# Binary and Number Systems #

Having a good idea of how binary works is the first step to understanding how IP addresses really operate.

Here, we’ll go over some core concepts of binary, how it compares to our usual numbering system (decimal), and some easy ways to understand how it works.

First off, let’s go over how decimal works deep-down.

## Decimal #

In our regular counting system, decimal, we use the digits `0-9` to represent the numbers zero to nine.

### Counting #

Let’s use this three-digit counter as an example. On the right, we’ll state the actual value that’s represented by the digits on the left.

``````000     - zero
``````

To count up, we increment (go up by one each time) from `0` to `1`, `1` to `2`, etc. Let’s count up to nine:

``````000     - zero
001     - one
002     - two
003     - three
004     - four
005     - five
006     - six
007     - seven
008     - eight
009     - nine
``````

However, there are no digits after `9`. To continue counting up to ten, we need to increment the next column and reset the rightmost one.

Let’s count up from 9:

``````009     - nine
010     - ten
``````

So we changed the middle digit from `0` to `1`, and reset the right digit from `9` back to `0`.

To continue counting up from ten, we do exactly the same with the rightmost number:

``````010     - ten
011     - eleven
012     - twelve
013     - thirteen
014     - fourteen
015     - fifteen
016     - sixteen
017     - seventeen
018     - eighteen
019     - nineteen
``````

And to count up to twenty, we do the same:

``````019     - nineteen
020     - twenty
``````

This is how decimal works, and specifically how counting works in decimal

Decimal can also be called “base 10” because it uses ten digits to represent values (`0`, `1`, `2`, `3`, `4`, `5`, `6`, `7`, `8`, and `9`).

## Binary #

Binary is similar to decimal, except that it uses a base-2 system (just the digits `0` and `1`).

Let’s keep the same three-digit counter, starting from zero:

``````000     - zero
``````

Counting up to one is simple:

``````000     - zero
001     - one
``````

However, going up to two requires us to increment the next digit and reset the rightmost one. In other words, because there aren’t any digits after `1`, we need to go to the next column.

``````001     - one
010     - two
``````

Let’s count up as far as we can:

``````010     - two
011     - three
100     - four
101     - five
110     - six
111     - seven
``````

Computers use binary because it meshes well with how they work deep down. Computers are based on electrical signals, and those signals can either be high (`1`), or low (`0`). Because of this, using a number system with only two values for each column makes sense.

### Bits and Bytes #

Sometimes when using binary, you’ll see the term “bits”. Bit simply stands for binary digit, and it used as a shorthand when talking about computers in particular. Up until now, our counter has used three bits, and four/eight bits are typical sizes used a lot when working with computers.

Collections of different numbers of bits can also have names. For instance, eight bits are called a byte, and four bits are called a nibble. As an example:

``````bit    =  0
nibble =  0000
byte   =  0000 0000
``````

When working with binary, bytes are sometimes written as groups of four bits. This just makes it easier to read, and easy to convert to other number systems (as you’ll see later).

## Base 10 vs Base 2 #

With base 10 (decimal), if we count all the way up, 3 digits lets us represent these values:

``````*** decimal ***
000     - zero
...
999     - nine-hundred and ninety-nine
``````

Or in other words, one thousand separate values.

With base 2 (binary), if we do the same we can represent these values:

``````*** binary ***
000     - zero
...
111     - seven
``````

Or, eight separate values.

In a single column, you can represent the base number of digits that number system has (two for binary, ten for decimal). With each new column, the number of values you can represent is simply multiplied by the base. For example:

``````*** decimal ***
9     one digit, ten values
99    two digits, one hundred values
999   three digits, one thousand values

*** binary ***
1     one digit, two values
11    two digits, four values
111   three digits, eight values
``````

This is useful to keep in mind when working with binary, particularly when working out network addresses.

## Converting #

To get the value from a binary number, you can count up how large each column is based on whether it’s a `0` or a `1`.

As an example, let’s get the value from the binary number `1011`. The top line shows the value each column represents.

``````1011 = 1 0 1 1 (split into columns)

values:  8 | 4 | 2 | 1
binary:  1 | 0 | 1 | 1

8 + 0 + 2 + 1  =  eleven
``````

So the binary number `1011` represents eleven, or:

``````1011 in base 2  =  11 in base 10
``````

Being able to convert binary numbers to real values is very useful, and in the next section we’ll be going over IP addresses and subnets.

Try going through some of these examples on a piece of paper, to see whether you’re able to convert them to the proper values. Once you’ve finished, try cross-checking your answers with an online converter or Google:

``````1001 = ?      00101101 = ?      11111100 = ?      00010000 = ?
0110 = ?      11011101 = ?      00011111 = ?      00110001 = ?
``````

## Base 16 #

There is also a similar number system that’s often used in computing – “base 16”, or hexidecimal. It uses these 16 digits:

``````0 1 2 3 4 5 6 7 8 9 A B C D E F
``````

The useful thing with base 16 is that each column can represent 16 separate values. Similarly, four columns of binary can also represent 16 separate values. Because of this, hexidecimal can easily – and very compactly – represent binary digits.

Here’s a chart of values represented in Binary, Hex, and Decimal, which can be useful when converting between them: ## Overall #

• Decimal uses ten digits (`0-9`) in each column to represent values, and because of this it’s called “base 10”.
• Binary uses two digits (`0,1`) in each column to represent values, and it’s called “base 2”.
• Hexidecimal uses sixteen digits (`0-9,A-F`) in each column to represent values, and it’s called “base 16”.
• Counting, based on columns, work the same way in decimal as it does in binary (as it also does in hexidecimal).