Problem 19 - Discrete Mathematics with Applications

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. $$ 237_{\text { sixteen }} $$

Step-by-Step Solution

Answer

The binary representation of \(237_{16}\) is \(100110111_{2}\).

Step 1: Break Down Hexadecimal Representation

We need to break down the given hexadecimal number \(237_{16}\) into individual digits: \(2\), \(3\), and \(7\).

Step 2: Convert Each Hexadecimal Digit to Binary Representation

Next, we need to convert each hexadecimal digit to its binary representation. We can use the following conversions: - \(2_{16}\) is equal to \(0010_{2}\). - \(3_{16}\) is equal to \(0011_{2}\). - \(7_{16}\) is equal to \(0111_{2}\).

Step 3: Combine the Binary Representations

Finally, we can combine the binary representations of each hexadecimal digit to find the binary representation of the given number: \(0010_{2}0011_{2}0111_{2}\). So, \(237_{16}\) is equal to \(0010_{2}0011_{2}0111_{2}\) or \(100110111_{2}\).

Key Concepts

Binary Representation

In computer science, the binary system is a fundamental way of representing numbers using only two digits: 0 and 1. Each digit, known as a bit, represents a power of 2, increasing from right to left. For example, the binary number 101 represents:

• 1 in the 4's place (2^2)
• 0 in the 2's place (2^1)
• 1 in the 1's place (2^0)

Therefore, 101 in binary equals 5 in decimal.
Binary representation is crucial for digital electronics and computer systems because it aligns naturally with digital circuits using switches that are either on (1) or off (0). This makes it a universal language across digital devices.

Hexadecimal Conversion

Hexadecimal, or hex for short, is a base-16 numbering system that is often used in computing to simplify binary coding. It uses sixteen distinct symbols: 0-9 for values zero to nine, and A-F for values ten to fifteen.
Converting a hexadecimal number to binary involves translating each hex digit into a four-bit binary equivalent. For instance:

• The hex digit 2 is 0010 in binary.
• The hex digit 3 is 0011 in binary.
• The hex digit 7 is 0111 in binary.

Thus, a hexadecimal number, like 237, becomes 001000110111 when converted to binary by mapping each hex digit to a corresponding binary sequence. This method allows for a more compact and readable representation of binary data.

Octal Numbering

Octal is a base-8 numbering system which uses digits 0 to 7. It's quite useful for representing binary numbers in a more compressed form, where every octal digit can be directly converted from a 3-bit binary sequence.
For instance, let's convert the binary 111 to octal:

• 111 in binary equals 7 in octal (the largest 3-bit number).

Using octal can simplify notations and arithmetic for binary-based systems, especially when the datasets consist of large binary sequences. This system can help reduce complexity compared to long binary strings.

Integer Conversions

The process of integer conversions involves changing numbers from one base system to another. Converting between decimal, binary, octal, and hexadecimal formats is essential in programming and computer systems troubleshooting.
For example, take a number in decimal, such as 255. To convert this to binary, divide the number by 2, while recording the remainder, until you reach zero:

• 255 ÷ 2 = 127, remainder 1
• 127 ÷ 2 = 63, remainder 1
• and so on...

Repeating this process gives the binary value 11111111.
Such conversions are fundamental skills, enabling developers to read, interpret, and debug computer-encoded numbers effectively. They also aid in understanding how systems manage data at a lower level.