Question 1: Convert 2 base-10 numbers - (106 and -2) - to 8-bit binary numbers. Then, add the two binary numbers. Show your arithmetic.
Question 2: Convert 2 base-10 numbers - (106 and 90) - to 8-bit binary numbers. Then subtract the second number from the first number. Show your arithmetic
Question 3: Is this an efficient way to add/subtract? Are there any shortcuts?
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Answer 1:
To convert the base-10 number 106 to an 8-bit binary number, we need to divide the number by 2 repeatedly until we reach 0 and record the remainders.
106 divided by 2 is equal to 53 with a remainder of 0.
53 divided by 2 is equal to 26 with a remainder of 1.
26 divided by 2 is equal to 13 with a remainder of 0.
13 divided by 2 is equal to 6 with a remainder of 1.
6 divided by 2 is equal to 3 with a remainder of 0.
3 divided by 2 is equal to 1 with a remainder of 1.
1 divided by 2 is equal to 0 with a remainder of 1.
Reading the remainders in reverse order, the binary representation of 106 in 8 bits is 01101010.
To convert the base-10 number -2 to an 8-bit binary number, we take the absolute value of 2 (which is 2) and represent it in binary form: 00000010. However, since it is a negative number, we have to apply the two's complement. Inverting each bit and adding 1 gives us 11111110.
To add the two binary numbers, we align them and perform the addition:
01101010
+ 11111110
-----------
1 01111000
The result is 101111000 in binary.
Answer 2:
To convert the base-10 number 106 to an 8-bit binary number, we use the same process as before, resulting in 01101010.
To convert the base-10 number 90 to an 8-bit binary number, we divide repeatedly:
90 divided by 2 is equal to 45 with a remainder of 0.
45 divided by 2 is equal to 22 with a remainder of 1.
22 divided by 2 is equal to 11 with a remainder of 0.
11 divided by 2 is equal to 5 with a remainder of 1.
5 divided by 2 is equal to 2 with a remainder of 1.
2 divided by 2 is equal to 1 with a remainder of 0.
1 divided by 2 is equal to 0 with a remainder of 1.
Reading the remainders in reverse order, the binary representation of 90 in 8 bits is 01011010.
To subtract the second number from the first number, we perform the subtraction:
01101010
- 01011010
-----------
00010000
The result is 00010000 in binary.
Answer 3:
This method is one way to add/subtract binary numbers, and it is commonly used. However, for more efficient addition and subtraction, there are shortcuts available.
For addition, if the carry out of the most significant bit (MSB) is '1', it indicates overflow. In this case, the result is not represented accurately in 8 bits, and the use of a larger bit representation may be required.
For subtraction, we can make use of two's complement. To subtract a number, we can calculate the two's complement of the second number (subtrahend) and add it to the first number (minuend).
These shortcuts help reduce the amount of computation required, but the basic method of converting to binary and performing addition/subtraction is still used as the foundation.
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