Why does two's complement work?

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By HER Team | Thursday, 09 Sep 2021, 10:11 IST

You want to represent signed integers into the computer,two's complement is the way to do it.

The thought is to eliminate the subtraction operation by using an addition operation that replaces it. That’s how you can use both operations of addition and subtraction in same circuit.

This process is used to reduce the number of gates which will than reduce the size, power and eventually cost of the system.

The complement way is a revolutionary concept of representation that helps eliminate old approach to deal with decimal numbers.

How to find two's complement

A binary number can be converted into two’s complement with the help of simple algorithm.

All you have to do is to invert the binary number and add 1 to its least significant bit to get desired result.

Binary system is formed by two symbols 0 and 1 . Here only two results are possible.

There are two types of complement that can be given to binary number:

One's complement
Two’s complement

One complement is relatively simple in which you have to simply invert the given binary number and you will get One’s complement as a result.

We have described two’s complement above now we are going to give some examples to make it easy to understand.

Example number 1

What is two’s complement of the binary number 10101110?

Now all you have to do is invert every bit in the binary number that will be 01010001.

Now you have to add 1 to its least significant bit.

01010001+1 equals 01010010.

There you have your answer:

Example number 2

What is two’s complement for binary number 10001.001?

Now you have to apply the same rule that says invert every bit of a binary number so you will get 01110.110 than all you have to do is add 1 to its least significant bit.

So 01110.110+1 equals 01110.111

Here you have your answer.

Now in the table below we will give you both one’s and two’s complement for 3-bit binary numbers.

Binary number	One’s complement	Two’s complement
111	000	001
110	001	010
101	010	011
100	011	100
000	000	000
001	110	111
010	101	110
011	100	101

Usefulness of two’s complement in binary number system

Two’s complement is a very useful tool to represent signed binary number. A lot of arithmetic operations like addition; subtraction etc also uses two’s complement system.

Signed binary number with two’s complement.

We have two types of number:

Positive
Negative

Positive number representation is pretty basic as simple binary numbers but Negative numbers use two’s complement for its representation. First you have to write the number that is with the positive sign and than you will take two’s complement of it to get the required answer.

For example, the representation of +5 and -5 are as follows:

+5 would be presented in sign magnitude method

While – 5 would take two steps.

Take two’s complement of +5 that will be 1 1011 the 1^st digit from left is always 1 when it’s the negative figure.

Two complement calculators

Not every one has the understanding or time to do all these complex mathematical calculations as they are kind boggling as well as very much time consuming.

One wrong calculation and every thing results in false way that means you have to do it all again and all the time you have spent doing it is a total waste of precious time and effort.

You unfortunately have to redo every single calculation again which is just too much work in 1^st place and still you can’t insure if it’s absolutely right or not.

So, in order to make two’s complement system easy for you we bring you an online two’s complement calculator. This not only make sure that the result is absolutely correct but also saves you a lot of time and hard mental effort as not everyone is master at it.