Bits. Binary Digits. 0 or 1

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Emil Jennings
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1 Data Representation

2 Bits Binary Digits 0 or 1 Everything stored in a computer is stored as bits. Bits can mean different things depending on how the software or hardware interpret the bits Bits are usually grouped into chunks of eight bits A chuck of eight bits is called a byte

3 Digitization Convertor Computing Device Convertor Input such as sound, light, pressing a key Output such as sound, light, moving a robot arm

4 Bits Consider the follow byte ( 8 bits) What could it represent? The integer 90 The letter Z (ASCII) Red part of color deep reddish brown The opcode of an instruction The sequence of boolean values false true false true true false true false

5 Common Questions How many bits are needed to represent a set of values? How many bits do I need to represent the values in a set with N elements? Ceiling(log 2 N) If N is 8 then 3 bits are required If N is 11 then 4 bits are required

6 Common Questions How many values can be represented with a given number of bits? How many values can be represented with N bits? 2 N If N is 5 the 32 values can be represented If N is 7 then 128 values can be represented

7 Maximum Patterns for a Given Number of Bits Number of Bits Maximum Number of Pattern N 2 N

8 Counting In Binary Binary Decimal

9 Character Code Mapping between numbers and characters ASCII Unicode

10 ASCII Codes Letter Symbol, Decimal, Binary A, 65, B, 66, C, 67, D, 68, E, 69, F, 70, G, 71, H, 72, I, 73, Symbol, Decimal, Binary a, 97, b, 98, c, 99, d, 100, e, 101, f, 102, g,103, h,104, i, 105,

11 ASCII Codes Digits and Other Symbol, Decimal, Binary 0, 48, , 49, , 50, , 51, , 52, , 53, , 54, , 55, , 56, , 57, Symbol, Decimal, Binary Line Feed, 10, Space, 32,

12 ASCII Character Code

13 ASCII Character Code

14 Example Problem Convert the word Computer into a sequence of numbers representing the ASCII characters codes of the letters.

15 Example Problem Answer Computer

16 Example Problem Convert the following sequence of numbers to text using the ASCII character code. To make the sequence easier to read a space is used to separate numbers

17 Example Problem Answer Code!

18 Practice Problems Convert the sequence of letters in your name (first last) into a sequence of numbers using the ASCII character code. The code for a space is 32 Convert the following sequence of numbers to text using the ASCII character code. To make the sequence easier to read a space is used to separate numbers

19 Place-Value Notation Consider the base 10 number 7438 This number (sequence of digits) represents 7* * * * = 7438 Since the number is a base 10 number the digit in position i is multiplied by 10 i Positions are numbered from right to left beginning at 0 This works for any base A base n number must have n different digits

20 Place-Valued Notation Most common bases used in computer science Base 2 (binary) Base 8 (octal) Base 10 (decimal) Base 16 (Hexadecimal)

21 Base 2 Two Digits 0 and 1 Consider the base 2 number This number represents 1* * * * * *2 0 Since there are only two digits it is common to write the above expression using only the powers of 2 that are multiplied by This represents to base 10 number = 39 To indicate the base of a number a subscript is sometimes used at the end of the number

22 Powers of 2 Formula Value

23 Base 8 Eight digits 0, 1, 2, 3, 4, 5, 6, 7 Consider the base 8 number 5072 The number represents 5 * * * *8 0 The is equivalent to the base 10 number 5 * * * * 1 = 2618

24 Powers of 8 Formula Value

25 Base 16 Sixteen digits Consider the base 16 number B74D This number represents B* * * D* * * * *16 0 This is equivalent to the base 10 number 11* * * *1 = 46925

26 Powers of 16 Formula Value

27 Base 10 Base 2 Base 8 Base A B C D E F

28 Conversion Problems Convert between bases For example given the base 2 number find the equivalent base 10 and base 16 number Base Base 16 C9

29 Conversion Problems Base 2 -> Base 10 Base 2 -> Base 16 Base 10 -> Base 2 Base 10 -> Base 16 Base 16 -> Base 2 Base 16 -> Base 10

30 Convert Base 2 to Base 16 last4bits(x) last4bits( ) 1101 bitsexcludinglast4(x) bitsexcludinglast4( ) 1100

31 Convert base 2 to base 16 concatenate(a, B) concatenate(6, 547) 6547 base16digit(x) base16digit(1100) C

32 Algorithm to Convert Base 2 to Base 16 Let X be the base 2 number we want to convert to base 16 Give B the value of the empty string Do the following Give Y the value of last4bits(x) Give Z the value base16digit(y) Give B the value concatenate(z, B) Given X the value of bitsexcludinglast4(x) Until X is empty B is the base 16 representation of the original X

33 Algorithm to convert base 10 representation to binary representation Let X be the base 10 representation to convert Give B the value of the empty string Do the following Give Y the value of the integer quotient of X / 2 Give Z the value of the remainder of X / 2 Give B the value of concatenate(z, B) Give X the value of Y Until X is 0 B is now the binary representation of the original value of X

34 Example: convert to binary X Y Z B

35 Practice Problems Convert the following base 10 numbers to base How can you check your answer?

36 How Many Dots?

CS & IT Conversions. Magnitude 10,000 1,

CS & IT Conversions. Magnitude 10,000 1, CS & IT Conversions There are several number systems that you will use when working with computers. These include decimal, binary, octal, and hexadecimal. Knowing how to convert between these number systems