Two's Complement Calculator

How to work with negative numbers in binary? - 2's complement representation

In the binary system, all numbers are a combination of two digits, 0 or 1. Each digit corresponds to a successive power of 2, starting on the right.

For example, 12 in binary is 1100, as 12 = 8 + 4 = 1*2³ + 1*2² + 0*2¹ + 0*2⁰ (using scientific notation). An extended version of the binary system is the hexadecimal system (which uses base 16 instead of base 2). The latter is frequency used in many computer softwares and systems.

Learning about binary leads to many natural questions arising - what about negative numbers in the binary system? Or how do I subtract binary numbers? As we can only use 1 to show that something is present, or 0 to mean that there is a lack of that thing, there are two main approaches:

1. Two's complement representation, or, in other words, signed notation - the first bit tells about the sign. The convention is that a number with a leading 1 is negative, while a leading 0 denotes a positive value. In an 8-bit representation, we can write any number from -128 to 127. The name comes from the fact that a negative number is a two's complement of a positive one.

2. Unsigned notation - a representation that supports only positive values. Its advantage over the signed one is that, within the same 8-bit system, we can get any number from 0 up to 255.

As long as we need to add or multiply positive numbers, the unsigned notation is good enough. But, usually, the more practical solution is to work with negative numbers as well. A useful thing about the 2's complement representation is that subtraction is equivalent to an addition of a negative number, which we can handle.

Turning two's complement to decimal

Our 2's complement calculator can also work the other way around - converting any two's complement to its decimal value. Let's try to convert 1011 1011, a signed binary, to decimal. There are two useful methods that help you find the outcome:

1. Convert this signed binary into a decimal, like normal, but multiply the leading digit by -1 instead of 1. Starting from the right:

   decimal = 1*2⁰ + 1*2¹ + 0*2² + 1*2³ + 1*2⁴ + 1*2⁵ + 0*2⁶ - 1*2⁷

   decimal = 1 + 2 + 8 + 16 + 32 - 128 = -69

2. We can see that the first digit is 1, so our number is negative. First, find its two's complement, then convert the value to a decimal, and come back to the original value:

   1. Reverse digits, 1011 1011 → 0100 0100.
   2. Add a unity, 0100 0100 + 1 = 0100 0101.
   3. Convert to a decimal (starting from the right), decimal = 1*2⁰ + 0*2¹ + 1*2² + 0*2³ + 0*2⁴ + 0*2⁵ + 1*2⁶ + 0*2⁷.
   4. decimal = 1 + 4 + 64 = 69.
   5. As 69 is the absolute value of our initial (negative) binary, add a minus sign in front of it.
   6. 1011 1011 is -69 in two's complement binary notation.