1. 1 Data Representation Characters, Integers and Real Numbers Binary Number System Octal Number System Hexadecimal Number System Powered by DeSiaMore

2. 2 Data Representation Binary Representation Only two things, need only 1 bit Want to represent Sweet and Sour  1 for Sweet  0 for Sour Want to represent more than 2 things  Need more bits For 4 things, like 4 colors need 2 bits  00 - blueyellow - 10  01 - redblack - 11 Powered by DeSiaMore

3. 3 Data Representation Binary Representation If you have  More than 4 things need more than 2 bits  8 things, need 3 bits  16 things, need 4 bits  32 things, need 5 bits In general, if you want to represent 2 n things  need n bits Binary means each time we add a bit, we double the things we can represent Powered by DeSiaMore

4. 4 Data Representation Representing Numbers Integer - is a whole number, 0, 1, 2, 3, 4 … n To store integers in the computer, we only need the number of bits for the magnitude of the number So, for integers up to 256 which = 2 8  we need 8 bits Higher integers will require more bits  9 bits can store integers up to 512  10 bits can store integers up to 1024 Powered by DeSiaMore

5. 5 Representing Numbers For integers `always include 0  Zero takes up one of our choices Means we can actually store numbers up to 255 in bits Binary Decimal 0000 0000 0 0000 0001 1 0000 0010 2 0000 0011 3 0000 0100 4 … 1111 1111 255 Maximum in bits + 1 Add 1 10000 0000 256 Powered by DeSiaMore

6. 6 Representing Numbers Real Numbers or “Floating Point”  More complicated to store real numbers  Will have a value and a fractional part  Example: 104.3 -35.21 3.5346 0.999 10000.98 8.6547  Numbers that have decimal points are real numbers Powered by DeSiaMore

7. 7 Representing Numbers Real Numbers  Example: 3 2 7 5. 6 1 2 10 3 10 2 10 1 10 0 10 -1 10 -2 10 -3 To the left of decimal point: 1’s,10’s,100’s,1000’s To the right of point: 10th’s, 100th’s, 1000th’s The decimal point has a special name it’s called the Radix point Inside the computer we store the value as an integer and then include information about Radix Point Radix Point Powered by DeSiaMore

8. 8 Representing Numbers Real Numbers  Composed of three parts 1. Sign + or - 2. Mantissa - the number value 3. Exponent - tells how the radix point gets moved  Example: 148.69  Sign: +  Exponent: -2  Mantissa: 14869 Powered by DeSiaMore

9. 9 Representing Numbers Real Numbers  More Examples: -2345.1  Sign: -  Exponent:-1  Mantissa:23451  Example:677.00  Sign:+  Exponent:0  Mantissa:677 Powered by DeSiaMore

10. 10 Representing Numbers Real Numbers  Equation for defining a real number: Sign * mantissa * 10 exponent Decimal point “floats” according to the exponent Examples using the above equation (assume 5 digits per number): 12001.00+12001 X 10 0 -520.05-52005 X 10 -2 0.62200+62200 X 10 -5 21.998+21998 X 10 -3 Powered by DeSiaMore

11. 11 Representing Numbers Real Numbers  When storing a real number, PC uses 64 bits  1 bit for the sign  11 bits for the exponent  52 bits for the mantissa Powered by DeSiaMore

12. 12 Representing Characters Storing Characters  A text document that you type in a word processor is decomposed into paragraphs, sentences and words  Individual characters are actually stored in the computer Powered by DeSiaMore

13. 13 Representing Characters To represent a document  Must represent every character  How do you do this?  Documents are in a given language  In any language there are a finite number of characters  Want to list each character and assign it a unique binary string or pattern Powered by DeSiaMore

14. 14 Representing Characters English - How many characters?  26 - Upper case letters  26 - Lower case letter  0 - digits  Punctuation characters  Non-printing control codes  The number of unique things determines the number of bits needed Powered by DeSiaMore

15. 15 Representing Characters Computer manufacturers found it easier to standardize the way characters are represented Recall: ASCII, EBCDIC and Unicode  ASCII - Allows for 256 characters  That means can represent 256 different characters  How many bits then per character? Powered by DeSiaMore

16. 16 Representing Characters ASCII  Needs bits to represent all the characters  Charts are printed in different ways  Hex, binary and decimal  Easiest to read in decimal  However, numbers are stored in binary in the computer  Note: ASCII codes are in order  Means can sort characters and numbers  A, B, C, D, E, F, …  a, b, c, d, e, f, …  1, 2, 3, 4, 5, …. Powered by DeSiaMore

17. 17 ASCII Codes Powered by DeSiaMore

18. 18 Representing Characters ASCII  First 32 characters are non-printing  ESC key, Backspace key, Tab, Enter key plus others  ASCII is ok for English, but what about other languages? Like Chinese?  Chinese has many more characters far greater than 256 Powered by DeSiaMore

19. 19 Representing Characters Unicode  Goal of the Unicode representation is to create a character set that can represent every language used in the world  Unicode is a large extension of the ASCII character set  Comparison between ASCII and Unicode Unicode ASCII 16 bits/character8 bits/character 65,000 characters256 characters Superset of ASCII Powered by DeSiaMore

20. 20 Representing Characters Unicode  Superset of ASCII  ASCII takes up the first 256 characters of Unicode  Still evolving, currently 40,000 characters have been encoded! Powered by DeSiaMore

21. 21 Number Systems Powered by DeSiaMore

22. 22 Number Systems All number systems have a base or radix Decimal system – the base is 10 Different number systems have different bases  Binary the base = 2  Time the base = 60 Base affects counting  Counting in Decimal – ranges from 0 -  Counting in base 60 – ranges from 0 - 59 Powered by DeSiaMore

23. 23 Number Systems Number systems are positional Know how big the number is from its position in the number Example: 943  You know that 9=900, 4 = 40, 3 = 3 Powered by DeSiaMore

24. 24 Number Systems Positional Notation  A more structured way to say this:  Rightmost digit = value x base 0 power  Next digit to the left = value x base 1 power  Next digit to the left = value x base 2 power ….. Example: 943 base 10 9 X 10 2 = 9 X 100 = 900 4 X 10 1 = 4 X 10 = 40 + 3 X 10 0 = 3 X 1 = 3 943 We will extend this concept to the other number systems: Octal and Hexadecimal Powered by DeSiaMore

25. 25 Computer Number Systems  Actually 3 number systems used for computers  Binary - base 2  Octal – base 8  Hexadecimal - base 16 Digits in each system  Binary: 0,1  Octal: 0 - 7  Hexadecimal - 0 -, and A,B,C,D,E,F  A = 10, B=11, C=12, D=13, E=14, F=15 Powered by DeSiaMore

26. 26 Computer Number Systems  Why other number systems? Binary numbers are hard to read – ok for computers Even small numbers need lots of digits Example:  Decimal number = 14  Binary number = 1110 Other number systems are for people  Makes it easier to read and work with digital numbers Powered by DeSiaMore

27. 27 Binary Number Chart Powered by DeSiaMore

28. 28 Computer Number Systems  Why other number systems? Turns out that  Octal and Hexadecimal make it easy to convert between the different number systems  Binary to Octal, Hexadecimal  Octal, Hexadecimal to Decimal Powered by DeSiaMore

29. 29 Number System Conversions Binary Number System and Decimal Decimal and Binary Binary and Octal and hexadecimal Binary Arithmetic Powered by DeSiaMore

30. 30 Converting Between Number Systems Useful to convert between number systems People more familiar with decimal numbers Should know how to do this even if you always use a calculator to do the conversion  Binary to octal, hex or decimal  Decimal to binary  Octal or hex to decimal Powered by DeSiaMore

31. 31 Converting Between Number Systems Example: Binary to Decimal 1 0 1 1 2 3 2 2 2 1 2 0 1 x 2 3 = 1 x 8 = 8 0 x 2 2 = 0 x 4 = 0 1 x 2 1 = 1 x 2 = 2 1 x 2 0 = 1 x 1 = + 1 11 Powered by DeSiaMore

32. 32 Converting Between Number Systems Octal Conversion 3 2 1 8 210 3 x 8 2 = 3 x 64 = 192 2 x 8 1 = 2 x 8 = 16 + 1 x 8 0 = 1 x 1 = 1 209 10 Powered by DeSiaMore

33. 33 Number Systems  Converting Between Number Systems Octal Conversion 2 3 3 1 8 3210 2 x 8 3 = 2 x 512 = 1024 3 x 8 2 = 3 x 64 = 192 + 3 x 8 1 = 3 x 8 = 24 1 x 8 0 = 1 x 1 = 1 1241 10 Powered by DeSiaMore

34. 34 Number Systems  Converting Between Number Systems Hexadecimal Conversion A B C 16 16 2 16 1 16 0 A x 16 2 = 10 x 256 = 2560 B x 16 1 = 11 x 16 = 176 + C x 16 0 = 12 x 1 = 1 2748 10 Powered by DeSiaMore

35. 35 Number Systems  Converting Between Number Systems Binary to decimal  Number is large, takes a long time  Shortcut:  Convert from binary to octal (or hex)  Convert from octal (or hex) to decimal Powered by DeSiaMore

36. 36 Number Systems  Converting Between Number Systems  Convert a binary number to Octal - Easy 111101100 = 482 in decimal 1. Break up the number into groups of three starting at the right 2. Each group of three forms a binary number 3. For each digit - you only need to know binary numbers up through 7 4. Memorize or look up on a chart Powered by DeSiaMore

37. 37 Converting from Binary to Octal Example: 111|101|100 100 = 4 101 = 5 111 = 7 754 octal Example: 1|010|110 110 = 6 010 = 2 001 = 1 126 octal Powered by DeSiaMore

38. 38 Convert from Octal to Decimal 754 octal 7 x 8 2 = 7 x 64 = 448 5 x 8 1 = 5 x 8 = 40 4 x 8 0 = 4 x 1 = 4 482 10 126 octal 1 x 8 2 = 1 x 64 = 64 2 x 8 1 = 2 x 8 = 16 6 x 8 0 = 6 x 1 = 6 86 10 Powered by DeSiaMore

39. 39  Converting from Binary to Hex 1. Starting from the rightmost digit, group into 4 instead of 3, convert each group of 4 into a hex digit 2. Example: 1010110 0110 = 6 0101 = 5 3. Now convert to decimal as before 5 x 16 1 = 5 x 16 = 80 6 x 16 0 = 6 x 1 = 6 86 10 Powered by DeSiaMore

40. 40 Converting Decimal to Other Base Systems Rules for converting decimals involve division Divide the decimal by the other base  Example: Convert to binary, divide by 2  Get a quotient and a remainder  Remainder becomes the new digit in the number  Quotient replaces the original number  Do this until quotient = 0 Powered by DeSiaMore

41. 41 Converting Decimal to Other Base Systems  Algorithm for Converting Decimal to other Base Systems While quotient is not zero  Divide decimal number by new base  Make remainder the next digit to the left in the answer  Replace decimal number with the quotient Powered by DeSiaMore

42. 42 Converting Decimal to Other Base Systems  Decimal to Binary Example: 45 decimal 45 / 2 = 22 quotient with 1 remainder 22/ 2 = 11 quotient with 0 remainder 11/2 = 5 quotient with 1 remainder 5/2 = 2 quotient with 1 remainder 2/2 = 1 quotient with 0 remainder 1/2 = 0 quotient with 1 remainder Final answer: 101101 binary Powered by DeSiaMore

43. 43 Converting Decimal to Other Base Systems Decimal to octal Example: 321 decimal 321/8 = 40 is quotient and 1 is remainder 40/8 = 5 is quotient and 0 is remainder 5/8 = 0 is quotient and 5 is remainder Final Answer: 501 octal Powered by DeSiaMore

44. 44 Converting Decimal to Other Base Systems  Decimal to hex Example: 43 decimal 43/16 = 2 is quotient and 11 is remainder, 11=B in hex 2/16 = 0 is quotient and 2 is remainder Final answer: 2B in hex Powered by DeSiaMore

45. 45 Binary Arithmetic Similar to decimal - same rules, different digits Basic idea is: 0 + 1 = 1, 1 + 1 = 2, 2 + 1 = 3 until we have 9 + 1 = what? No symbol for 10 in decimal, reuse the same digits Rely on position to tell what the meaning is Rightmost digit becomes 0 and carry 1 to next position: 9 + 1 10 Powered by DeSiaMore

46. 46 Binary Arithmetic In Binary addition there is the same idea Run out of digits a lot sooner 0 + 1 = 1 and 1 + 1 = 0 with a carry of 1 Repeat this for a large number with multiple columns Rules of Binary Addition: 0 + 1 = 1 1 + 1 = 0 with a carry of 1 1 + 1 + 1 = 1 with a carry of 1 Powered by DeSiaMore

47. 47 Binary Arithmetic Add two numbers Examples 10110 46 1010 10 + 11011 27 + 1111 15 101 5 1110 14 + 101 5 + 111 7 Powered by DeSiaMore

48. 48 Binary Arithmetic Binary Subtraction  Similar to decimal  Rules for borrowing 0 - 0 = 0 1 - 1 = 0 0 - 1 = Can’t do this unless you borrow from the next column with a 1 Powered by DeSiaMore

49. 49 Binary Arithmetic Binary Subtraction  Cross out the 1 and put a zero in its place  In decimal, you reduce the number by 1  In binary, you cross out the 1 and put a zero  The 1 you borrowed becomes: 10 - 1 = 1 Why? Powered by DeSiaMore

50. 50 Binary Arithmetic Binary Subtraction  10-1 = 1 because 10 is 2 in binary and 2 - 1 = 1  Examples: 1011 11 110010 50 - 100 4 - 10101 21 1000 8 1001001 73 - 11 3 - 10111 23 Powered by DeSiaMore

51. 51 Binary Arithmetic Multiplication  Rules  1 x 0 = 0  0 x 0 = 0  1 x 1 = 1 11010 26101110 46 10101 21 x 101 5 x 10 2 x 111 7 ------------------ ------------------- ----------------- Powered by DeSiaMore

52. 52 Converting Numbers with Fractions Powered by DeSiaMore

53. 53 Signed Numbers Powered by DeSiaMore

54. 54 BCD Powered by DeSiaMore