Two’s Complement

Introduction

In this post, I want to talk about two’s complement.

You might wonder, “Is it necessary to write about such a simple topic?” But even after reading many articles, there were parts I personally didn’t fully understand.

Here are the questions I had:

What is two’s complement? What is one’s complement? What is a complement?

Of course, the methods for finding one’s and two’s complement are widely known. For example:

Finding Complements

Given the binary number 0110:

One’s Complement

The one’s complement of 0110 is obtained by flipping each bit.

That is, 1001.

Two’s Complement

Add 1 to the one’s complement.

That is, 1010.

But what exactly is a complement, and why do we call the results above one’s and two’s complement? Let’s resolve this question in this post.

What is a Complement?

Let’s first understand what a complement is. That will help us understand what one’s and two’s complement mean.

Definition

A complement is a number that “completes” another number to a certain value.

In Korean, it’s written as:

• 補數 (보수): to help (보), number (수)

In English:

• complement: to complete or supplement

Example

• The 10’s complement of 1 is 9.
  • How much do you need to add to 1 to get 10?
  • It’s 9, because 1 + 9 = 10.
• The 15’s complement of 4 is 11.
  • How much do you need to add to 4 to get 15?
  • It’s 11, because 4 + 11 = 15.

Revisiting One’s Complement

Let’s look at one’s complement in binary again.

The one’s complement of 0110 is, as above, 1001.

So, according to the earlier examples, shouldn’t 0110 + 1001 = 1?

But if you add them, you get 1111!

This doesn’t match the earlier definition. This was confusing, but it turns out there was a misunderstanding about the term “one’s complement”…

Ones’ Complement (Plural)

How do you say “one’s complement” in English?

ones’ complement

In English, the possessive of the plural “ones” is used, not the singular “one”.

From the English Wikipedia on ones’ complement:

“ones’ complement” (note this is possessive of the plural “ones”, not of a singular “one”)

So, for a 4-bit number, the string of ones is 1111.

Therefore, the ones’ complement of 0110 is 1001.

And 0110 + 1001 = 1111.

That’s why the method for finding the ones’ complement is to flip all the bits of the number.

Revisiting Two’s Complement

So what about two’s complement? Why does adding 1 to the ones’ complement give the two’s complement?

If you add 1 to the ones’ complement of 0110 (1001), you get 1010.

And 0110 + 1010 = 10000.

If it’s two’s complement, shouldn’t the result be 2? But for 4 bits, you get 16 (which is 2^4).

Two’s Complement as a Radix Complement

From the English Wikipedia on two’s complement:

Two’s complement is an example of a radix complement. The ’two’ in the name refers to the term which, expanded fully in an N-bit system, is actually “two to the power of N” - 2^N

So, two’s complement is a type of radix complement, and the “two” refers to 2^N in an N-bit system.

Given 0110 is 4 bits, N = 4, so two’s complement means the complement to 2^4.

Therefore, the complement to 2^4 is 1010. 0110 + 1010 = 10000 = 16 (in decimal).

That’s why adding 1 to the ones’ complement gives the two’s complement.

Negative Numbers in Computers

Now, it’s easier to understand why two’s complement is used to represent negative numbers in computers. There are some constraints:

1. Computers must represent negative numbers using only bits (0 and 1).
2. Computers only perform addition.
3. Adding a positive number and its negative counterpart should result in 0 (e.g., 10 + (-10) = 0).

Let’s use the earlier example, 0110:

If you add the value represented by xxxx, you should get 0000. But how can adding to 0110 result in zero?

The key is that computers can discard the carry bit.

If you treat 0000 as 10000, you can determine what xxxx should be.

Here, the carry (1) is discarded.

Therefore, the two’s complement of 0110 (which is 1010) is used to represent the negative value.

So, adding these two gives 0000—which is zero!