Question

Express the following bit patterns in hexadecimal notation: a. 10110100101101001011 b. 000111100001 c. 1111111011011011

Short answer

a. B4B4B; b. 1E1; c. FEDB.

Step-by-step solution

Understand the pattern grouping

To convert a binary number to hexadecimal, first, we need to group the binary digits into sets of four, starting from the right. If the number of binary digits is not a multiple of four, add zeros at the beginning to make it so.

Group binary numbers for part a

The binary number given is 10110100101101001011. Group these binary digits into sets of four from the right: 1011 0100 1011 0100 1011.

Convert binary groups to hexadecimal for part a

Convert each 4-bit binary group into a hexadecimal digit. For 1011, the hexadecimal equivalent is B. Thus, the groups convert to: 1011 -> B, 0100 -> 4, 1011 -> B, 0100 -> 4, 1011 -> B. Therefore, the hexadecimal notation is B4B4B.

Group binary numbers for part b

The binary number is 000111100001. Group these into sets of four: 0001 1110 0001.

Convert binary groups to hexadecimal for part b

Convert each group: 0001 -> 1, 1110 -> E, 0001 -> 1. Hence, the hexadecimal notation is 1E1.

Group binary numbers for part c

The binary number is 1111111011011011. Group these into: 1111 1110 1101 1011.

Convert binary groups to hexadecimal for part c

Convert each group: 1111 -> F, 1110 -> E, 1101 -> D, 1011 -> B. Thus, the hexadecimal notation is FEDB.

Key concepts

Bit Patterns

In digital systems, bit patterns represent information using sequences of bits, where each bit can be either 0 or 1. These bits form the foundation of binary numbers used in computer systems.
A bit pattern's complexity can vary, depending on the number of bits involved. The length of the bit pattern impacts how we convert it to other number systems, like hexadecimal.
When visualizing a bit pattern, each bit holds a place value, starting from the rightmost bit, which is the least significant bit (LSB). As you move left, each successive bit represents an increasing power of 2.

• The LSB represents \(2^0\) or 1.
• The second bit from the right represents \(2^1\) or 2.
• This continues, with each position representing a higher power of 2.

By understanding the positional values in a bit pattern, you can easily grasp how these sequences encode data and form the basis for binary to hexadecimal conversion.

Hexadecimal Notation

Hexadecimal notation is a base-16 numbering system used in computing to simplify the representation of binary data. This system uses 16 symbols: the numbers 0 to 9 represent values zero to nine, and the letters A to F represent values ten to fifteen.
This form of notation significantly reduces the length of binary sequences, making them more readable and easier to manage. For example, one hexadecimal digit is equivalent to a 4-bit binary sequence.
Hexadecimal notation is predominantly used in programming and computer system design for tasks such as memory addressing and color codes. One of its key features is its ability to compactly express large binary numbers with fewer digits.

• It aligns with the byte structure of computers, where one byte consists of 8 bits, conveniently expressed as two hexadecimal digits.
• Hexadecimal is also highly readable compared to the longer binary form.

Overall, hexadecimal notation offers a streamlined way to present complex binary sequences efficiently.

Binary Group Conversion

Binary group conversion is the process of translating binary numbers into other forms, particularly hexadecimal, through grouping and mapping each binary group to a corresponding hexadecimal digit. This involves several clear steps.
First, separate the binary number into groups of four bits, starting from the right. If the number of bits isn't a multiple of four, padding with zeros on the left ensures complete groups.
This method works because each 4-bit group can directly map to one hexadecimal digit, given that hexadecimal uses 16 values (0 to 15).

• Take the binary number 1011, for instance, which converts to the hexadecimal B.
• Each binary group from left to right is handled similarly, translating bit patterns to a manageable hexadecimal form.

By this conversion, binary numbers retain their values while becoming more manageable and concise in hexadecimal notation.