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Farhad Rahimi Klie
Farhad Rahimi Klie

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Understanding Computer Binary: The Foundation of All Computing

#architecture #computerscience #basic #programming

Binary is the foundation of all modern computing systems. Every program you write, every image you see, every database record, and every network packet ultimately resolves to binary data—a sequence of 0s and 1s.

1. What Is Binary?

Binary is a base-2 number system that uses only two digits:

0 and 1
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System Base Digits
Decimal 10 0–9
Binary 2 0–1
Octal 8 0–7
Hexadecimal 16 0–9, A–F

Computers use binary because electronic circuits have two stable states:

  • Low voltage (OFF) → 0
  • High voltage (ON) → 1

2. Binary at the Hardware Level (Physical Architecture)

2.1 Transistors

A transistor acts as a switch:

State Voltage Binary
OFF Low 0
ON High 1

Modern CPUs contain billions of transistors, each storing or manipulating binary data.

2.2 Logic Gates (Binary Logic)

Binary data is processed using logic gates.

Gate Operation Truth Table
AND 1 if both are 1 1 & 1 = 1
OR 1 if any is 1 `1 0 = 1`
NOT Inverts !1 = 0
XOR Different → 1 1 ^ 0 = 1

Example:

A = 1
B = 0
A AND B = 0
A OR B  = 1
A XOR B = 1
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These gates form adders, multiplexers, registers, and ALUs.

3. Binary Data Units

Unit Size
Bit 1 binary digit
Nibble 4 bits
Byte 8 bits
Word CPU-dependent (32 or 64 bits)

Example byte:

10101100
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4. Binary Number Representation

4.1 Positional Value (Base-2)

Binary digits represent powers of 2:

Binary:  1 0 1 1
Index:   3 2 1 0
Value:   8 0 2 1
Result:  11 (decimal)
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4.2 Decimal to Binary Conversion

Example: Convert 13 → Binary

13 / 2 = 6 remainder 1
6 / 2 = 3 remainder 0
3 / 2 = 1 remainder 1
1 / 2 = 0 remainder 1

Binary = 1101
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4.3 Binary to Decimal

Example:

10101 = (1×16) + (0×8) + (1×4) + (0×2) + (1×1)
      = 21
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5. Signed Binary Numbers

5.1 Unsigned Binary

Only positive values.

5.2 Signed Binary (Two’s Complement)

Most systems use Two’s Complement.

Steps to represent -5 in 8 bits:

  1. 5 → 00000101
  2. Invert → 11111010
  3. Add 1 → 11111011

Range:

8-bit signed → -128 to +127
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6. Binary Arithmetic

6.1 Binary Addition 

  1011
+ 0110
------
 10001
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Rules:

  • 0+0=0
  • 1+0=1
  • 1+1=10 (carry)

6.2 Binary Subtraction

Uses two’s complement internally.

6.3 Binary Multiplication

Shift-and-add method.

101 × 11
= 101
+1010
-----
1111
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7. Binary in CPU Architecture

7.1 Registers

Registers store binary values:

RAX = 0000000000001010
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7.2 ALU (Arithmetic Logic Unit)

ALU performs:

  • Binary arithmetic
  • Bitwise logic
  • Comparisons

7.3 Instruction Encoding

Machine code:

10101010 00001010
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Assembly:

MOV AX, 10
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8. Binary Memory Representation

8.1 RAM

Memory is a sequence of bytes:

Address   Data
0x1000 → 01010101
0x1001 → 11100010
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8.2 Endianness

Type Order
Little Endian Least byte first
Big Endian Most byte first

Example (0x12345678):

Little → 78 56 34 12
Big    → 12 34 56 78
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9. Binary Data Types

9.1 Integers 

int x = 5; // 00000101
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9.2 Floating Point (IEEE-754)

Structure:

[ Sign ][ Exponent ][ Mantissa ]
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32-bit float:

  • 1 sign bit
  • 8 exponent bits
  • 23 mantissa bits

10. Bitwise Operations (Binary Syntax)

Operator Meaning
& AND
` ` OR
^ XOR
~ NOT
<< Left shift
>> Right shift

Example in C:

int a = 5;   // 0101
int b = 3;   // 0011

int c = a & b; // 0001
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11. Binary Encodings

11.1 ASCII 

'A' → 65 → 01000001
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11.2 Unicode / UTF-8

Supports all languages.

'€' → 11100010 10000010 10101100
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12. Binary Files and Storage

All files are binary:

  • Images
  • Videos
  • PDFs
  • Executables

Example file header:

PDF → 25 50 44 46
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13. Binary in Networking

Network packets are binary streams.

Example TCP header fields:

Source Port → 16 bits
Destination → 16 bits
Flags → 6 bits
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14. Binary Compression

Compression algorithms manipulate bits:

  • Huffman Encoding
  • LZ77
  • DEFLATE

15. Binary Security & Cryptography

Encryption operates on binary blocks:

  • AES → 128-bit blocks
  • RSA → binary exponentiation
  • Hashes → bit mixing

16. Binary in Programming Languages

Python 

x = 10
print(bin(x))  # 0b1010
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JavaScript 

let x = 5;
console.log(x.toString(2)); // "101"
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C++ 

int x = 5;
std::bitset<8> b(x); // 00000101
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17. Why Binary Is Irreplaceable

Binary provides:

  • Electrical stability
  • Noise tolerance
  • Logical clarity
  • Hardware simplicity
  • Universal abstraction

No modern digital system operates without binary.

18. Summary

Binary is:

  • The lowest-level language of computers
  • The bridge between physics and software
  • The foundation of CPU, memory, networking, files, and security

Understanding binary deeply makes you:

  • A better programmer
  • A stronger system designer
  • A more confident low-level engineer

Final Thought

Every abstraction eventually collapses into binary.
If you understand binary, you understand computing at its core.

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