Solve the following operation with binary numbers. Check your answer by converting the binary numbers to base 10 numbers, doing the operation on the base 10 numbers, and converting the answer back to base 2.
1012
- 112
To subtract binary numbers, we can use the borrowing method, similar to subtracting decimal numbers.
Starting from the rightmost column, we subtract the corresponding digits of the two binary numbers. If the result is 0 or positive, we write the result in the answer. If the result is negative, we borrow 1 from the next higher column. After borrowing, we add 2 to the digit from which we borrowed and subtract the second digit. This process continues until we subtract all the digits.
1012
- 112
-------
101
To convert the binary numbers to base 10 numbers, we can use the place value method. The rightmost digit has a place value of 2^0 = 1, and the place value doubles as we move to the left.
1012 = (1 * 2^2) + (0 * 2^1) + (1 * 2^0) = 4 + 0 + 1 = 5
112 = (1 * 2^2) + (1 * 2^1) + (0 * 2^0) = 4 + 2 + 0 = 6
When we subtract the base 10 numbers, we get 5 - 6 = -1.
To convert the answer back to base 2, we can divide the base 10 number by 2 and keep track of the remainders. The remainder of each division is a digit in the binary number, starting from the least significant digit.
-1 ÷ 2 = -0 remainder 1
Since the division results in a negative number, we need to adjust the sign of the remainder by adding 2^1 = 2 to it.
-0 remainder 1 ==> 2 remainder 1
Therefore, the answer in binary is 12.
To solve the subtraction operation with binary numbers, we need to align the numbers vertically and perform the subtraction bit by bit.
First, let's align the numbers:
1012
- 112
Next, start with the rightmost bit and subtract:
1012
- 112
------
0010
So, the binary subtraction operation 1012 - 112 equals 0010.
To check our answer, let's convert the binary numbers to base 10:
1012 in base 2 is equal to 1 * 2^2 + 0 * 2^1 + 1 * 2^0 = 4 + 0 + 1 = 5 in base 10.
112 in base 2 is equal to 1 * 2^2 + 1 * 2^1 + 2^0 = 4 + 2 + 1 = 7 in base 10.
Now, let's perform the subtraction in base 10:
5 - 7 = -2
Finally, let's convert the answer back to base 2. Since the result is negative, we need to represent it in binary using two's complement. In two's complement, we flip the bits and add 1:
-2 in base 10 is represented as 1110 in base 2.
Therefore, our answer is 1110 in base 2.
To summarize:
Binary subtraction operation:
1012 - 112 = 0010
Conversion to base 10:
1012 = 5 in base 10
112 = 7 in base 10
Subtraction in base 10:
5 - 7 = -2
Conversion back to base 2 (using two's complement):
-2 in base 10 = 1110 in base 2