In my experience of teaching networking many students struggle with IP addresses because they lack an basic understanding of binary numbers.

An understanding of binary numbers, and how to convert between binary and decimal is essential for anyone involved in computers and networking.

### What You will Learn

- Basics of Number Bases-Base 10, Base 2 and Base 16
- How to convert binary numbers to decimal and vice versa
- How to convert binary numbers to Hexadecimal and vice versa,
- How to convert Hexadecimal to decimal and vice versa,

### Overview of Base 10 or Decimal Numbers

Before we talk about binary numbers we will look in more detail at our normal decimal numbering system.

The principals are the same for all numbering systems, and they are easier to learn using a system that you are more familiar with.

Firstly our decimal system uses 10 as a **base** and the numbers range from **0 to 9**

Let’s look at a few numbers

We start with the three digit number **129** (one hundred and twenty nine).

This made up of **100 +20 +9 =129**

If we look at the diagram below we see that as we move from right to left the columns increase by a factor of 10.

The 2 in the second column isn’t 2 but is 2*10=20 and the 1 in the third column isn’t 1 but 1*10*10=100.

means 10 raised to the power of 0. This is equal to 1 and represents our units column.

The short table below shows a few more entries using the power notation.

When writing decimal numbers we rarely write column values above the numbers as we already know what they are so we simply write:

**129** and not

I have introduced the power notation because it is fundamental to understanding binary numbers.

The minimum number possible with three digits is** 000** and the maximum is** 999. **For numbers larger than **999** we need a 4th column which with be the 1000s column.

### Binary Numbers

Binary numbers are base 2 numbers, and have only two values – **0 and 1.**

If we look at a binary number like 101, then we can again assign column values as we did with our decimal number, but this time we use 2, and not 10 as the base.

So binary **101** binary has 1 in the units column,0 in the 2s column and 1 in the 4s column.

Again if we work our way from right to left then:

The 1 is a 1 as it is in the units column but the next 1 is not 1 but 1*4=4

Binary numbers use base 2 and so the columns are

### Binary To Decimal Conversion

Let’s look at a few binary numbers and convert them to decimal

We start with the three digit binary number** 101** (see image above

The number can be converted to decimal by multiplying out as follows:

1*1 + 0*2 + 1*4 = 5

The maximum value we can have with three binary digits is 111 = decimal 7 calculated as follows-

1*1 + 1*2 + 1*4

### More Examples:

1011 binary = 1*1+1*2+0*4+1*8=11

1111 binary = 1*1+1*2+1*4+1*8=15

### Try It Yourself

1001 binary = ?

1100 binary = ?

### Converting from Decimal to Binary

How do you convert a decimal number to a binary number.

Example what is** decimal 10** in binary.

The way I do it is by using the following table of 2 multiples.

1,2,4,8,16,32 etc

Here’s an handy chart

Now we just find the nearest number to 10 which is 8 =2^{3}

now 10-8=2 =2^{1.}

so we have 1 eight , No fours, 1 two, No units = 1010.

**Example 2**: Decimal 13 to binary

1 eight , 1 four, 0 two, 1 units = 1101.

**Example 3**: Decimal 7 to binary

0 eight , 1 four, 1 two, 1 units = 0111.

**Answers to try it yourself questions**

1001 binary = 9

1100 binary = 12

### Video

### Bytes and Octets and Hexadecimal Numbers

In computers and networking 8 bit numbers are common.

An 8 bit number is known as an **octet**, and also more commonly it is called a **byte**. See Wiki for details.

### Binary to Decimal and Decimal to Binary Conversion

An 8 bit binary number can represent a maximum of decimal **255**= binary **11111111**.

Calculated as follows:

1*128 +1*64+1*32+1*16+1*8+1*4+1*2+1+1 = **decimal 255**

Here is another 8 bit binary number –**01101011. **

To convert it to decimal we write the number with the column numbers above, as follows:

if we convert our columns to decimal equivalents using the following chart.

then

**01101011** = 1*1 +1*2+0*4=1*8+0*16=1*32+1*64+0*128

=64+32+8+2+1= 107

**Notice** it consists purely of 1’s and 0’s.

To convert this number into decimal we need to understand what each 1 represents.

If we write the **column value**s above the numbers then it becomes easy to convert the binary number to decimal.

We can convert a decimal number into binary by following a similar procedure as follows:

The procedure is to subtract the number with largest power of two from the decimal number. So if we start with a simple example of decimal 5.

The** largest power of two numbe**r that we can subtract is **4** which is

2^{2}.

So 5-4 =1 =2^{0} so our binary number is **101= 2 ^{2}+ 2^{0}**.

Let’s do another example and convert decimal 14 to binary.

**14**= 8+4+2=**2 ^{3}+ 2^{2}+ 2^{1 }** =1110

A final larger example convert decimal 200 to binary

**200**=128+64+8=**2 ^{7}+ 2^{6}+ 2^{3 }** =

**11001000**

Once you are happy with the process then you can use a caculator like the one on windows.

This converts binary numbers to decimal

and this converts decimal numbers to binary

### Understanding Hexadecimal numbers

A hexadecimal number (base 16) requires **4 bits** and and has a maximum value of **15**. It uses the symbols **0-9,A,B,C,D,E,F**.

They are represented in binary form as follows:

0000=0

0001=1

0010=2

0011=3

0100=4

..

1010=A

1011=B

1100=C

1101=D

1110=E

1111=F

**A byte** (8 bits) can be represented as **two hexadecimal numbers.**

so

**FF**=binary 11111111 and decimal 255

** F0**=11110000 binary and decimal 240

**Questions**

What is

- FE in binary
- 01001011 in hexadecimal
- 01001011 in decimal
- 200 in binary

#### Answers

- 254
- 8B
- 75
- 11001000

**Resources and Related Articles:**