Bin Oct Dec Hex

There are a lot of systems in use besides base-10 or decimal:

For example, base-12 in packaging and shipping: dozen eggs in a carton, a gross of eggs in a case of 12 cartons.

You'll use at least four numeral systems as a developer:

Decimal (base-10):
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
No need to explain this further here.
Binary (base-2):
0, 1
Octal (base-8):
0, 1, 2, 3, 4, 5, 6, 7
Hexadecimal (base-16):
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f

Computer technology is built on binary arithmetic because our hardware represents information as two different states of matter (a low voltage and a high voltage).

Number systems based on powers of two fit nicely in this world.
Because 8 and 16 are powers of 2 (2³ and 2⁴) these are very common in computing.

Binary¶

Though we seldom use binary numbers directly, understanding and being able to count in binary is a necessary skill.

place: 8421   decimal value
       ----
          0   [0]
          1   [1]
         10   [2] (There are 10 kinds of people in the world: those who understand binary and those who don't.)
         11   [3]
        100   [4]
        101   [5]
        110   [6]
        111   [7]
       1000   [8]
       1001   [9]
       1010   [10]
       1011   [11]
       1100   [12]
       1101   [13]
       1110   [14]
       1111   [15]
      10000   [16]

Octal¶

The value of any 8-bit byte can be represented with three octal digits.

Practice Exercise¶

File permission modes are often represented using octal digits (0 - 7), one each for the file's owner, members of the file's group, and all others.

In each of the three numbers, each bit (binary digit) represents whether a permission is enabled or disabled.

The permissions are read, write, and execute.

owner  group  other
-----  -----  -----
r w x  r w x  r w x    permissions
4 2 1  4 2 1  4 2 1    place
1 1 0  1 0 0  1 0 0    binary
    6      4      4    octal

chmod 644 myFile.txt   # Read-write for me, read-only for everyone else.

Hexadecimal¶

The value of any 8-bit byte can be represented with two hexadecimal digits (0 - F).

ASCII characters are often encoded in hexadecimal.

The value of any two-byte (16-bit) short or char can be represented with four hexadecimal digits.

Java char values (other than ASCII characters) are given in hexadecimal.
The Unicode code charts list character values in hex.
Java unicode escapes also use hexadecimal.

A 32-bit int can be represented with eight hexadecimal digits.

Hexadecimal is used for network MAC addresses.
IP addresses and their netmasks are often represented in hexadecimal.
IPv6 addresses are always given in hex.
Hexadecimal color values are common in HTML and CSS.

Practice Exercise¶

For extremely large numbers like encryption keys, or for encoding binary data as text for transmission via text-based protocols like email, base-64 is commonly used. In base-64 the digits include: * The uppercase letters: A - Z (26) * The lowercase letters: a - z (another 26, 52 digits so far) * The digits: 0 - 9 (10 of these, 62 digits, need two more) * + and /

To use this, the data is broken into groups of six bits - a six-bit number can be represented by a single base-64 digit (the largest 6-bit number is 63) - and then encoded with these digits.

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