Data Representation: notes

Why binary

Computers store everything as bits (binary digits): 0 or 1, off or on. Numbers, text, pictures and sound are all turned into patterns of bits.

Place values

An 8-bit number has place values 128, 64, 32, 16, 8, 4, 2, 1.

• Binary to denary: add the place values that have a 1. Example: 00101101 = 32 + 8 + 4 + 1 = 45.
• Denary to binary: subtract the biggest place value that fits, write 1, carry on. Example: 13 = 8 + 4 + 1 = 00001101.
• Hexadecimal (base 16) uses 0 to 9 then A to F (A = 10, F = 15). One hex digit stands for 4 bits. Example: 1111 0010 = F2.
• Octal is base 8, using digits 0 to 7.

Binary addition

Rules: 0 + 0 = 0, 0 + 1 = 1, 1 + 1 = 0 carry 1, 1 + 1 + 1 = 1 carry 1. If the answer needs a ninth bit in an 8-bit register, that is an overflow.

Binary subtraction

Rules: 0 - 0 = 0, 1 - 0 = 1, 1 - 1 = 0, and 0 - 1 = 1 after borrowing 1 from the next column.

Units of storage

• 1 nibble = 4 bits; 1 byte = 8 bits
• 1 kilobyte (KB) = 1024 bytes; 1 megabyte (MB) = 1024 KB; 1 gigabyte (GB) = 1024 MB; 1 terabyte (TB) = 1024 GB

Logic gates

• AND: output 1 only when both inputs are 1.
• OR: output 1 when at least one input is 1.
• NOT: one input; the output is the opposite.
• NAND is AND then NOT; NOR is OR then NOT.
• A truth table lists every input combination. Two inputs give 4 rows; three inputs give 8 rows.