## Base-n

Other examples of powers of 2 base numbers:

### Base-4 (Quarternary) Numbers:

• In accordance with the above, the base-4 number system consists of 4 digits: 0, 1, 2 and 3.
• Since a 2-bit number can only represent one of 0, 1, 2, or 3, any 2-bit number can be represented by a single base-4 number.
• To convert a binary number to base-4, group the binary number from right to left in groups of 2 bits and represent their values in base-4
• So, a binary number 1011 converts to base-4 as the following: 102+ 112 or 234
• Only 2 digits are now required to represent any 4-bit number.

### Base-8 (Octal) Numbers:

• The base-8 number system consists of 8 digits: 0, 1, 2, 3, 4, 5, 6, and 7.
• Since a 3-bit number can only represent one of 0, 1, 2, 3, 4, 5, 6, or 7, any 3-bit number can be represented by a single base-8 number.
• To convert a binary number to octal, simply group the binary numberfrom right to left in groups of 3 bits and represent their values in base-8

• So for example, a binary number 001011 converts to base-8 as the following: 0012+ 0112 or 138
• Note that the two zeros in the 32s and 16s column are only place holders so even if the original binary number was 1011, the resulting base-8 value would be the same.
• Only 2 digits are required to represent any 6-bit number.

• Base-16 notation which has exactly 16 digits - 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.
• Note: since values greater than 9 would require two digits to be represented in e.g. decimal 10, under this system such values are represented by a single alpha character, e.g., B16/Hex= 1110
• Typically in computing, hexadecimal (Hex, for short) numbers are employed as a short hand for representing any 4 bit pattern using only a single character because there are exactly 16 possible 4-bit combinations.

Thus, any 4-bit number can be represented by a single base hex number.

• To convert a binary number to hexadecimal (hex), simply group the binary number from right to left (from the least significant digit to the most significant) in groups of 4 bits and represent their values in base-16
• So, a binary number 1011 converts to base-16 as the following:

(1x8)+(0x4)+(1x2)+(1x1) or 8+2+1 or B16 or Hex or 1110

Therefore, only 2 hex digits are required to represent any 8-digit binary number.

**How many base-32 digits would be required to represent any 7-bit number?