Convert Hexadecimal to Binary Step-by-Step

Converting a hexadecimal (base-16) number directly to a binary (base-2) number is a fast and simple process because 16 is an exact power of 2 (\(2^4 = 16\)). This mathematical relationship means that each individual hexadecimal digit corresponds directly to a unique 4-bit binary sequence (known as a nibble). This guide provides the complete step-by-step procedure for performing this direct conversion, along with the standard reference mapping and practical examples.

Hexadecimal to 4-Bit Binary Reference Chart

To convert directly without converting to decimal first, use the standard 4-bit binary values for each hexadecimal digit:

• 0 = 0000
• 1 = 0001
• 2 = 0010
• 3 = 0011
• 4 = 0100
• 5 = 0101
• 6 = 0110
• 7 = 0111
• 8 = 1000
• 9 = 1001
• A (10) = 1010
• B (11) = 1011
• C (12) = 1100
• D (13) = 1101
• E (14) = 1110
• F (15) = 1111

Step-by-Step Conversion Procedure

Step 1: Separate the Hexadecimal Digits

Write down the hexadecimal number and separate each digit individually. If the number includes a radix point (hexadecimal point), keep the point in its relative position.

Step 2: Convert Each Digit to a 4-Bit Binary Group

Replace every single hexadecimal digit with its exact 4-digit binary equivalent from the reference table.

• Important: Always write the full 4 bits for each digit, including leading zeros (e.g., convert 3 to 0011, not 11).

Step 3: Concatenate the Binary Groups

Join all 4-bit binary groups together in the exact order of the original hexadecimal digits. If there is a radix point, place it between the converted groups where it originally appeared.

Step 4: Remove Unnecessary Leading Zeros

Remove any leading zeros from the very front of the whole number part to produce the final, clean binary representation. (Do not remove trailing zeros after a radix point unless simplifying precision).

Example 1: Converting a Whole Number (4B7₁₆)

1. Separate digits: 4 | B | 7
2. Convert each to 4 bits:
   • 4 \(\rightarrow\) 0100
   • B \(\rightarrow\) 1011
   • 7 \(\rightarrow\) 0111
3. Combine the groups: 010010110111
4. Drop the leading zero: 10010110111₂

Result: 4B7₁₆ = 10010110111₂

Example 2: Converting with a Fractional Part (2E.8₁₆)

1. Separate digits: 2 | E | . | 8
2. Convert each to 4 bits:
   • 2 \(\rightarrow\) 0010
   • E \(\rightarrow\) 1110
   • . \(\rightarrow\) .
   • 8 \(\rightarrow\) 1000
3. Combine the groups: 00101110.1000
4. Clean up leading/trailing zeros: 101110.1₂

Result: 2E.8₁₆ = 101110.1₂