Page 25 - ARM 64 Bit Assembly Language

8Chapter 1

Table 1.2: The first 21 integers (starting with 0) in various bases.
Base
16 10 9 8 7 6 5 4 3 2
0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1 1
2 2 2 2 2 2 2 2 2 10
3 3 3 3 3 3 3 3 10 11
4 4 4 4 4 4 4 10 11 100
5 5 5 5 5 5 10 11 12 101
6 6 6 6 6 10 11 12 20 110
7 7 7 7 10 11 12 13 21 111
8 8 8 10 11 12 13 20 22 1000
9 9 10 11 12 13 14 21 100 1001
A 10 11 12 13 14 20 22 101 1010
B 11 12 13 14 15 21 23 102 1011
C 12 13 14 15 20 22 30 110 1100
D 13 14 15 16 21 23 31 111 1101
E 14 15 16 20 22 24 32 112 1110
F 15 16 17 21 23 30 33 120 1111
10 16 17 20 22 24 31 100 121 10000
11 17 18 21 23 25 32 101 122 10001
12 18 20 22 24 30 33 102 200 10010
13 19 21 23 25 31 34 103 201 10011
14 20 22 24 26 32 40 110 202 10100

matical operations using strings of digits between 0 and 9 seems natural. However, there are
other ways to count and perform arithmetic, such as Roman numerals, unary systems, and
Chinese numerals. With a little practice, it is possible to become as proficient at performing
mathematics with other number systems as with the Hindu-Arabic system.
The Hindu-Arabic system is a base ten or radix ten system, because it uses the ten digits 0, 1,
2, 3, 4, 5, 6, 7, 8, and 9. For our purposes, the words radix and base are equivalent, and refer
to the number of individual digits available in the numbering system. The Hindu-Arabic sys-
tem is also a positional system, or a place-value notation, because the value of each digit in a
number depends on its position in the number. The radix ten Hindu-Arabic system is only one
of an infinite family of closely related positional systems. The members of this family differ
only in the radix used (and therefore, the number of characters used). For bases greater than
base ten, characters are borrowed from the alphabet and used to represent digits. For example,
the first column in Table 1.2 shows the character “A” being used as a single digit representa-
tion for the number ten.