Networking Basics

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Base 2 number system


This page will discuss the number system that computers use to recognize and process data, which is binary, or Base 2.
The binary system uses only two symbols, which are 0 and 1. The position of each digit from right to left in a binary number represents the base number 2 raised to a power or exponent. These place values are, from right to left, 20, 21, 22, 23, 24, 25, 26, and 27, or 1, 2, 4, 8, 16, 32, 64, and 128 respectively.

Here is an example:

101102 =
(1 x 2^4 = 16) + (0 x 2^3 = 0) + (1 x 2^2 = 4) + (1 x 2^1 = 2) + (0 x 2^0 = 0) = 22 (16 + 0 + 4 + 2 + 0)

This example shows that the binary number 10110 is equal to the decimal number 22.

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Converting decimal numbers to 8-bit binary numbers


This page will teach you how to convert decimal numbers to binary numbers.


Conversion exercise:

Use the example below to convert the decimal number 168 to a binary number:

128 is less than 168 so the left most bit in the binary number is a 1. 168 - 128 = 40.
64 is not less than or equal to 40 so the second bit from the left is a 0.
32 is less than 40 so the third bit from the left is a 1. 40 - 32 = 8.
16 is not less than or equal to 8 so the fourth bit from the left is a 0.
8 is equal to 8 so the fifth bit from the left is a 1. 8 - 8 = 0. Therefore, the bits to the right are all 0.

This example shows that the decimal number 168 is equal to the binary number 10101000.

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Converting 8-bit binary numbers to decimal numbers

This page will teach you how to convert binary numbers to decimal numbers.

Here is an example:

Convert the binary number 01110000 to a decimal number.


NOTE:
Work from right to left. Remember that anything raised to the 0 power is 1.

0 x 20 = 0

0 x 2^1 = 0

0 x 2^2 = 0

0 x 2^3 = 0

1 x 2^4 = 16

1 x 2^5 = 32

1 x 2^6 = 64

0 x 2^7 = 0

__________

= 112

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Four-octet dotted decimal representation of 32-bit binary numbers

This page will explain how binary numbers are represented in dotted decimal notation.

Currently, addresses assigned to computers on the Internet are 32-bit binary numbers. To make it easier to work with these addresses, the 32-bit binary number is broken into a series of decimal numbers. First the binary number is split into four groups of eight binary digits. Then each group of eight bits, or octet, is converted into its decimal equivalent. This conversion can be performed as shown on the previous page.

When written, the complete binary number is represented as four groups of decimal digits separated by periods. This is called dotted decimal notation and provides a compact and easy way to refer to 32-bit addresses. This representation is used frequently later in this course, so it is necessary to understand it. For dotted decimal to binary conversions, remember that each group of one to three decimal digits represents a group of eight binary digits. If the decimal number that is being converted is less than 128, zeros will be needed to be added to the left of the equivalent binary number until there are a total of eight bits.

Try the following conversions for practice:

Convert 200.114.6.51 to its 32-bit binary equivalent.

Convert 10000000 01011101 00001111 10101010 to its dotted decimal equivalent.

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Hexadecimal

This page will teach you about the hexadecimal number system. You will also learn how hexadecimal is used to represent binary and decimal numbers.

The hexadecimal or Base 16 number system is commonly used to represent binary numbers in a more readable form. Computers perform computations in binary. However, there are several instances when the binary output of a computer is expressed in hexadecimal to make it easier to read.

The configuration register in routers often requires hexadecimal to binary and binary to hexadecimal conversions. routers have a configuration register that is 16 bits long. The 16-bit binary number can be represented as a four-digit hexadecimal number. For example, 0010000100000010 in binary equals 2102 in hexadecimal. A hexadecimal number is often indicated with a 0x. For example, the hexadecimal number 2102 would be written as 0x2102.

Like the binary and decimal systems, the hexadecimal system is based on the use of symbols, powers, and positions. The symbols that hexadecimal uses are the digits 0 through 9 and the letters A through F.

All combinations of four binary digits can be represented with one hexadecimal symbol. These values require one or two decimal symbols. Two hexadecimal digits can efficiently represent any combination of eight binary digits. The decimal representation of an eight-bit binary number will require either two or three decimal digits. Since one hexadecimal digit always represents four binary digits, hexadecimal symbols are easier to use than decimal symbols when working with large binary numbers. Using hexadecimal representation also reduces the confusion of reading long strings of binary numbers and the amount of space it takes to write binary numbers. Remember that 0x may be used to indicate a hexadecimal value. The hexadecimal number 5D might be written as 0x5D.

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Boolean or binary logic

This page will introduce Boolean logic and explain how it is used.

Boolean logic is based on digital circuitry that accepts one or two incoming voltages. Based on the input voltages, output voltage is generated. For computers the voltage difference is represented as an ON or OFF state. These two states are associated with a binary 1 or 0.

Boolean logic is a binary logic that allows two numbers to be compared and makes a choice based on the numbers. These choices are the logical AND, OR, and NOT. With the exception of the NOT, Boolean operations have the same function. They accept two numbers, which are 1 and 0, and generate a result based on the logic rule.

The NOT operation takes the value that is presented and inverts it. A 1 becomes a 0 and a 0 becomes a 1. Remember that the logic gates are electronic devices built specifically for this purpose. The logic rule that they follow is whatever the input is, the output is the opposite.

The AND operation compares two input values. If both values are 1, the logic gate generates a 1 as the output. Otherwise it outputs a 0. There are four combinations of input values. Three of these combinations generate a 0, and one combination generates a 1.

The OR operation also takes two input values. If at least one of the input values is 1, the output value is 1. Again there are four combinations of input values. Three combinations generate a 1 and the fourth generates a 0.

The two networking operations that use Boolean logic are subnetwork and wildcard masking. The masking operations are used to filter addresses. The addresses identify the devices on the network and can be grouped together or controlled by other network operations.

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IP addresses and network masks

This page will explain the relationship between IP addresses and network masks.

When IP addresses are assigned to computers, some of the bits on the left side of the 32-bit IP number represent a network. The number of bits designated depends on the address class. The bits left over in the 32-bit IP address identify a particular computer on the network. A computer is referred to as a host. The IP address of a computer consists of a network and a host part.

To inform a computer how the 32-bit IP address has been split, a second 32-bit number called a subnetwork mask is used. This mask is a guide that determines how the IP address is interpreted. It indicates how many of the bits are used to identify the network of the computer. The subnetwork mask sequentially fills in the 1s from the left side of the mask. A subnet mask will always be all 1s until the network address is identified and then it will be all 0s to the end of the mask. The bits in the subnet mask that are 0 identify the computer or host.

Some examples of subnet masks are as follows:

11111111000000000000000000000000
written in dotted decimal as 255.0.0.0

11111111111111110000000000000000
written in dotted decimal as 255.255.0.0

In the first example, the first eight bits from the left represent the network portion of the address, and the last 24 bits represent the host portion of the address. In the second example the first 16 bits represent the network portion of the address, and the last 16 bits represent the host portion of the address.

The IP address 10.34.23.134
in binary form is 00001010.00100010.00010111.10000110.

A Boolean AND of the IP address 10.34.23.134 and the subnet mask 255.0.0.0 produces the network address of this host:

00001010.00100010.00010111.10000110
11111111.00000000.00000000.00000000
00001010.00000000.00000000.00000000

The dotted decimal conversion is 10.0.0.0 which is the network portion of the IP address when the 255.0.0.0 mask is used.

A Boolean AND of the IP address 10.34.23.134 and the subnet mask 255.255.0.0 produces the network address of this host:

00001010.00100010.00010111.10000110
11111111.11111111.00000000.00000000
00001010.00100010.00000000.00000000

The dotted decimal conversion is 10.34.0.0 which is the network portion of the IP address when the 255.255.0.0 mask is used.

This is a brief illustration of the effect that a network mask has on an IP address. The importance of masking will become much clearer as more work with IP addresses is done. For right now it is only important that the concept of the mask is understood.