Двоичные числа

Everything is bits

• Deep down, a computer knows only two states: on and off, written 1 and 0.
• A bit 位 is a single 0 or 1 — the smallest piece of data.
• Two states are easy to build from a wire (voltage or none) and hard to confuse.
• Counting with two digits is binary 二进制 (base 2); everyday decimal 十进制 (base 10) uses ten.

Place values and conversion

• In binary, each position is a power of two 2 的幂, doubling leftward: $\dots, 8, 4, 2, 1$.
• To read binary → decimal, add the position values where there is a 1.
• To go decimal → binary, keep subtracting the largest power of two that fits.

An 8-bit place-value chart: the 1s sit under the values that add to the number

Fixed bits and overflow

• Computers store numbers in a fixed number of bits 固定位数, such as 8 or 16.
• With $n$ bits you can make $2^n$ patterns, from $0$ up to $2^n - 1$.
• If a result is too big to fit, the extra bits are lost — an overflow error 溢出错误.
• Worked idea. With 4 bits the largest value is $2^4 - 1 = 15$; adding $15 + 1$ needs a 5th bit that does not exist, so it overflows.

Analog to digital

• Analog 模拟 data is smooth and continuous, like a real sound wave.
• To store it, a computer takes many quick measurements; each is a sample 采样.
• Each sample is rounded to the nearest storable value, so the copy only approximates the original.
• More samples and more bits per sample give a closer copy — the result is digital 数字 data.

Bytes

• Bits are tiny, so we group them: eight bits make one byte 字节.
• A byte is the common unit of data size — enough for one letter of English text.
• Bigger units build on it: ~1000 bytes is a kilobyte, ~a million is a megabyte.
• Worked idea. A 200-letter message at 1 byte each is 200 bytes, or $200 \times 8 = 1600$ bits.