Binary, Decimal, Hex,
Octal & Text

How Binary Numbers Work

Binary is base-2, using only 0 and 1. Each position represents a power of 2:

• Position 0 (rightmost): 2⁰=1 · Position 1: 2¹=2 · Position 2: 2²=4 · Position 3: 2³=8

To convert binary to decimal: multiply each digit by its positional value and add. Binary 1011=(1×8)+(0×4)+(1×2)+(1×1)=11. Binary 11111111=128+64+32+16+8+4+2+1=255.

Why Computers Use Binary

A transistor — the fundamental building block of modern processors — has two stable states: conducting (1) and not conducting (0). These map directly to binary's two digits. Distinguishing two voltage levels is far more reliable than distinguishing ten. Binary also enables logic gates — AND, OR, NOT, XOR — which implement all arithmetic and logic through simple transistor combinations. Modern CPUs contain billions of such transistors operating billions of times per second.

Hexadecimal — Binary's Shorthand

Hexadecimal (base 16) uses digits 0–9 and A–F (A=10, B=11...F=15). Each hex digit represents exactly 4 binary bits:

• 0000=0 · 1010=A · 1111=F
• An 8-bit byte = two hex digits (00 to FF = 0 to 255)
• A 32-bit integer = eight hex digits (00000000 to FFFFFFFF)

Hex appears everywhere: web colors (#FF5733), memory addresses (0x7FFE4A20), file signatures (JPEGs start with FF D8 FF). The 0x prefix indicates hexadecimal — 0xFF=255, 0x1A=26.

Bits, Bytes, and Data Storage

• Bit — a single binary digit: 0 or 1
• Nibble — 4 bits; one hex digit (0–F)
• Byte — 8 bits; 256 possible values (0–255). One ASCII character.
• Kilobyte (KB) — 1,024 bytes (2¹⁰). One text page ≈ 2 KB.
• Megabyte (MB) — 1,048,576 bytes (2²⁰). An MP3 song ≈ 3–5 MB.
• Gigabyte (GB) — 2³⁰ bytes. A smartphone photo ≈ 2–5 MB; a film ≈ 4–25 GB.
• Terabyte (TB) — 2⁴⁰ bytes. Large hard drives hold 1–8 TB.

Two's Complement — How Computers Handle Negative Numbers

To negate a binary number using two's complement: flip all bits, then add 1. In 8-bit: +1=00000001. Flip: 11111110. Add 1: 11111111=−1. The leftmost bit is the sign bit (0=positive, 1=negative). This allows subtraction to be performed using the same addition hardware as positive numbers — a key CPU design simplification.

ASCII — Text in Binary

ASCII maps characters to numbers (0–127). A=65=01000001, a=97=01100001, space=32=00100000. The case difference in ASCII between uppercase and lowercase letters is exactly 32 (00100000) — a single bit toggle switches case. ASCII was extended to Unicode (UTF-8), which supports all world scripts and emoji using 1–4 bytes per character, while keeping ASCII characters as valid single-byte sequences for backward compatibility.

Octal — Base 8 and Unix Permissions

Octal (base 8) uses digits 0–7. Each octal digit represents exactly 3 binary bits. Today octal survives primarily in Unix/Linux file permissions. chmod 755 sets permissions in octal: 7 (111=rwx), 5 (101=r-x), 5 (101=r-x). The three digits represent permissions for owner, group, and others respectively. Understanding octal is essential for Unix/Linux system administration.

Bitwise Operations in Programming

• AND (&) — 1 if both bits are 1; used to mask bits
• OR (|) — 1 if either bit is 1; used to set bits
• XOR (^) — 1 if bits differ; useful for toggling
• NOT (~) — flips all bits (one's complement)
• Left shift (<<) — shifts left, multiplying by 2 per shift: 5<<1=10
• Right shift (>>) — shifts right, dividing by 2 per shift