Hexadecimal Number System Explained with Binary & Decimal Conversions (Base-16)

Introduction to Hexadecimal Numbers

The Hexadecimal Number System, commonly called Hex, is a base-16 numbering system widely used in computer and digital systems. It provides a compact and readable way to represent long binary numbers. Binary numbers use base-2 (0 and 1), while hexadecimal numbers use base-16. When working with large digital systems, binary numbers may contain 8, 16, or even 32 bits, making them long and difficult to read. To simplify this, binary digits are grouped into sets of four bits. Each 4-bit group can be represented by a single hexadecimal digit.

For example:

1101 0101 1100 1111

is easier to read than:

1101010111001111

Hexadecimal Digits

Since hexadecimal is a base-16 system, it uses sixteen distinct symbols:

• 0 to 9
• A, B, C, D, E, F

The letters represent decimal values:

Relationship Between Binary and Hexadecimal

One hexadecimal digit represents exactly four binary bits.

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Since 16 = 2⁴, hexadecimal has a direct relationship with binary numbers.

Place Value in Hexadecimal

Each digit in hexadecimal has a positional weight based on powers of 16:

16⁰, 16¹, 16², 16³ …

General expansion form:

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Binary to Hexadecimal Conversion (Example 1)

Convert binary number 11101010₂ to hexadecimal.

Step 1: Group into four bits from right.

1110 1010

Step 2: Convert each group.

• 1110 = 14 = E
• 1010 = 10 = A

Final Answer:

11101010₂ = EA₁₆

Hexadecimal to Binary Conversion (Example 2)

Convert 3FA7₁₆ into binary.

Replace each hex digit with its 4-bit binary equivalent:

3 = 0011
F = 1111
A = 1010
7 = 0111

Final Answer:

3FA7₁₆ = 0011111110100111₂

Hexadecimal to Decimal Conversion

Convert 3FA7₁₆ into decimal.

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Substitute decimal values (F = 15, A = 10):

• 3 × 4096 = 12288
• 15 × 256 = 3840
• 10 × 16 = 160
• 7 × 1 = 7

Add all values:

12288 + 3840 + 160 + 7 = 16295

Therefore:

3FA7₁₆ = 16295₁₀

Counting in Hexadecimal

Counting sequence:

0, 1, 2, … 9, A, B, C, D, E, F, 10, 11, 12 … 1F, 20 …

Note:

• 10₁₆ = 16 (decimal)
• FF₁₆ = 255 (decimal)
• FFF₁₆ = 4095 (decimal)

Adding Leading Zeros in Binary

If the binary number is not a multiple of four bits, add zeros to the left (MSB side).

Example:

11001011011001

Add leading zeros:

0011 0010 1101 1001

Convert to hexadecimal:

32D9₁₆

Advantages of Hexadecimal Numbers

• More compact than binary.
• Easier to read and write.
• Reduces errors in large binary numbers.
• Widely used in memory addressing and programming.
• Quick conversion between binary and hexadecimal.

Summary

• Hexadecimal is a base-16 numbering system.
• It uses digits 0–9 and letters A–F.
• One hex digit equals 4 binary bits.
• 16 = 2⁴, which creates a direct relationship with binary.
• It is widely used in computers, digital systems, and memory addressing.

The Hexadecimal Number System makes it easy to represent long binary numbers in a shorter, more readable format. It is one of the most important numbering systems used in modern computing.