Last modified on January 29th, 2024

chapter outline

 

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 NumberBinary Number
00000
10001
20010
30011
40100
50101
60110
70111
81000
91001
A1010
B1011
C1100
D1101
E1110
F1111

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

1F5C
0001111101011100

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

Convert (AB)16 into the binary number.

Solution:

A → 1010
B → 1011
(AB)16 → (10101011)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 Value5C
Decimal Value5 × 16112 × 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.

Convert (64)16 to binary number.

Solution:

By converting (64)10 into its corresponding decimal, we get
(6 × 161) + (4 × 160) = 96 + 4 = 100
Now, by converting (100)10 into its corresponding binary, we get
Thus, (64)16 = (1100100)2.

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