Understanding Number Systems: Binary, Hexadecimal, and Denary

Table of Contents

1. Introduction to Number Systems
2. Binary (Base-2)
3. Denary/Decimal (Base-10)
4. Hexadecimal (Base-16)
5. Conversions Between Number Systems
6. Why Do Programmers Use Hexadecimal?
7. Summary & Key Takeaways

---

1. Introduction to Number Systems

Numbers can be represented in different bases (or radix). The most common ones in computing are:

System	Base	Digits Used	Example
Binary	2	0, 1	1010
Denary	10	0-9	42
Hexadecimal	16	0-9, A-F	2F

• Binary: Used by computers (everything is 0s and 1s).
• Denary: Humans use this daily (standard counting).
• Hexadecimal: A compact way to represent binary for programmers.

---

2. Binary (Base-2)

• Only two digits: 0 and 1.
• Each digit is called a bit (8 bits = 1 byte).
• Used in computers because transistors (hardware) operate in two states: ON (1) and OFF (0).

Binary to Denary Conversion

Each binary digit represents a power of 2, starting from the right (LSB → MSB).

Example: 1011 in binary → Denary?

Bit Position	3	2	1	0
Binary Digit	1	0	1	1
Calculation	1×2³	0×2²	1×2¹	1×2⁰
Value	8	0	2	1

Total = 8 + 0 + 2 + 1 = 11 (Denary)

---

3. Denary/Decimal (Base-10)

• Uses digits 0-9.
• How humans naturally count.

Denary to Binary Conversion

Divide the number by 2 repeatedly and note remainders.

Example: Convert 13 to binary.

Division	Quotient	Remainder
13 ÷ 2	6	1 (LSB)
6 ÷ 2	3	0
3 ÷ 2	1	1
1 ÷ 2	0	1 (MSB)

Read remainders from bottom-up → 1101 (Binary)

---

4. Hexadecimal (Base-16)

• Uses digits 0-9 and letters A-F (where A=10, B=11, …, F=15).
• Shorter representation of binary (1 hex digit = 4 binary digits).

Hexadecimal to Denary Conversion

Each hex digit represents a power of 16.

Example: 2F in hex → Denary?

Digit	Position	Calculation	Value
2	1	2 × 16¹	32
F (15)	0	15 × 16⁰	15

Total = 32 + 15 = 47 (Denary)

Binary to Hexadecimal

Group binary digits into 4-bit nibbles and convert each to hex.

Example: 11010110 → Hex?

1. Split into nibbles: 1101 0110
2. Convert each:
   • 1101 = 13 → D
   • 0110 = 6 → 6

Result: D6 (Hex)

---

5. Conversions Between Number Systems

Quick Conversion Table

Denary	Binary (8-bit)	Hexadecimal
0	00000000	0
5	00000101	5
10	00001010	A
15	00001111	F
255	11111111	FF

---

6. Why Do Programmers Use Hexadecimal?

1. Compact Representation

• Writing long binary strings is tedious.
• Example:
  • Binary: 1101001110101101
  • Hex: D3AD (Much shorter!)

2. Easier to Read & Debug

• Memory addresses, color codes (#FF5733), and binary data are often represented in hex.

3. Direct Mapping to Binary

• Each hex digit = 4 binary digits (nibble).
• Makes binary manipulation easier.

4. Historical Use in Computing

• Early computers used hex for simplicity in low-level programming.

---

7. Summary & Key Takeaways

✔ Binary (Base-2) → Computers understand it.
✔ Denary (Base-10) → Humans use it daily.
✔ Hexadecimal (Base-16) → Shorthand for binary, used in programming.

Conversion Methods

From → To	Method
Binary → Denary	Sum of powers of 2.
Denary → Binary	Divide by 2, track remainders.
Binary → Hex	Group into 4-bit nibbles.
Hex → Denary	Sum of powers of 16.

Why Hex?

• Shorter than binary.
• Easier to read & debug.
• Directly maps to binary (1 hex digit = 4 bits).

---

Final Thought

Hexadecimal is like a “compressed binary”—it makes life easier for programmers while keeping close ties to how computers work.

🚀 Now you can convert and understand these number systems like a pro!

(Need a cheat sheet? Here’s a quick reference table below!)

Quick Reference Table

System	Base	Digits	Example
Binary	2	0,1	1010
Denary	10	0-9	42
Hex	16	0-9, A-F	2F