Problem 18

Question

The techniques explained in Exercises $9-12$ are reversible; that is, the octal and hexadecimal representations of integers can be used to find their binary representations. For example, $$ 345_\mathrm{eight}=011100 \quad 101_{\mathrm{two}}=11100101_{\mathrm{two}} $$ Using this technique, rewrite each number in base two. $$ 237_{\text { eight }} $$

Step-by-Step Solution

Verified

Answer

The binary representation of the given octal number $237_{\text {eight}}$ is $010011111_{\text {two}}$.

1Step 1: Write down the octal number

We are given the octal number $237_{\text {eight}}$.

2Step 2: Convert each octal digit to binary

Replace each octal digit with its equivalent binary triplet using the following table: Octal | Binary ----- | ------ 0 | 000 1 | 001 2 | 010 3 | 011 4 | 100 5 | 101 6 | 110 7 | 111 So, for the octal number $237_{\text {eight}}$: - The binary representation of 2 is 010 - The binary representation of 3 is 011 - The binary representation of 7 is 111

3Step 3: Construct the binary representation

Concatenate the binary triplets obtained in Step 2 to form the binary representation of the octal number: $237_{\text {eight}} = 010~011~111_{\text {two}}$ Therefore, the binary representation of the given octal number is $010011111_{\text {two}}$.

Key Concepts

Binary RepresentationOctal SystemHexadecimal SystemReversible TechniquesBase Two Conversion

Binary Representation

Binary representation is all about using the base-2 numeral system to denote numbers. In this system, we only have two possible digits, 0 and 1. Whereas in the decimal system we typically make use of ten different digits from 0 to 9. Each digit in a binary number represents a power of 2, starting from 2 raised to the power of 0 (the rightmost digit).

The rightmost digit is the least significant bit (LSB).
The leftmost digit is the most significant bit (MSB).

For example, the binary number 101 represents the decimal number 5, as it is equivalent to: \[1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 4 + 0 + 1 = 5\]Binary numbers are essential in computing and electronics, as electronic devices use these simple signals to operate.

Octal System

The octal system is a base-8 numeral system. It is formed using the digits 0 to 7. This system is particularly beneficial when compressing binary representations, as each octal digit corresponds to exactly three binary bits.
Here's how it works:

Each digit in an octal number can be represented as a binary triplet.
For example, the octal digit 5 corresponds to the binary digits 101.

The octal system provides an efficient means to represent binary numbers in a more compact form. It is particularly useful in certain computing settings, where easier readability and error-checking is required. For the octal number 237, converting each digit to binary gives us the binary number 010011111.

Hexadecimal System

The hexadecimal system, or hex, is a base-16 numeral system. It uses sixteen distinct symbols, the numbers 0 to 9 and the letters A to F, to represent values. Like the octal system, it simplifies binary representations by grouping binary digits into larger units.

Each hex digit represents four binary bits, or a "nibble."
The digit 'A', for example, corresponds to the binary 1010.

When converting between binary and hex, the process involves replacing every four binary digits with their corresponding hex digit. This system is widely used in computing for address locations and compact data stream representation, providing a balance between human readability and binary-friendly operations.

Reversible Techniques

Reversible techniques in numeral systems allow for easy conversion back and forth between representations. These techniques are essential in computing for tasks like encoding, decoding, and memory allocation.

For the octal system, you can easily convert to binary by replacing each digit with a three-bit binary equivalent.
Similarly, for the hexadecimal system, each digit is directly convertible to a four-bit binary system.

The key to mastering these conversions is practice. As demonstrated, the octal number 237 converts to the binary series 010011111. By breaking down these conversions, understanding and verification become straightforward, ensuring accuracy and facilitating error detection in computing tasks.

Base Two Conversion

Base two conversion involves transforming a number from its current base into the binary system. This conversion is fundamental in fields like digital electronics and computer science, as binary is the core language of machines.
A step-by-step guide to converting between bases includes:

Converting each digit of the original number into its binary equivalent based on its place value.
Combining these binary segments to form the final binary number.

For example, converting the octal 237 to binary:

The digit 2 is 010 in binary.
The digit 3 is 011 in binary.
The digit 7 is 111 in binary.
Concatenate these to get the binary number 010011111.

By understanding the place value and binary digit representation, conversions can be completed with ease and precision.

Problem 18

Other exercises in this chapter

Problem 17

Evaluate each sum, where $d$ is a positive integer. $$\sum_{d | 18}\left(\frac{1}{d}\right)$$

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Problem 18

Let $A=\left\langle a_{i j}\right)_{n \times n}$ and $B=\left(b_{i j}\right)_{n \times n}$ $A$ is less than or equal to $B$ denoted by $A \leq B,$ if

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Problem 18

Write an iterative algorithm to do the tasks. Let $A=\left(a_{i j}\right)_{n \times n}$ and $B=\left(b_{i j}\right)_{n \times n} . A$ is less than or equal

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Problem 18

Evaluate each sum, where $d$ is a positive integer. $$\sum_{d | 18}\left(\frac{18}{d}\right)$$

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