Most of our daily tasks are now digitalized and it happened after the invention of computers. Computers use a binary number system to understand the input and output. We interact with computers with the help of binary numbers 0 and 1 without even knowing. When we click on a specific folder on our computer or touch an icon on our mobile phone, it all binary happening in the background.

The issue is that we frequently require operations that are not clearly defined on universal rings. For example, we may wish to divide, compute remainders, take square roots, and compare two numbers to determine which is “larger.” We’re particularly curious whether a given number is bigger or less than zero. That’s where two’s complement comes into the picture.

If these processes are to be efficient, they must be straightforward to implement in hardware, hence the simpler the better.

## How does a computer represent numbers?

Remember that computers represent small negative numbers with enormously huge numbers. Numbers greater than 2m-1 have their highest bit set to one when using m bits for integers. This provides us the concept of using this bit to identify the sign. The signed integer range is now [-N/2, N/2-1]: half are negative, half are nonnegative. To determine whether a number is negative, simply examine its most important bit. If you used a different range, such as [-100, N-101], you would have to check much more than a single bit to determine whether a value is less than zero.

According to Wikipedia, there are several methods to represent signed integers on computers, which change depending on how one derives the representation of -x from x. Let’s look at how to change the sign of a number in this system.

Let us represent the number produced by flipping all bits of x by x: where x contains a zero, x contains a one, and when x contains a one, x contains a zero. This signifies that x + x will be made up entirely of ones. However, the number with all ones in its binary form is 2m-1, therefore x + x = 2m-1. From this, we get -x = x-2m+1. We have -x = x+1 since we are working in mod 2m. As a result, our derivation of signed integers corresponds to the usual two’s complement form.

## Role of Two’s Complement

According to the argument stated here, the only logical way to represent signed numbers is 2’s complement. Every other technique fails in some way: For example, 1’s complement contains two unique representations for 0 and necessitates separate circuits for addition and subtraction. Why would anyone bother with a different method of portraying numbers?

According to Wikipedia, the rationale is usability: Because early programmers had to deal with machine code and binary memory dumps all day, perceiving negative integers as sign-magnitude, or at the very least as 1’s complement, would have made their jobs a little simpler.There are several online tools that can be used to find 2’s complement of a binary number. One of them is 2’s complement calculator by allmath.com.

In the end, the 2’s complement approach triumphed because it was easier and less expensive to implement in hardware. Usability, on the other hand, has become a non-issue as programming tools have improved.

## Finding two’s complement of a binary number

Before we move onto calculating two’s complement, check out this two’s complement calculator. This tool will be very helpful if you quickly want to get the 2’s complement of a binary or decimal number.

Calculating 2’s complement of decimal or binary number requires binary addition. Most of the time, we convert a decimal number into a binary number. Then, we have to invert all numbers i.e., 1 to 0 and 0 to 1. After inverting the numbers, you have to add 1 in the inverted values to get a 2s complement.

Let us explain the manual process to find 2s complement with examples.

### Two’s complement of a binary number

- Assume the binary number (00110010)
_{2}. Let’s find 2s complement of this number.

Invert all zeros (0) into ones (1) and all ones (1) into zeros (0).

(00110010)_{2}= (11001101)_{2}

Add 1 into (1 1 0 0 1 1 0 1)_{2}.

(1 1 0 0 1 1 0 1)_{2}+ 1 = 00110011

00110011is two’s complement of (0 0 1 1 0 0 1 0)_{2}.

2. Another example will clear the ambiguities if you have any. Assume a binary number. In this case, the binary number will be,

(1 1 0 1 1 1 0 0)_{2}

Invert all zeros (0) into ones (1) and all ones (1) into zeros (0).

(1 1 0 1 1 1 0 0)_{2} = (0 0 1 0 0 0 1 1)_{2}

Add 1 into (0 0 1 0 0 0 1 1)_{2}.

(0 0 1 0 0 0 1 1)_{2 }+ 1 = 0010 0100

2’s complement =0010 0100

### Two’s complement of a decimal number

Suppose, we have a decimal number 30 ad we need to convert it into 2’s complement.

**30**

First of all, convert this decimal number into a binary number using a decimal to binary converter.

(30)_{10} = (00011110)_{2}

Now invert all zeros (0) into ones (1) and all ones (1) into zeros (0).

(00011110)_{2} = (11100001)_{2}

Add 1 into 11100001.

11100001+ 1 = 11100010

11100010 is two’s complement of 30.

## Conclusion

Hope this post constructively helped you. If you are a student, you can practice more using sample exercises. If you are a programmer or working in the computer hardware industry, the mentioned resources could be very handy for you.