimal
Long division base ten