Announcement!!! First exam next Thursday (I’m trying to give you a first exam before the drop date) I’ll post a sample exam over the weekend and will try to go over it on Tuesday…. It will cover everything we’ve gone over so far….some intro computers, data types; I/O; if then else; Boolean operators; number systems

A pause……number systems..because I have to… What you need to know … 1. binary number system 2. converting from decimal to binary and binary to decimal 3. Hexadecimal system -- conversion

Why binary

Advantages The advantages of binary: –Simple; easy to build. –Unambiguous signals (hence noise immunity). –Flawless copies can be made. –Anything that can be represented with some sort of pattern can be represented with patterns of bits.

More advantages Since data of all kinds is stored in computer memory (main and secondary) using the same electronic methods, this means that endless perfect copies can be made of any type of data or program.

In order to understand the binary numbering system lets first look at our decimal system. The decimal numbering system consists of the numbers 0 through After nine we place a 1 in the tens column and start again with 0. Which gives us 10. The decimal system is also known as base 10 because it is based on the 10 numbers 0 – 9.

Binary Numbers have only two digits 0 or 1 DecimalBinary Binary is known as Base 2

Converting binary – decimal As you can see it would take a lot of time to create charts to represent Binary numbers. An easier way is to use the powers of = = = = = = = = 1 Lets place the above calculations into a chart that will make it easy to convert a binary number to a decimal number.

Converting Decimal to Binary Multiply each digit weight by the base power (i.e. 2) at that unit position and add up all the products

Remember how we “really” convert decimal to decimal => 5 x 10 0 = 5 2 x 10 1 = 20 1 x 10 2 = Base 125

x 1 = 1 Total = x 0 = 0 64 x 0 = 0 32 x 1 = x 0 = 0 8 x 1 = 8 4 x 0 = 0 Use the chart to convert the binary number to decimal. Note: The bit to the far right is the Least Significant Bit (LSB) and will determine if the number is even or odd. 2 x 0 = 0

x 1 = 1 Total = x 0 = 0 64 x 1 = x 1 = x 1 = 16 8 x 1 = 8 4 x 1 = 4 Use the chart to convert the binary number to decimal. 2 x 1 = 2 Note: if consecutive bits from the right are all 1’s Then the answer is the next power of 2 minus 1 In this case 128 – 1 = 127

x 0 = 0 Total = x 1 = x 0 = 0 32 x 1 = x 1 = 16 8 x 1 = 8 4 x 0 = 0 Take a piece of paper and convert the binary number to decimal. 2 x 1 = 2

Quick exercises Convert from binary to decimal. Convert from binary to decimal. Convert from binary to decimal.

The largest number that can be represented using an 8 bit binary number is Remember the rule – if all the digits are 1 then the number is the next power of 2 minus – 1 = 255

So….we need something else

Which Digits Are Available in which Bases 17 Base Base digits 2 digits Base A B C D E F digits Note: Base 16 is also called “Hexadecimal” or “Hex”. Base 16 Cheat Sheet A 16 = B 16 = C 16 = D 16 = E 16 = F 16 = Add Placeholder

Hexadecimal Numbers - Example 16 0 place16 1 place16 2 place 3AB 16 This subscript denotes that this number is in Base 16 or “Hexadecimal” or “Hex”. 1’s place16’s place256’s place Note: 16 2 = 256

Hexadecimal Numbers - Example 3AB 16 1’s place16’s place256’s place So this number represents 3 two-hundred fifty-sixes 10 sixteens 11 ones Base 16 Cheat Sheet A 16 = B 16 = C 16 = D 16 = E 16 = F 16 = Mathematically, this is (3 x 256) + (10 x 16) + (11 x 1) = =

Converting Hex to Decimal – Again we use the technique: Multiply each bit by 16 n, where n is the weight (or power) of the bit The weight is the position of the bit, starting from 0 on the right Add the results

Example ABC 16 => C x 16 0 = 12 x 1 = 12 B x 16 1 = 11 x 16 = 176 A x 16 2 = 10 x 256 =

Exercises Convert 3F3 to decimal Convert AA1 to decimal Convert 11A to decimal

Why Hexadecimal Is Important What is the largest number you can represent using four binary digits? _ _ ==== = … the smallest number? _ _ = 0 10 What is the largest number you can represent using a single hexadecimal digit? Base 16 Cheat Sheet A 16 = B 16 = C 16 = D 16 = E 16 = F 16 = _ 16 F = … the smallest number? _ 16 0 = 0 10 Note: You can represent the same range of values with a single hexadecimal digit that you can represent using four binary digits!

Why Hexadecimal Is Important It can take a lot of digits to represent numbers in binary. Example: = Long strings of digits can be difficult to work with or look at. Also, being only 1’s and 0’s, it becomes easy to insert or delete a digit when copying by hand.

Converting Decimal to Binary Technique –Divide by two, keep track of the remainder –First remainder is bit 0 (LSB, least significant bit) –Second remainder is bit 1 –Etc.

Converting Decimal to Binary Example: We want to convert to binary. 125 / 2 = 62 R 1 62 / 2 = 31 R 0 31 / 2 = 15 R 1 15 / 2 = 7 R 1 7 / 2 = 3 R 1 3 / 2 = 1 R 1 1 / 2 = 0 R =

Exercises Convert 33 decimal to Binary. Convert 51 decimal to Binary. Convert 19 decimal to Binary.

Converting Decimal to HEX Technique –Divide by 16, keep track of the remainder –First remainder is bit 0 (LSB, least significant bit) –Second remainder is bit 1 –Etc

Converting Decimal to Hex Example: We want to convert to hex. 125 / 16 = 7 R 13 7 / 16 = 0 R = 7D 16 Base 16 Cheat Sheet A 16 = B 16 = C 16 = D 16 = E 16 = F 16 = 15 10

Exercises Convert 28 to Hex Convert 346 to Hex Convert 117 to Hex

Converting Hex to Binary Technique –Convert each hexadecimal digit to a 4 bit equivalent binary representation (chop it into 4 digit representation

Converting Binary Numbers to Hex Recall the example binary number from the previous slide: First, split the binary number into groups of four digits, starting with the least significant digit. Next, convert each group of four binary digits to a single hex digit. CA52 Base 16 Cheat Sheet A 16 = B 16 = C 16 = D 16 = E 16 = F 16 = Put the single hex digits together in the order in which they were found, and you’re done! 16

Example 1 0 A F AF16 =

Exercise Convert to hex Convert AA1 (hex) to binary

Summary All programs and data are stored in binary as it maps directly to on/off signals Hexadecimal is base 16, every four binary digits can be represented by one Hex digit (Shorthand for computers) To convert any number base to decimal – Multiply each digit weight by the base power at that unit position and add up all the products To convert decimal to any base(2,16..) –Keep Dividing the decimal number by the base until you reach zero, keeping the remainders each time. Read from the bottom up.

Problems: Convert 101 to binary Convert to decimal Convert to hexadecimal

Real-world problems Convert the following MAC address to decimal, keeping a colon between each byte. fe:fd:00:00:5c:a4 Convert the following IP address to binary and hexadecimal, keeping a dot between each byte (remember a byte is 8 bits)

More real-world problems Convert the following subnet mask to binary, keeping a dot between each byte