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Convert decimal numbers to 8-bit 2’s complement and perform binary subtraction using 8-bit 2’s complement. The instructions include converting negative decimal numbers into 8-bit 2’s complement form through binary conversion, zero-padding, inverting bits, and adding one. Additionally, they cover how to handle binary subtraction by converting subtraction into addition of negative numbers in 8-bit 2’s complement form, then adding and discarding overflow if it occurs.

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The process of converting decimal numbers to 8-bit two’s complement notation is fundamental in computer science, especially in computer architecture and digital logic design. This representation allows negative numbers to be stored and manipulated efficiently within binary systems and provides a standard way to perform binary arithmetic operations, including addition and subtraction. Understanding this system is essential for students and professionals working with low-level programming, hardware design, and digital electronics.

To convert a decimal number into 8-bit 2’s complement form, the process begins with converting the number to binary. For positive numbers, this involves straightforward binary conversion, followed by zero-padding to ensure the binary number is 8 bits long. For example, converting the decimal number 47 involves converting it to binary (101111) and adding a leading zero to make it 8 bits (00101111). For negative numbers, the process is slightly more involved; it requires converting the positive equivalent to binary, then inverting all bits and adding one to complete the two’s complement form.

For example, consider the conversion process of -47. First, convert 47 to binary: 101111. Then, pad it to 8 bits: 00101111. Next, invert all bits: 11010000. Finally, add 1: 11010001. This binary number, 11010001, correctly represents -47 in 8-bit two’s complement notation. Students can practice similar conversions for -5 and -21, which involve the same steps: convert positive, invert, and add one. For -67, which exceeds the magnitude of 8 bits, the same process applies, but it’s crucial to recognize that -67 is within the range of 8-bit two’s complement numbers, which is from -128 to 127. Its binary form after inversion and addition will be 10111111.

Binary subtraction in digital systems is often managed using addition and two’s complement arithmetic due to the complexities of direct subtraction at the hardware level. To subtract one number from another, the subtracted number is first converted to its two’s complement form, which effectively represents its negative. Then, binary addition is performed between the minuend and this two’s complement. If the sum produces an overflow (a ninth bit), this can be ignored or discarded, as only the last 8 bits are relevant in an 8-bit system. This technique simplifies hardware implementation as it reduces subtraction to an addition operation, which is easier to perform and optimize in digital circuits.

An example of this process is calculating 150 - 47. First, convert 150 to binary: 10010110. Since 150 exceeds the range of an 8-bit signed number, it wraps around due to overflow, but for simplicity, assuming the system can represent 150, the operation can still proceed in an extended form, or with two’s complement representation if we treat the negative with overflow correction. The number 47 in 8-bit two’s complement form is 00101111. To perform subtraction, convert 47 into its two’s complement negative form: invert bits (11010000), and add 1 (11010001). Adding this to 150's binary representation, the sum yields 10010110 + 11010001 = 01100111, ignoring overflow. The result, 01100111, in decimal corresponds to 103, confirming that 150 - 47 = 103.

Practicing such conversions and arithmetic operations enhances understanding of how computers perform calculations at the binary level. It also underscores the importance of binary number systems and two’s complement arithmetic in enabling efficient and reliable digital computation. Mastery of these concepts is crucial for designing digital systems, debugging low-level code, and understanding the inner workings of processors and digital devices.

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