Hexadecimal to Binary

A hexadecimal-to-binary conversion is done to convert a hexadecimal number (base 16) to its equivalent binary number (base 2). The methods used to convert a hexadecimal number to its binary counterpart are discussed below.

Direct Method: Using Table

The hexadecimal number system has 16 digits from 0 to 9 and A to F, which are directly represented with their corresponding binary numbers using only 4 bits.

Let us convert (1F5C)₁₆ into its corresponding binary number.

Step 1: Grouping 1F5C into individual digits, we have 1, F, 5, and C.

Step 2: Now, we use the conversion table below to find each hexadecimal digit’s corresponding binary number.

Hexadecimal Number	Binary Number
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

By converting each hexadecimal digit of (1F5C)₁₆ to its equivalent binary, we get

1	F	5	C
0001	1111	0101	1100

Here, we observe that each hex digit gives us a 4-bit binary number, so we must keep track of the leading zeros to maintain the correct number of digits while writing in the binary form.

Step 3: On writing the equivalent binary, we get (1F5C)₁₆ = (0001111101011100)₂ or (1111101011100)₂.

For Fractional Hexadecimal Numbers

Similarly, we can convert the fractional hexadecimal values into their corresponding binary numbers. 

Converting (0.A0F)₁₆ into its equivalent binary, we get

For the Integral Part:

0 → 0000

For the Fractional Part:

A → 1010, 0 → 0000, and F → 1111

Thus, (0.0A0F)₁₆ = (0.101000001111)₂

Indirect Method: Without Using Table

There is another way to represent each digit in the hexadecimal number system to its corresponding binary number without using the conversion table. 

Let us convert a hexadecimal number (5C)₁₆ into its corresponding binary without using the conversion table.

First, we convert 5C into a decimal number, and then decimal to the corresponding binary number.

Step 1: Hexadecimal to Decimal

While converting (5C)₁₆ to its respective decimal number, we multiply each digit (from right to left) by the corresponding powers of 16, as shown.

Hexadecimal Value	5	C
Decimal Value	5 × 161	12 × 160

Now, on adding the values, we get the decimal number

(5 × 16¹) + (12 × 16⁰) = 80 + 12 = 92

Step 2: Decimal to Binary

Now, converting (92)₁₀ into its binary equivalent, we get

Thus, (5C)₁₆ = (1011100)₂.