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Farhad Rahimi Klie
Farhad Rahimi Klie

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Binary and Hexadecimal Number Systems — A Complete Guide for Low-Level Programming

#bit #bitmanipulation #lowlevel #programming

When you work close to hardware—whether in C, C++, embedded systems, operating systems, or reverse engineering—you must think like the machine. And machines do not understand decimal numbers. They operate using binary (base-2). To make binary manageable for humans, we use hexadecimal (base-16).

This article gives you a complete, step-by-step understanding of both systems, with practical insight for real programming scenarios.

1. Why Number Systems Matter in Low-Level Programming

At the hardware level:

  • Memory stores bits (0 and 1)
  • CPU processes binary instructions
  • Registers, pointers, addresses → all binary

But binary is long and hard to read. That’s where hexadecimal comes in—it’s a compact human-friendly representation of binary.

2. Binary Number System (Base 2)

2.1 Definition

Binary uses only two digits:

0 and 1
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Each position represents a power of 2.

2.2 Positional Value

Example:

1011₂
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Break it down:

1×2³ + 0×2² + 1×2¹ + 1×2⁰
= 8 + 0 + 2 + 1
= 11₁₀
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2.3 Binary Table (Important to Memorize)

Power Value
2⁰ 1
2¹ 2
2² 4
2³ 8
2⁴ 16
2⁵ 32
2⁶ 64
2⁷ 128

These are critical for bit manipulation.

2.4 Binary in Memory

A bit = 0 or 1
A byte = 8 bits

Example:

00001010 → 10 in decimal
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2.5 Binary Representation of Numbers

Positive Numbers

Stored directly:

5 → 00000101
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Negative Numbers (Two’s Complement)

Steps:

  1. Convert to binary
  2. Invert bits
  3. Add 1

Example: -5 (8-bit)

5      = 00000101
Invert = 11111010
+1     = 11111011
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2.6 Binary Operations

AND (&) 

1 & 1 = 1
1 & 0 = 0
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OR (|) 

1 | 0 = 1
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XOR (^) 

1 ^ 1 = 0
1 ^ 0 = 1
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NOT (~) 

~00000101 = 11111010
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2.7 Example in C 

#include <stdio.h>

int main() {
    int a = 5;   // 00000101
    int b = 3;   // 00000011

    printf("AND: %d\n", a & b); // 1
    printf("OR : %d\n", a | b); // 7
    printf("XOR: %d\n", a ^ b); // 6

    return 0;
}
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3. Hexadecimal Number System (Base 16)

3.1 Definition

Hexadecimal uses:

0–9 and A–F
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Hex Decimal
A 10
B 11
C 12
D 13
E 14
F 15

3.2 Why Hexadecimal?

Binary is long:

11111111
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Hex makes it short:

FF
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So:

1 byte = 8 bits = 2 hex digits
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3.3 Positional Value

Example:

2F₁₆
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2×16¹ + 15×16⁰
= 32 + 15
= 47₁₀
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3.4 Binary ↔ Hex Conversion (CRITICAL SKILL)

Rule:

Group binary into 4 bits

Example 1: Binary → Hex 

10101111
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Group:

1010 1111
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Convert:

1010 = A
1111 = F
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Result:

AF₁₆
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Example 2: Hex → Binary 

3C
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Convert each:

3 = 0011
C = 1100
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Result:

00111100
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3.5 Hex in Programming

C Example: 

int x = 0xFF;  // 255
int y = 0x1A;  // 26
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Memory Address: 

0x7ffeefbff5c8
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4. Binary vs Hexadecimal (Comparison)

Feature Binary Hexadecimal
Base 2 16
Digits 0,1 0–9, A–F
Length Long Short
Usage Machine level Human-friendly

5. Real Use Cases in Low-Level Programming

5.1 Memory Inspection

Debugger output:

0x7fff → address
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5.2 Bit Masking 

int flags = 0x0F; // 00001111
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5.3 Embedded Systems

Registers:

0xFF00 → hardware control
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5.4 Networking

IP (hex view):

C0 A8 01 01 → 192.168.1.1
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5.5 Assembly Language 

MOV AX, 0x1F
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6. Conversion Methods (Step-by-Step)

6.1 Decimal → Binary

Divide by 2 repeatedly:

13 ÷ 2 = 6 r1
6 ÷ 2  = 3 r0
3 ÷ 2  = 1 r1
1 ÷ 2  = 0 r1

Result: 1101
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6.2 Decimal → Hex

Divide by 16:

47 ÷ 16 = 2 r15 (F)
2 ÷ 16  = 0 r2

Result: 2F
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6.3 Binary → Decimal

Multiply powers of 2.

6.4 Hex → Decimal

Multiply powers of 16.

7. Important Patterns to Master

7.1 Hex ↔ Binary Table (Must Memorize)

Hex Binary
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111
8 1000
9 1001
A 1010
B 1011
C 1100
D 1101
E 1110
F 1111

8. Common Mistakes

  • Mixing base systems (e.g., treating hex as decimal)
  • Forgetting grouping in binary → hex
  • Ignoring leading zeros
  • Confusing ASCII vs numeric values

9. How to Master This (Practical Strategy)

  1. Memorize powers of 2
  2. Memorize hex ↔ binary table
  3. Practice conversions daily
  4. Use C programs to verify results
  5. Read memory using debuggers (gdb)

10. Final Insight

Binary is the language of machines.
Hexadecimal is the language of programmers working close to machines.

If you master both:

  • Bit manipulation becomes easy
  • Debugging becomes powerful
  • System programming becomes natural

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