Binary arithmetic

Doing maths with 1s and 0s

• Computers add, shift and store negative numbers — all in binary.
• An 8-bit register 寄存器 has limits, so results can overflow 溢出.
• Two's complement 补码 lets us store negatives.

Binary addition and overflow

• Add column by column from the right, carrying a 1: $1+1 = 10$ (write 0, carry 1).
• Example: 01110110 (118) + 00110000 (48) = 10100110 (166).
• An 8-bit register holds 0–255. A result above 255 needs a 9th bit that won't fit — this is overflow, and the stored answer is wrong.

An 8-bit place-value chart. The 1s sit under the values that add up to 150 (128 + 16 + 4 + 2).

Logical binary shift

• A logical shift 逻辑移位 moves all bits left or right; bits off the end are lost and zeros fill the gap.
• A left shift multiplies by 2 per place; a right shift divides by 2 per place.
• Example: 00110101 (53) left-shifted by 2 → 11010100 (212) = $53 \times 4$.

Each hexadecimal digit becomes its own 4-bit group (nibble): F = 1111, 0 = 0000, 8 = 1000.

Two's complement (negatives)

• In two's complement, the most significant bit 最高有效位 has a negative value: -128 64 32 16 8 4 2 1.
• MSB 0 → positive; MSB 1 → negative.
• To make a number negative: write the positive, flip every bit, then add 1. e.g. $+40$ = 00101000 → flip 11010111 → +1 → 11011000 = $-40$.
• Range of 8-bit two's complement: −128 to +127.

The most significant bit is worth −128, so 11011000 = −128 + 64 + 16 + 8 = −40

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