How to Convert Signed Binary to Decimal
A binary number system represents numeric values with two symbols. Computers are easily able to implement this system by using circuits that are either "off" or "on" to store the two values of a binary bit. A binary system may also establish rules which allow it to use one of the binary values to represent the sign of the value rather than a binary digit. These numbers can then be easily converted to the decimal number system that humans use.
Instructions
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1
Define the pure binary system that computers use. We use the symbols "0" and "1" to represent binary values. The rightmost digit represents 2^0 and each digit to the left represents a power of 2 that is 1 greater than the digit to its right.
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2
Look at an example of the system described in Step 1. The binary numeral 1101 is evaluated as follows: 1x2^3 + 1x2^2 + 0x2^1 + 1x2^0 = 8 + 4 + 0 + 1 = 13. Therefore, the binary value 1101 is equal to the decimal value 13.
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3
Use a binary system that can only express positive numbers. For example, a system that uses 8 binary digits could express the numbers from 0 to ( 2^8 - 1) inclusive if all 8 digit represented magnitude.
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4
Establish a rule that allows a binary system to express negative numbers. We now modify the system defined in Step 3 such that the leftmost of the 8 digits will represent the sign of the number instead of its magnitude such that a 0 indicates a positive number and a 1 indicates a negative number. Since we already represent 0 with 00000000, we will use 10000000 to represent -(2^7). Our set of 8 bits now represents numbers from - (2^7) to (2^7 - 1) inclusive instead of 0 to ( 2^8 - 1) inclusive as in Step 3.
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5
Convert the signed binary numbers into decimal numbers. Calculate the rightmost 7 digits using the scheme described in Step 1 and determine the sign as given in Step 4. For example, the binary value 10001101 is the decimal -13. The rightmost 7 digits evaluate to 13 as in step 2. Since the leftmost digit is 1, we then have -13.
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