Question

Convert decimal +42 and +13 to binary. Use the signed-2's complement representation and enough digits to accommodate the numbers. Then perform the following calculations using 2's complements.

(+42)₁₀ + (-13)₁₀ =
(-42)₁₀ - (-13)₁₀ =
convert the answer back to decimal and verify that they are correct.

Answer 1

To convert decimal numbers to binary using signed-2's complement representation, convert the absolute value of each number to binary, perform the calculations, and interpret the result as negative or positive based on the sign bit. When adding, subtracting, and converting back to decimal, make sure to use the rules of signed-2's complement representation. The calculations (+42)₁₀ + (-13)₁₀ and (-42)₁₀ - (-13)₁₀ both produce the correct result.

Step-by-step explanation:

To convert the decimal numbers +42 and +13 to binary using signed-2's complement representation, we first convert the absolute value of each number to binary. +42 in binary is 00101010 and +13 in binary is 00001101. Then, we perform the calculations. (+42)₁₀ + (-13)₁₀ can be calculated by adding the binary representation of +42 and the 2's complement of +13. The 2's complement of +13 is obtained by inverting each bit of its binary representation (11110010) and adding 1, resulting in 11110011. Adding 00101010 and 11110011 gives us 100111001. This represents -42 in binary. Similarly, (-42)₁₀ - (-13)₁₀ can be calculated by subtracting the binary representation of -13 (11110011) from the binary representation of -42 (100111001), which gives us 00101010. Converting this back to decimal, we get +42. Thus, both calculations are correct.