Number representations and codes

22.5 °C, held in bits

• A thermostat shows 22.5 °C, but the microcontroller behind it holds only bits: two voltage levels, 0 and 1.
• How does 22.5 become a pattern a machine can add, compare and store? And how do negative numbers fit? This lesson is that translation.

Binary: read the weights

• In binary 二进制 each position has a weight: 128, 64, 32, 16, 8, 4, 2, 1 for 8 bits. A pattern's value is the sum of the weights under its 1s.
• To convert an integer, divide by 2 and collect the remainders, or take the largest weights that fit: $156 = 128 + 16 + 8 + 4 = 10011100_{2}$.

Hexadecimal and fractions

• Hexadecimal 十六进制 is binary's shorthand: each hex digit is exactly 4 bits. Group from the right: $1001\ 1100 = \text{9C}_{16}$.
• Fractions convert by doubling and keeping the integer part: $0.625 \to 1.25\ (1) \to 0.5\ (0) \to 1.0\ (1)$, so $0.625 = 0.101_{2}$, which is $0.5 + 0.125$. The thermostat's $22.5$ is $10110.1_{2}$.

Two's complement: invert and add one

• Two's complement 二进制补码 negates a number by inverting every bit and adding 1. In 8 bits, $5 = 00000101$, so $-5 = 11111010 + 1 = 11111011$.
• Why it works: modulo $2^{N}$, the negative of $b$ is $2^{N} - b = (2^{N} - 1 - b) + 1$, and $2^{N} - 1 - b$ is $b$ with every bit flipped. So the ordinary adder also subtracts.

The range, and reading a pattern

• N bits hold the range 范围 $-2^{N-1}$ to $2^{N-1} - 1$: −128 to +127 in 8 bits. The range is lopsided: −128 exists, +128 does not.
• A leading 1 means negative. To decode, read the bits as unsigned and subtract $2^{N}$: $10011100$ is 156 unsigned, and $156 - 256 = -100$.

Overflow: the sign rule

• Overflow 溢出 happens when the true result is outside the range. Detect it by sign: two positives giving a negative, or two negatives giving a positive.
• A carry out of the top bit is not, by itself, overflow. In circuit terms, overflow is when the carry into the top bit differs from the carry out.

Three common slips

Recap