Converting Between Decimal, Binary, Octal, And Hexadecimal Number Systems

by ADMIN 74 views

When you are trying to convert decimal numbers to binary, it's crucial to address both the integer and fractional parts separately. This meticulous approach ensures precision and clarity in the conversion process. Breaking down the process into manageable steps not only simplifies the calculations but also minimizes the chances of errors, leading to a more accurate final binary representation. Let's delve into the step-by-step conversion of the decimal number (98.72)₁₀ into its binary equivalent.

First, we'll convert the integer part, 98, to binary. This involves successive division by 2 and noting the remainders. The remainders, read in reverse order, will give us the binary equivalent of the integer part. Here’s how it works:

  • 98 ÷ 2 = 49, Remainder = 0
  • 49 ÷ 2 = 24, Remainder = 1
  • 24 ÷ 2 = 12, Remainder = 0
  • 12 ÷ 2 = 6, Remainder = 0
  • 6 ÷ 2 = 3, Remainder = 0
  • 3 ÷ 2 = 1, Remainder = 1
  • 1 ÷ 2 = 0, Remainder = 1

Reading the remainders in reverse order, we get 1100010. This is the binary representation of 98.

Next, we convert the fractional part, 0.72, to binary. This involves successive multiplication by 2. The integer parts of the results, read in the order they are obtained, will give us the binary equivalent of the fractional part. Let's break down this process:

    1. 72 × 2 = 1.44, Integer part = 1
    1. 44 × 2 = 0.88, Integer part = 0
    1. 88 × 2 = 1.76, Integer part = 1
    1. 76 × 2 = 1.52, Integer part = 1
    1. 52 × 2 = 1.04, Integer part = 1
    1. 04 × 2 = 0.08, Integer part = 0
    1. 08 × 2 = 0.16, Integer part = 0

Reading the integer parts in order, we get 0.1011100... This is the binary representation of 0.72, truncated to seven decimal places for practical purposes. The binary representation of fractional numbers often results in a non-terminating sequence, so we typically round or truncate after a certain number of places, depending on the required precision.

Combining the binary representations of the integer and fractional parts, we get (1100010.1011100)₂. This is the binary equivalent of the decimal number (98.72)₁₀. Understanding this conversion process is a fundamental skill in computer science and digital electronics, as it forms the basis for representing numerical data in binary systems, the language of computers. Mastering this conversion allows for a deeper understanding of how computers process and store numerical information.

Converting a decimal number like (283.81)₁₀ to octal (base 8) is a critical skill in understanding different number systems, which are fundamental in computing and digital electronics. The octal system, with its base of 8, uses digits from 0 to 7, making it a more compact representation than binary but still easily convertible to it. This conversion involves handling the integer and fractional parts separately to ensure accuracy and clarity. Let’s break down the process step by step to make it easy to understand and apply.

To convert the integer part, 283, to octal, we use successive division by 8, noting the remainders at each step. This method efficiently breaks down the decimal number into its octal components. The remainders, when read in reverse order, form the octal equivalent of the decimal integer. Here’s how it’s done:

  • 283 ÷ 8 = 35, Remainder = 3
  • 35 ÷ 8 = 4, Remainder = 3
  • 4 ÷ 8 = 0, Remainder = 4

Reading the remainders in reverse order, we get 433. So, the octal representation of 283 is (433)₈.

Next, we convert the fractional part, 0.81, to octal. This involves successive multiplication by 8. We take the integer part of the result at each step and continue multiplying the fractional part until we achieve the desired precision or the fractional part becomes zero. This method accurately captures the octal representation of the decimal fraction.

    1. 81 × 8 = 6.48, Integer part = 6
    1. 48 × 8 = 3.84, Integer part = 3
    1. 84 × 8 = 6.72, Integer part = 6
    1. 72 × 8 = 5.76, Integer part = 5
    1. 76 × 8 = 6.08, Integer part = 6

Reading the integer parts in the order they are obtained, we get 0.63656... This is the octal representation of 0.81, truncated to five octal places for practical use. In many cases, the fractional part in octal form can be non-terminating, necessitating truncation or rounding based on the required level of precision.

Combining the octal representations of the integer and fractional parts, we get (433.63656)₈. This is the octal equivalent of the decimal number (283.81)₁₀. Understanding this conversion process is crucial for working with systems that use octal representation, often found in older computer systems and some digital applications. The ability to seamlessly convert between decimal and octal systems enhances one’s understanding of numerical representations and their practical applications in technology.

When converting the decimal number (193.77)₁₀ to hexadecimal (base 16), it's important to understand that the hexadecimal system uses 16 symbols: 0-9 and A-F, where A represents 10, B represents 11, and so on, up to F, which represents 15. This system is widely used in computer science for memory addressing and data representation due to its compactness and ease of conversion with binary. The conversion process involves treating the integer and fractional parts separately, ensuring accuracy in the final hexadecimal representation. Let's walk through the process step by step.

First, we convert the integer part, 193, to hexadecimal. This is done by successive division by 16, noting the remainders at each step. The remainders, when read in reverse order, give the hexadecimal equivalent. This method efficiently breaks down the decimal number into its hexadecimal components, making the conversion straightforward.

  • 193 ÷ 16 = 12, Remainder = 1 (1 = 1 in hexadecimal)
  • 12 ÷ 16 = 0, Remainder = 12 (12 = C in hexadecimal)

Reading the remainders in reverse order, we get C1. So, the hexadecimal representation of 193 is (C1)₁₆.

Next, we convert the fractional part, 0.77, to hexadecimal. This involves successive multiplication by 16. We take the integer part of the result at each step, converting it to its hexadecimal equivalent if it is 10 or greater, and continue multiplying the fractional part. This accurately captures the hexadecimal representation of the decimal fraction.

    1. 77 × 16 = 12.32, Integer part = 12 (12 = C in hexadecimal)
    1. 32 × 16 = 5.12, Integer part = 5 (5 = 5 in hexadecimal)
    1. 12 × 16 = 1.92, Integer part = 1 (1 = 1 in hexadecimal)
    1. 92 × 16 = 14.72, Integer part = 14 (14 = E in hexadecimal)
    1. 72 × 16 = 11.52, Integer part = 11 (11 = B in hexadecimal)

Reading the integer parts in the order they are obtained, we get 0.C51EB... This is the hexadecimal representation of 0.77, truncated to five hexadecimal places. The hexadecimal representation of fractional numbers is often non-terminating, so truncation or rounding is necessary based on the desired precision.

Combining the hexadecimal representations of the integer and fractional parts, we get (C1.C51EB)₁₆. This is the hexadecimal equivalent of the decimal number (193.77)₁₀. Understanding this conversion is essential for anyone working with computer systems, as hexadecimal is commonly used in programming, data representation, and memory addressing. The ability to convert seamlessly between decimal and hexadecimal systems enhances one’s grasp of how data is managed and manipulated in computing environments.

Converting a number from octal (base 8) to binary (base 2) is a fundamental skill in digital electronics and computer systems. The octal system, using digits 0-7, is often used as a more human-friendly shorthand for binary because each octal digit maps directly to a 3-bit binary sequence. This direct mapping makes the conversion process straightforward and efficient. Let’s convert (77.13)₈ to binary, step by step, to illustrate this process.

The key to converting octal to binary is to replace each octal digit with its 3-bit binary equivalent. This method works because 8 is a power of 2 (8 = 2³), allowing for this simple and direct substitution. This approach not only simplifies the conversion but also reduces the chances of error, making it a preferred method in practical applications.

  • Convert the integer part, 77, to binary:
    • The first digit, 7, in octal is equivalent to 111 in binary.
    • The second digit, 7, in octal is also equivalent to 111 in binary.
    • Combining these, we get 111111 as the binary representation of the integer part.
  • Convert the fractional part, 0.13, to binary:
    • The first digit after the decimal point, 1, in octal is equivalent to 001 in binary.
    • The second digit, 3, in octal is equivalent to 011 in binary.
    • Combining these, we get 0.001011 as the binary representation of the fractional part.

Now, we combine the binary representations of the integer and fractional parts. The integer part (77)₈ converts to (111111)₂, and the fractional part (0.13)₈ converts to (0.001011)₂. Therefore, the complete binary representation of (77.13)₈ is (111111.001011)₂. This direct conversion method is a powerful tool for anyone working with digital systems, as it provides a quick and accurate way to translate between octal and binary representations.

In summary, these conversions between decimal, binary, octal, and hexadecimal number systems are fundamental concepts in computer science and digital electronics. Understanding these conversions is crucial for working with computer hardware, software, and various data representations. Each conversion method—successive division, multiplication, and direct substitution—provides a unique approach to translating numbers between different bases, enhancing one’s ability to work effectively in the digital world. The ability to fluently convert between these systems not only improves technical proficiency but also fosters a deeper understanding of how numerical data is represented and manipulated in computing environments.