Number systems and conversions

The terabyte that was not a terabyte

• In 2005 a Californian bought a hard drive labelled 80 GB. His computer reported 74.5 GB. He sued, and the manufacturers eventually paid out: they had counted a gigabyte as a thousand million bytes, and the operating system had counted it as $2^{30}$.
• Neither was lying. There are two families of prefix, one built on powers of ten and one on powers of two, and they drift further apart the bigger the number gets.
• The whole of data representation rests on that idea: a computer stores nothing but 0s and 1s, and everything else is an agreement about how to read them.
• This lesson is the three number systems, the conversions between them, and the two prefix families.

The three number systems

• Denary 十进制 (base 10) uses digits 0 to 9, with place values that are powers of ten. It is the everyday system.
• Binary 二进制 (base 2) uses 0 and 1, with place values that are powers of two. One digit is a bit 位; 4 bits are a nibble 半字节; 8 bits are a byte 字节.
• Hexadecimal 十六进制 (base 16) uses 0 to 9 then A to F for 10 to 15. One hex digit stands for exactly 4 bits, which is why it is the shorthand programmers use for binary.

Every system is place value; only the base changes

Denary to binary

• Method 1, place values: write the powers of two, $128, 64, 32, 16, 8, 4, 2, 1$, and subtract the largest that fits, repeating with what is left. Put a 1 under each one you used.
• Method 2, repeated division: divide by 2 over and over, writing down each remainder, then read the remainders bottom-up.
• $42 = 32 + 8 + 2$, so the bits under 32, 8 and 2 are 1 and the rest are 0: 0010 1010.

Subtract the biggest power of two that fits, and repeat

Worked example: denary 200 to binary and hex

Binary and hex, both directions

• Binary to hex: group the bits into nibbles from the right, padding the leftmost group with zeros, and convert each nibble. 0010 0010 1110 becomes 2 2 E, so 22E.
• Hex to binary: replace each hex digit with its own 4-bit pattern. No arithmetic needed.
• Hex to denary: multiply each digit by its place value, $16^2 = 256$, $16^1 = 16$, $16^0 = 1$. 22E $= 2 \times 256 + 2 \times 16 + 14 = 558$.

How many bits does a value need?

• Exam questions fix the register width 寄存器宽度 (8, 12 or 16 bits), and you must pad with leading zeros to that width. $558$ in 12 bits is 0010 0010 1110, never 10 0010 1110.
• An unsigned integer from 0 to $2^n - 1$ needs $n$ bits: 200 needs 8 bits (8 bits reach 255), 1000 needs 10 bits (10 bits reach 1023), 16 needs 5 bits (4 bits stop at 15).
• One hexadecimal digit needs 4 bits, one BCD digit needs 4 bits, and one ASCII character needs 7 bits, or 8 for extended ASCII.

Worked example: the minimum number of bits

Binary prefixes and decimal prefixes

• Decimal prefixes are powers of ten and are used for drive capacities and network speeds: kilo $= 10^3$, mega $= 10^6$, giga $= 10^9$, tera $= 10^{12}$.
• Binary prefixes are powers of two and are used for memory sizes: kibi 二进制千 (Ki) $= 2^{10} = 1024$, mebi (Mi) $= 2^{20}$, gibi (Gi) $= 2^{30}$, tebi (Ti) $= 2^{40}$.
• So a "1 TB" drive holds $10^{12}$ bytes, but an operating system reporting in TiB divides by $2^{40}$ and shows about 0.91. Nothing has been lost; two different units were used.

Where hexadecimal is actually used

• Memory addresses and machine code, because one hex digit is exactly one nibble and a byte is exactly two hex digits, so a dump is readable at a glance.
• Colour codes in web pages: #FF8800 is one byte each of red, green and blue. MAC addresses and error codes likewise.
• The reason is always the same: hex is a shorthand for binary that a person can read and write without mistakes, not a system the computer itself uses.

Marks that slip away

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