Ma216 Hw3i Suggest You Read Carefully Through Chapter 1 Sect
Ma216 Hw3i Suggest You Read Carefully Through Chapter 1 Section 3 Of
Review Chapter 1, Section 3 of Wrean's textbook, focusing on the quotient and remainder method for converting decimal numbers to different bases. Complete exercises 1–12 and 17–28 on pages 29–30, and check your answers against the solutions on pages 31–32. Specifically, perform conversions of decimal numbers to binary, octal, and hexadecimal using the quotient and remainder method, and document your step-by-step process. For practice, convert the decimal number 999 to binary, octal, and hexadecimal using this method. Ensure your work accurately illustrates the division process, remainders, and the final converted number.
Paper For Above instruction
Number systems form the foundation of digital computing, making it essential for students and professionals in computer science and engineering to understand various base conversions thoroughly. Among the various methods available for converting decimal numbers into binary, octal, and hexadecimal systems, the quotient and remainder method stands out for its clarity and systematic approach. This method involves division by the target base, recording the remainders, and reading them in reverse order to obtain the converted number.
The initial step in applying the quotient and remainder method is to divide the decimal number by the base to which conversion is desired. In the case of binary conversion, the base is 2; for octal, 8; and for hexadecimal, 16. After dividing, the quotient is recorded, and the division process repeats with this new quotient until it reaches zero. The remainders recorded at each step form the digits of the converted number when read in reverse order.
For example, converting decimal 31 to binary involves dividing 31 by 2 repeatedly and recording the remainders. The process is as follows: 31 divided by 2 gives a quotient of 15 and a remainder of 1; 15 divided by 2 gives 7 and 1; 7 divided by 2 gives 3 and 1; 3 divided by 2 gives 1 and 1; 1 divided by 2 gives 0 and 1. Reading the remainders from bottom to top results in 11111, which is the binary representation of 31.
Similarly, to convert 500 to octal, divide by 8 repeatedly: 500 divided by 8 yields a quotient of 62 and a remainder of 4; 62 divided by 8 yields 7 and 6; 7 divided by 8 yields 0 and 7. Reading the remainders from bottom to top yields 764, the octal equivalent of 500.
Converting decimal 250 to hexadecimal involves dividing by 16: 250 divided by 16 yields 15 and a remainder of 10 (represented as A in hexadecimal); 15 divided by 16 yields 0 and 15. Reading remainders from bottom to top yields FA, the hexadecimal equivalent of 250.
Applying this systematic approach to the decimal number 999: converting to binary involves successive divisions by 2, recording and reversing remainders, resulting in 1111100111. For octal, dividing 999 by 8 repeatedly yields remainders that assemble into 1757. For hexadecimal, successive divisions yield the hexadecimal representation 3E7. These conversions exemplify the utility of the quotient and remainder method in base conversions, establishing a critical skill for digital logic design and computer architecture.
References
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