Encoding Schemes and Number System

How text is coded (ASCII, ISCII, Unicode) and how numbers are written in binary, octal, decimal and hexadecimal, with conversions including fractional parts.

Key ideas

An encoding scheme gives every character a unique code: ASCII uses 7 bits (128 characters), ISCII adds the Indian scripts, and Unicode covers every script.
In UTF-8 a character takes 1 to 4 bytes (an English letter 1, a Devanagari letter 3); in UTF-32 every character takes 4 bytes.
Binary has base 2 (digits 0, 1), octal base 8 (0 to 7), decimal base 10, and hexadecimal base 16 (0 to 9, then A to F).
Decimal to another base: divide by the base again and again, then read the remainders from the bottom up.
Any base to decimal: multiply each digit by its place value (a power of the base) and add.
One octal digit is exactly 3 bits and one hexadecimal digit 4 bits, so group the bits outwards from the point.
A decimal fraction becomes binary by multiplying by 2 again and again, taking the whole parts from the top down.

Remember

ASCII codes	'A' = 65, 'a' = 97, '0' = 48, space = 32
Hexadecimal digits	A = 10, B = 11, C = 12, D = 13, E = 14, F = 15
To decimal	(157)₈ = 1 × 64 + 5 × 8 + 7 = 111; (2F)₁₆ = 2 × 16 + 15 = 47
Decimal to binary	13 ÷ 2 gives remainders 1, 0, 1, 1; read upwards: (1101)₂
Binary to octal and hex	10111010 → 010 111 010 = (272)₈; 1011 1010 = (BA)₁₆
After the point	places 1/2, 1/4, 1/8: (101.11)₂ = 5 + 1/2 + 1/4 = 5.75
Fraction to binary	0.625 × 2 = 1.25 → 1; 0.25 × 2 = 0.5 → 0; 0.5 × 2 = 1.0 → 1: 0.101
Message size	20 English letters + 10 Devanagari: UTF-8 20 + 30 = 50 bytes; UTF-32 120 bytes

Watch out

Reading the remainders from the top: the last remainder is the leftmost digit.
Grouping bits from the left end: group from the point, adding 0s at the outer ends.
Padding a fraction on the wrong side: .101 becomes .1010 for hexadecimal, which is .A.
Taking 'a' as 65: capital letters start at 65, small letters at 97.

Worked example

Convert (10111100.101)₂ into (a) decimal (b) octal (c) hexadecimal.

Working: 188 + 5/8; octal: groups of three on each side of the point; hexadecimal: groups of four, padding the fraction with a 0 (1010)

Answer: (a) 188.625; (b) 274.5; (c) BC.A