##### Binary Number System

**Summary**

This topic calculate numbers between decimal and binary systems. Start learning CCNA 200-301 for free right now!!

Table of Contents

## Binary and IPv4 Addresses

IPv4 addresses begin as binary, a series of only 1s and 0s. These are difficult to manage, so network administrators must convert them to decimal. This topic shows you a few ways to do this.

Binary is a numbering system that consists of the digits 0 and 1 called bits. In contrast, the decimal numbering system consists of 10 digits consisting of the digits 0 – 9.

Binary is important for us to understand because hosts, servers, and network devices use binary addressing. Specifically, they use binary IPv4 addresses, as shown in the figure, to identify each other.

Each address consists of a string of 32 bits, divided into four sections called octets. Each octet contains 8 bits (or 1 byte) separated with a dot. For example, PC1 in the figure is assigned IPv4 address 11000000.10101000.00001010.00001010. Its default gateway address would be that of R1 Gigabit Ethernet interface 11000000.10101000.00001010.00000001.

Binary works well with hosts and network devices. However, it is very challenging for humans to work with.

For ease of use by people, IPv4 addresses are commonly expressed in dotted decimal notation. PC1 is assigned the IPv4 address 192.168.10.10, and its default gateway address is 192.168.10.1, as shown in the figure.

For a solid understanding of network addressing, it is necessary to know binary addressing and gain practical skills converting between binary and dotted decimal IPv4 addresses. This section will cover how to convert between base two (binary) and base 10 (decimal) numbering systems.

## Video – Converting Between Binary and Decimal Numbering Systems

Click Play in the figure for a video demonstrating how to convert between binary and decimal numbering systems.

## Binary Positional Notation

Learning to convert binary to decimal requires an understanding of positional notation. Positional notation means that a digit represents different values depending on the “position” the digit occupies in the sequence of numbers. You already know the most common numbering system, the decimal (base 10) notation system.

The decimal positional notation system operates as described in the table.

Radix | 10 | 10 | 10 | 10 |
---|---|---|---|---|

Position in Number | 3 | 2 | 1 | 0 |

Calculate | (10^{3}) |
(10^{2}) |
(10^{1}) |
(10^{0}) |

Position value | 1000 | 100 | 10 | 1 |

The following bullets describe each row of the table.

- Row 1, Radix is the number base. Decimal notation is based on 10, therefore the radix is 10.
- Row 2, Position in number considers the position of the decimal number starting with, from right to left, 0 (1st position), 1 (2nd position), 2 (3rd position), 3 (4th position). These numbers also represent the exponential value use to calculate the positional value in the 4th row.
- Row 3 calculates the positional value by taking the radix and raising it by the exponential value of its position in row 2.

**Note:**n^{0}is = 1. - Row 4 positional value represents units of thousands, hundreds, tens, and ones.

To use the positional system, match a given number to its positional value. The example in the table illustrates how positional notation is used with the decimal number 1234.

Thousands | Hundreds | Tens | Ones | |
---|---|---|---|---|

Positional Value | 1000 | 100 | 10 | 1 |

Decimal Number (1234) | 1 | 2 | 3 | 4 |

Calculate | 1 x 1000 | 2 x 100 | 3 x 10 | 4 x 1 |

Add them up… | 1000 | + 200 | + 30 | + 4 |

Result |
1,234 |

In contrast, the binary positional notation operates as described in the table.

Radix | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
---|---|---|---|---|---|---|---|---|

Position in Number | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |

Calculate | (2^{7}) |
(2^{6}) |
(2^{5}) |
(2^{4}) |
(2^{3}) |
(2^{2}) |
(2^{1}) |
(2^{0}) |

Position value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

The following bullets describe each row of the table.

- Row 1, Radix is the number base. Binary notation is based on 2, therefore the radix is 2.
- Row 2, Position in number considers the position of the binary number starting with, from right to left, 0 (1st position), 1 (2nd position), 2 (3rd position), 3 (4th position). These numbers also represent the exponential value use to calculate the positional value in the 4th row.
- Row 3 calculates the positional value by taking the radix and raising it by the exponential value of its position in row 2.

**Note:**n^{0}is = 1. - Row 4 positional value represents units of ones, twos, fours, eights, etc.

The example in the table illustrates how a binary number 11000000 corresponds to the number 192. If the binary number had been 10101000, then the corresponding decimal number would be 168.

Positional Value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|---|

Binary Number (11000000) | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

Calculate | 1 x 128 | 1 x 64 | 0 x 32 | 0 x 16 | 0 x 8 | 0 x 4 | 0 x 2 | 0 x 1 |

Add Them Up.. | 128 | + 64 | + 0 | + 0 | + 0 | + 0 | + 0 | + 0 |

Result |
192 |

## Convert Binary to Decimal

To convert a binary IPv4 address to its dotted decimal equivalent, divide the IPv4 address into four 8-bit octets. Next apply the binary positional value to the first octet binary number and calculate accordingly.

For example, consider that 11000000.10101000.00001011.00001010 is the binary IPv4 address of a host. To convert the binary address to decimal, start with the first octet, as shown in the table. Enter the 8-bit binary number under the positional value of row 1 and then calculate to produce the decimal number 192. This number goes into the first octet of the dotted decimal notation.

Positional Value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|---|

Binary Number (11000000) | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |

Calculate | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

Add Them Up… | 128 | + 64 | + 0 | + 0 | + 0 | + 0 | + 0 | + 0 |

Result |
192 |

Next convert the second octet of 10101000 as shown in the table. The resulting decimal value is 168, and it goes into the second octet.

Positional Value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|---|

Binary Number (10101000) | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 0 |

Calculate | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

Add Them Up… | 128 | + 0 | + 32 | + 0 | + 8 | + 0 | + 0 | + 0 |

Result |
168 |

Convert the third octet of 00001011 as shown in the table.

Positional Value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|---|

Binary Number (00001011) | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 1 |

Calculate | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

Add Them Up… | 0 | + 0 | + 0 | + 0 | + 8 | + 0 | + 2 | + 1 |

Result |
11 |

Convert the fourth octet of 00001010 as shown in the table. This completes the IP address and produces **192.168.11.10**.

Positional Value | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
---|---|---|---|---|---|---|---|---|

Binary Number (00001010) | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 0 |

Calculate | 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |

Add Them Up… | 0 | + 0 | + 0 | + 0 | + 8 | + 0 | + 2 | + 0 |

Result |
10 |

## Decimal to Binary Conversion

It is also necessary to understand how to convert a dotted decimal IPv4 address to binary. A useful tool is the binary positional value table.

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