Problem 10 - Discrete Mathematics with Applications

Question

The binary representation of an integer can conveniently be used to find its octal representation. Group the bits in threes from right to left and replace each group with the corresponding octal digit. For example, $$243=11110011_{\text {two }}=011 \quad 110 \quad 011_{\text {two }}=363_{\text {eight }}$$'Using this short cut, rewrite each binary number as an octal integer. 11011 two

Step-by-Step Solution

Answer

The given binary number is \(11011_{\text{two}}\). First, group the binary digits in threes from right to left, giving \(\underline{011}\underline{100}\text{two}\). Then, convert each group of binary digits to its equivalent octal digit: \(011_\text{two} = 3_\text{eight}\) and \(100_\text{two} = 4_\text{eight}\). Finally, write the octal representation of the binary number: \(11011_\text{two} = 34_\text{eight}\).

Step 1: Write down the given binary number

The given binary number is: \[11011_{\text{two}}\]

Step 2: Group the binary digits in threes from right to left

First, we need to group the binary digits in threes starting from the rightmost side. If there are not enough digits on the leftmost side, you can add zeros to make the group complete: \[\underline{011}\underline100\text{two}\]

Step 3: Convert each group of binary digits into its octal equivalent

Now, we will convert each group of binary digits to its equivalent octal digit: \[ \begin{cases} 011_\text{two} = 3_\text{eight} \\ 100_\text{two} = 4_\text{eight} \end{cases} \]

Step 4: Write the octal representation of the binary number

Place the octal digits obtained in the previous step in the same order to get the final octal representation: \[11011_\text{two} = 34_\text{eight}\] So, the octal representation of the given binary number \(11011_\text{two}\) is \(34_\text{eight}\).

Key Concepts

Binary Number System

In digital electronics and computer science, the binary number system is a cornerstone. It's built on only two digits: 0 and 1, known as bits. This base-2 system is the language of computers, where each bit represents an on (1) or off (0) state. Binary numbers follow place value, similar to the more familiar decimal system, but instead of tens, each position to the left increases in powers of two. For instance, the binary number 1011 represents the decimal value of 11, calculated as \(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0\).

Understanding the significance of each place value is crucial for tasks like binary to octal conversion, and knowing how to manipulate these binary digits forms the basis of most computer operations and digital logic.

Octal Number System

The octal number system operates on a base-8 structure, utilizing the digits 0 through 7. The system is less prevalent today but was historically significant in computing due to its ease of conversion with binary, as one octal digit neatly represents three binary digits. For example, the binary sequence 101 corresponds to the octal digit 5. In octal, each position's value increases in powers of eight, moving from right to left. While we may not use the octal system in everyday life, understanding its structure is beneficial for various coding and computer system applications.

Integer Representation

Integer representation in computing refers to the various ways integers can be encoded within computers. In most systems, integers are represented in binary form, which can easily be converted into other number systems for operations and human readability. The representation includes not just positive numbers but also negative numbers and zero, often using schemes like Two's Complement for negative values. For conversions between systems, the integrity and value of the integer must be maintained, whether represented in binary, octal, decimal, or hexadecimal.

Number Base Conversion

Number base conversion is a process of translating numbers from one base to another, such as from binary (base-2) to octal (base-8). This skill is useful for programmers, electronic engineers, and mathematicians dealing with different systems. Conversions are done using specific methods tailored to the bases involved. For binary to octal conversion, as seen in the provided exercise, the binary number is grouped in sets of three bits starting from the right, and each triplet is replaced with its corresponding octal digit. This method does not require the binary number to be first converted to decimal, which makes the process more efficient and less error-prone. Understanding and applying base conversion methods is essential in fields involving digital systems and computations.