Summary of "Representing Numbers and Letters with Binary: Crash Course Computer Science #4"

Representing numbers and letters with binary

Main ideas and concepts

Binary is the base‑2 number system used by computers: each digit (bit) is either 0 or 1. Larger values are formed by using more bits, the same way decimal (base 10) uses more digits.
The place‑value principle applies to any base:
- In decimal, each column is 10× the one to its right (ones, tens, hundreds).
- In binary, each column is 2× the one to its right (ones, twos, fours, eights, …).
Bits, bytes, and storage units:
- A bit is a single binary digit (0 or 1).
- A byte = 8 bits. Common prefixes: kilo-, mega-, giga-, tera- (KB, MB, GB, TB). Note: some contexts use factors of 1000, others use 1024 (2^10); both usages exist.
- 8‑bit values can represent 256 distinct values, ranging 0–255 (2^8).
Word size and addressing:
- Systems are often described as 8‑, 32‑, or 64‑bit, indicating the chunk size they operate on.
- A 32‑bit unsigned integer has a maximum value of 2^32 − 1 (≈ 4.29 billion). A 64‑bit unsigned integer max is 2^64 − 1 (≈ 1.84×10^19).
- Increasing bit‑widths enable larger address spaces, which is important for supporting more RAM.
Signed integers:
- A simple conceptual approach reserves the most significant bit as a sign (0 = positive, 1 = negative) and uses the remaining bits for magnitude. In practice, most systems use two’s complement because it simplifies arithmetic.
Floating‑point numbers:
- Represent non‑integer values by separating a significand (mantissa) and an exponent, similar to scientific notation (for example, 625.9 = 0.6259 × 10^3).
- IEEE 754 is the dominant standard. A common 32‑bit (single‑precision) layout has 1 sign bit, 8 exponent bits, and 23 significand bits.
Text and character encoding:
- Text is represented as numbers; each character maps to a numeric code.
- Early schemes included simple numeric assignments (A = 1, B = 2, …) and historical five‑bit codes (Francis Bacon).
- ASCII (1963) is a 7‑bit code (128 values) for English letters, digits, punctuation, and control characters (for example, a = 97, A = 65).
- Extending to 8 bits enabled national/extended character sets (values 128–255) but led to many incompatible encodings.
- Incompatible encodings and the demands of many‑character languages produced mojibake (scrambled text).
- Unicode (1992) unified encoding with a common space (commonly using 16‑bit code units and beyond) to cover characters from most scripts, symbols, emoji, etc. Modern Unicode supports well over 1,000,000 code points and more than 120,000 characters.
Everything in computers (text, images, audio, video, programs) is ultimately sequences of bits interpreted according to agreed formats and standards so different systems can interoperate.

Everything in a computer — text, images, audio, video, and programs — is ultimately sequences of bits that follow agreed formats/standards.

Methodologies — step‑by‑step processes

Converting a binary number to decimal (place‑value method)
- Write the binary digits with their positions (rightmost is position 0).
- For each 1 bit, compute 2^position and multiply by the bit (1 or 0).
- Sum those values to get the decimal equivalent.
- Example: 10110111
  - 1×2^7 + 0×2^6 + 1×2^5 + 1×2^4 + 0×2^3 + 1×2^2 + 1×2^1 + 1×2^0
  - = 128 + 0 + 32 + 16 + 0 + 4 + 2 + 1 = 183 (decimal).
Binary addition (columnwise with carry, analogous to decimal addition)
- Add the rightmost column bits first.
- If a column sums to 2, write 0 and carry 1 to the next column (because 2 in binary is 10).
- If a column plus carry sums to 3, write 1 and carry 1 (3 in binary is 11).
- Continue leftward, including carries each time.
- Example: adding binary representations of decimal 183 and 19 yields 11001010, which equals decimal 202.
Representing signed integers (simple sign‑bit explanation)
- Reserve the most significant bit as a sign (0 = positive, 1 = negative).
- Use remaining bits to store magnitude. This limits the representable range to about ±(2^(n−1) − 1) in this simplified view.
- Note: two’s complement is the typical practical representation because it simplifies arithmetic operations.
Representing floating‑point numbers (IEEE 754 conceptual steps)
- Express the number in normalized binary scientific notation: significand × 2^exponent.
- Store:
  - 1 sign bit,
  - a fixed number of exponent bits (biased exponent),
  - a fixed number of significand bits (fractional part; normalized numbers often have an implicit leading 1).
- Example: a 32‑bit float uses 1 sign bit, 8 exponent bits, and 23 significand bits.
Representing text characters
- Map each character to a numeric code using a standard encoding.
- Examples:
  - ASCII (7‑bit) maps letters/digits/symbols to numbers (e.g., a → 97).
  - 8‑bit extensions provided national characters but were inconsistent across systems.
  - Unicode assigns unique code points to characters to enable cross‑language interoperability.

Other noteworthy details and examples

8‑bit systems historically provided 256 distinct values and were common in older computers, graphics, and audio.
Memory and addressability drove the shift from 32‑bit to 64‑bit systems because larger physical memory requires larger address spaces.
Practical numeric ranges (approximate):
- 8‑bit max: 255.
- 32‑bit signed range: about ±2 billion (using 31 bits for magnitude in the sign‑bit view).
- 64‑bit unsigned max: about 1.84×10^19 (≈ 18.4 quintillion).
Many file formats (MP3, GIF, etc.) are binary encodings that interpret sequences of bits according to well‑defined formats.