Question

What bit pattern is represented by each of the following dotted decimal patterns? a. \(0.0\) b. \(26.19 .1\) c. \(8.12 .20 .13\)

Short answer

a. 00000000 00000000 b. 00011010 00010011 00000001 c. 00001000 00001100 00010100 00001101

Step-by-step solution

Understanding Dotted Decimal Notation

Dotted decimal notation is a way of writing bytes separated by dots. For the purpose of this exercise, we assume each group (byte) in the pattern can be expanded into 8 binary digits.

Converting 0 to Binary

The decimal number 0 converts to the binary number 00000000. It has 8 bits, all of which are zeros.

Converting 26 to Binary

The decimal number 26 in binary is 11010. To make it fit into a standard byte (8 bits), we add three leading zeros, making it 00011010.

Converting 19 to Binary

The decimal number 19 in binary is 10011. Similar to the previous step, add three leading zeros to make it 8 bits: 00010011.

Converting 1 to Binary

The decimal number 1 in binary is 1. We add seven leading zeros to make it 8 bits: 00000001.

Converting 8 to Binary

The decimal number 8 in binary is 1000. Add four leading zeros to make it 8 bits: 00001000.

Converting 12 to Binary

The decimal number 12 in binary is 1100. Add four leading zeros to make it 8 bits: 00001100.

Converting 20 to Binary

The decimal number 20 in binary is 10100. We add three leading zeros to make it a complete byte: 00010100.

Converting 13 to Binary

The decimal number 13 in binary is 1101. By adding four leading zeros, it becomes: 00001101.

Conclusion and Format

For each dotted decimal statement, we represent it as a continuous string of bytes (8-bits each) in binary, with each part separated by a space for clarity.

Key concepts

dotted decimal notation

Dotted decimal notation is a method used to express numeric values, commonly used for IP addresses in computer networks. In this notation, numbers are divided into groups separated by dots. Each group represents a byte—in other words, a section of 8 bits.
When interpreting dotted decimal numbers, each segment stands alone as a discrete byte.

• For example, the IP address `192.168.1.1` is a common use case of dotted decimal notation.
• Similarly, in the exercise, `26.19.1` is interpreted as three distinct bytes, each converting separately into binary form.

This division helps facilitate a more manageable way to parse and understand long sequences of binary numbers.

byte representation

A byte is a standard unit of digital data that consists of 8 bits. It provides a straightforward way to encapsulate binary information.
Each byte can represent values ranging from 0 to 255 in decimal form. This range is significant because:

• It covers the typical range needed for character encoding and various computer operations.
• In networking, each section of an IPv4 address is represented as a byte, which is why each number in dotted decimal notation falls within this range.

When converting a decimal number to fit within a byte's structure, it's necessary to ensure it has exactly 8 bits. If a number has fewer than 8 bits in its binary form, leading zeros are added to complete the byte.

binary number system

The binary number system is fundamental to computing, representing data in two distinct symbols: 0 and 1. Unlike the decimal system, which uses ten digits (0-9), binary solely relies on these two.
Why Binary is Crucial:

• Binary is the foundational language of computers, where all instructions and data are interpreted through sequences of bits.
• This system facilitates the conversion of high-level programming instructions into machine-readable forms.
• Binary numbers are integral in storing data efficiently, such as in the conversion exercises provided.

Each binary digit (bit) is a power of 2, starting from right (least significant bit) to left (most significant bit). Hence, a byte like `00011010` can be calculated by weighing each bit's contribution as powers of 2.

decimal to binary conversion

Decimal to binary conversion is a critical process in understanding digital systems. It involves breaking down a decimal number into its binary equivalent, which computers can utilize.
Steps for Conversion:

• Determine the largest power of 2 less than or equal to the decimal number.
• Subtract this value from the decimal number, marking a `1` for the corresponding binary position.
• Repeat the process with the remainder until you've broken down the decimal number completely.

For instance, converting the decimal number 20: - The largest power of 2 ≤ 20 is 16, so the binary starts with `1`. - Subtract 16 from 20 to get 4. - Continue by finding 4 in binary (`100`). - Altogether, 20 converts to `10100` in binary.
In exercises like these, ensure every conversion fills a byte by adding necessary leading zeros, reaching an 8-bit format, crucial for network and storage systems.