systems use the same methods as the decimal system. Web Links Base 10 (Decimal) Numbering System http://www.psinvention.com/zoetic/ base10.htm

Content 1.2 Network Math 1.2.4 Base 2 number system Computers recognize and process data using the binary, or Base 2, numbering system. The binary system uses only two symbols, 0 and 1, instead of the ten symbols used in the decimal numbering system. The position, or place, of each digit from right to left in a binary number represents 2, the base number, raised to a power or exponent, starting from 0. 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.Example: 101102 = (1 x 24 = 16) + (0 x 23 = 0) + (1 x 22 = 4) + (1 x 21 = 2) + (0 x 20 = 0) = 22 (16 + 0 + 4 + 2 + 0) If the binary number (101102) is read left to right, there is a 1 in the 16s position, a 0 in the 8s position, a 1 in the 4s position, a 1 in the 2s position, and a 0 in the 1s position, which adds up to decimal number 22. Web Links Base 2 (Binary) Numbering System http://www.psinvention.com/zoetic/ base2.htm

Content 1.2 Network Math 1.2.5 Converting decimal numbers to 8-bit binary numbers There are several ways to convert decimal numbers to binary numbers. The flowchart in Figure describes one method. The process is trying to figure out which values of the power of 2 that add together to get the decimal number being converted to a binary number. This method is one of several methods that can be used. It is best to select one method and practice with it until it always produces the correct answer. Conversion exercise
Use the example below to convert the decimal number 168 to a binary number:

128 fits into 168. So the left most bit in the binary number is a 1. 168 - 128 leaves 40.
64 does not fit into 40. So the second bit in from the left is a 0.
32 fits into 40. So the third bit in from the left is a 1. 40 - 32 leaves 8.
16 does not fit into 8 so the fourth bit in from the left is a 0.
8 fits into 8. So the fifth bit in from the left is a 1. 8 - 8 leaves 0. So, the remaining bits to the right are all 0.

Result: Decimal 168 = 10101000 For more practice, try converting decimal 255 to binary. The answer should be 11111111. The number converter activity in Figure will provide more practice. Lab Activity Lab Exercise: Decimal to Binary ConversionIn this lab, the student will learn and practice to convert decimal values to binary values. Web Links Binary Numbers http://www.netlingo.com/more/binary.html

Content 1.2 Network Math 1.2.6 Converting 8-bit binary numbers to decimal numbers There are two basic ways to convert binary numbers to decimal numbers. The flowchart in Figure shows one example. Binary numbers can also be converted to decimal numbers by multiplying the binary digits by the base number of the system, which is Base 2, and raised to the exponent of its position. 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. Therefore 20 = 1 0 x 20 = 0 0 x 21 = 0 0 x 22 = 0 0 x 23 = 0 1 x 24 = 16 1 x 25 = 32 1 x 26 = 64 + 0 x 27= 0
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112 Note: The sum of the powers of 2 that have a 1 in their position The number converter activity will provide more practice. Lab Activity Lab Exercise: Binary to Decimal ConversionIn this lab, the student will learn and practice the process of converting binary values to decimal values. Web Links Binary Numbers http://www.netlingo.com/more/binary.html

Content 1.2 Network Math 1.2.7 Four-octet dotted decimal representation of 32-bit binary numbers 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. To do this, split the binary number into four groups of eight binary digits. Then convert each group of eight bits, also known as an octet into its decimal equivalent. Do this conversion exactly as was shown in the binary-to-decimal conversion topic on the previous page. When written, the complete binary number is represented as four groups of decimal digits separated by periods. This is referred to as dotted decimal notation and provides a compact, easy to remember way of referring to the 32 bit addresses. This representation is used frequently later in this course, so it is necessary to understand it. When converting to binary from dotted decimal, remember that each group, which consists 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. Example: Convert 200.114.6.51 to its 32-bit binary equivalent. Convert 10000000 01011101 00001111 10101010 to its dotted decimal equivalent. Web Links IP Addressing Architecture http://www2.rad.com/networks/1994/ ip_addr/tcpip2.htm

Content 1.2 Network Math 1.2.8 Hexadecimal Hexadecimal (hex) is used frequently when working with computers since it can be used to represent binary numbers in a more readable form. The computer performs computations in binary, but there are several instances when the binary output of a computer is expressed in hexadecimal to make it easier to read. Converting a hexadecimal number to binary, and a binary number to hexadecimal, is a common task when dealing with the configuration register in Cisco routers. Cisco 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 hex. The word hexadecimal is often abbreviated 0x when used with a value as shown with the above number: 0x2102. Like the binary and decimal systems, the hexadecimal system is based on the use of symbols, powers, and positions. The symbols that hex uses are 0 - 9, and A, B, C, D, E, and F. Notice that all possible combinations of four binary digits have only one hexadecimal symbol, where it takes two in decimal. The reason why hex is used is that two hexadecimal digits, as opposed to decimal that would require up to four digits, can efficiently represent any combination of eight binary digits. In allowing two decimal digits to represent four bits, using decimal could also cause confusion in reading a value. For example, the eight bit binary number 01110011 would be 115 if converted to decimal digits. Is that 11-5 or 1-15? If 11-5 is used, the binary number would be 1011 0101, which is not the number originally converted. Using hexadecimal, the conversion is 1F, which always converts back to 00011111. Hexadecimal reduces an eight bit number to just two hex digits. This reduces the confusion of reading long strings of binary numbers and the amount of space it takes to write binary numbers. Remember that hexadecimal is sometimes abbreviated 0x so hex 5D might be written as "0x5D". To convert from hex to binary, simply expand each hex digit into its four bit binary equivalent. Lab Activity Hexadecimal ConversionsIn this lab, the student will learn the process to convert hexadecimal values to decimal and binary values. Web Links The Hexadecimal Number System http://www.math.ohiou.edu/~just/hex.htm

Content 1.2 Network Math 1.2.9 Boolean or binary logic Boolean logic is based on digital circuitry that accepts one or two incoming voltages. Based on the input voltages, output voltage is generated. For the purpose of computers the voltage difference is associated as two states, on or off. These two states are in turn associated as a 1 or a 0, which are the two digits in the binary numbering system. Boolean logic is a binary logic that allows two numbers to be