Two's Compliment

Introduction

In National 5, we learned that all data within a computer is represented using a system called binary.

The binary system that we learned in National 5 has 1 major flaw! It can't represent negative numbers!

How do we store -1 in binary?

For this we use a system called two's compliment.

Two's Compliment

At National 5 level we learned that we use place values to denote the value of each binary digit.

At Higher, we use the two's compliment system.

In this system, the leftmost place value becomes a negative number.

For example, our place values for an 8-bit binary number would be as follows:

At Higher, we are not limited to 8-bit numbers and therefore the negative number might change! However, it is always the leftmost place value!

For example,

4 bits

6 bits

8 bits

12 bits

etc...

Binary to Denary

We can convert any two's compliment binary number into its decimal equivalent.

For example,

Convert the two's compliment binary number 1001 1101 to denary.

Denary to Binary

If we wish to take a denary number and convert it to two's compliment binary we can follow one of two methods.

Method 1 is similar to one of the methods used in National 5, we simply look for the numbers that add together to give us our target number and add 1's below them, putting zero's everywhere else.

Method 1 can be tricky to do as we are using negative numbers. Method 2 will take slightly longer but is easier once you learn the method!

Denary to Binary - Method 1

For example, convert the number -19 to an 8-bit two's compliment binary number

We first start by writing our place values.

We then look at our place values and try to find numbers which add to give use our target number, in this case -19. You can only use each number once!

We can see that -128, 64, 32, 8, 4 and 1 add to give us -19. We put 1's under these numbers.

We can then put 0's under all other numbers.

We can then put 0's under all other numbers.

Therefore:

19 -> 0001 0011

Denary to Binary - Method 2

A lot of people find method 1 quite difficult because of the negative number used in the calculation. Instead we can use method 2!

For example, if we wanted to convert the denary number -25 to binary, it would be quite difficult to intuatively find numbers that add up to give -25.

For example, if we wanted to convert the denary number -25 to binary, it would be quite difficult to intuatively find numbers that add up to give -25.

In this case, method 2 should be used. This method will work with any two's compliment denary to binary conversion.

Method

This method involves 3 steps:

1. Write the positive representation of the number
2. Flip the bits
3. Add 1

Example

Convert the denary number -25 to an 8-bit two's compliment binary number.

We first begin by writing the binary place values.

Step 1

We start by writing the the positive representation of the number. In our case this is 25.

Step 2

We then do what is known as 'flipping the bits'. This means we turn all of the 0's to 1's, and all the 1's to 0's.

Our current number is:

If we flip the bits, this becomes:

Notice that all of the 0's have become 1's, and all of the 1's have become 0's.

Step 3

Our 3rd step is to add 1. This is where we have to do a little bit of maths in binary.

If you know how to add in binary, you can skip to Step 3 - Continued.

How to do Maths in Binary

Binary maths works much like normal maths that you use everyday!

Let's take 0 + 0, in binary this is still the same.

0 + 0 = 0

What about 0 + 1? Well this is the same too!

0 + 1 = 1

What about 1 + 1? Well this would usually be 2, but 2 isn't written with a 2 in binary, it is written as 10!

1 + 1 = 10

Let's try one more, what about 11 + 1 in binary(3 + 1 in denary)?

11 + 1 = 100

We all know 3 + 1 is 4 (or 11 + 1 is 100 if written in binary), but we need to understand why!

Lets's think in denary for a minute. When we do 9 add 1, we write a 0 and carry the one to the column to the left resulting in 10.

We do the same in binary! If we do 1 + 1 it results in a 0 and the 1 is carried to the column to the left.

If there is also a 1 in this column, we will have to write a 0 and carry a 1 to the column to the left again.

For Example, lets do 0111 + 1:

1 + 1 is 0 (carry 1, so we write 0 and try to add 1 to the next column

1 + 1 is 0 (carry 1, so we write 0 and try to add 1 to the next column

Again, 1 + 1 is 0 (carry 1, so we write 0 and try to add 1 to the next column

This time we have 0 + 1, we know that this is simply 1.

Step 3 - Continued

So far we have:

• Written the positive representation of the number
• Flipped the bits

Our current number is:

we now have to add 1.

This gives us:

Therefore, our final answer is:

-25 -> 1110 0111

Denary to Binary - Method 2 (Another Example)

Convert the denary number -32 to two's compliment binary.

We will follow the 3 steps:

1. Write the positive representation of the number
2. Flip the bits
3. Add 1

Step 1

We start by writing the the positive representation of the number. In our case this is 32.

Step 2

We then 'flip the bits'. This means we turn all of the 0's to 1's, and all the 1's to 0's.

Our current number is:

If we flip the bits, this becomes:

Again, notice that all of the 0's have become 1's, and all of the 1's have become 0's.

Step 3

Our last step is to add 1.

This time we will have to carry the 1 as we add it to each column.:

This gives us:

Therefore, our final answer is:

-32 -> 1110 0000