Number Systems

1234  Decimal Notation

1011 (B)  1 + 1 * 2 + 0 * 4 + 1 * 8    = 11 (D)

General Rule:
  last digit * 2^0 + second last digit * 2^1 + third last digit * 2^2 ....

1 2 4 8 16 32 64 128 256


10111 (B) = 23 (D)

25 (D) = ? (B) = 00011001 (B)

25 / 2  = 12 R 1
12 / 2  =   6 R 0
  6 / 2  =   3 R 0
  3 / 2  =   1 R 1
  1 / 2  =   0 R 1


52 (D) = ? (B) = 00110100 (B)

52 / 2 = 26 R 0
26 / 2 = 13 R 0
13 / 2 =   6 R 1
  6 / 2 =   3 R 0
  3 / 2 =   1 R 1
  1 / 2 =   0 R 1

8 bit = 1 byte

char datatype: 256 different character, which are internally 0..255

short int datatype, 2 byte / 16 bits, -32.000..+32.000

int datatype, 4 bytes, 32 bits, -2.1 bio... + 2.1 bio



Octal - base 8

123 (O) = 83 (D)

3 * 8^0 = 3 * 1   = 3
2 * 8^1 = 2 * 8   = 16
1 * 8^2 = 1 * 64 = 64



83 (D) = ? (O)

83 / 8 = 10 R 3
10 / 8 =   1 R 2
  1 / 8 =   0 R 1

83 (D) = 1010011 (B)
83 / 2 = 41 R 1
41 / 2 = 20 R 1
20 / 2 = 10 R 0
10 / 2 =  5  R 0
  5 / 2 =  2  R 1
  2 / 2 =  1  R 0
  1 / 2 =  0  R 1

1010011 (B) = ? (O) = 001 010 011 (B)

001  1
010  2
011  3
100  4
101  5
110  6
111  7


15263 (O) = ? B = 001 101 010 110 011 = 1101010110011

In C++:
Any integer that start with a zero (0) is interpreted as an octal number!


Hexadecimal - Base 16
10 A
11 B
12 C
13 D
14 E
15 F


82 (H) = 130 (D)
2 * 1      = 2
8 * 16    = 128

82 (D) = 52 H
82 / 16 = 5 R 2
5   / 16 = 0 R 5

130 (D) = 82 (H)
130 / 16 = 8 R 2
8     / 16 = 0 R 8

8D (H) = 141 (D)
13 * 1 = 13
8 * 16 = 128

175 (D) = AF (H)
175 / 16 = 10 R 15 F
10 / 16  =    0 R 10 A

123 (H) = ? B = 0001 0010 0011 (B)

A (H) = ? B = 1010 (B)

ABCDEF (H) = ? B 1010 1011 1100 1101 1110 1111 (B)

the 0x prefix indicates that the number is given in hexadecimal notation
example: 0x123 0xaf 0xF6 0xaF

Stream manipulator hex, dec and oct can be used to print in different notations