1. Computer Math CPS120: Lecture 3

2. Binary computers have storage units called binary digits or bits: Low Voltage = 0 High Voltage = 1 all bits have 0 or 1 22 Binary and Computers

3. 8 bits = 1 byte The number of bytes in a word determines the word length of the computer: 32-bit machines 64-bit machines etc. 23 Binary and Computers

4. Binary Representations One bit can be either 0 or 1. Therefore, one bit can represent only two things. To represent more than two things, we need multiple bits. Two bits can represent four things because there are four combinations of 0 and 1 that can be made from two bits: 00, 01, 10,11.

5. Binary Representations (Cont’d) If we want to represent more than four things, we need more than two bits. Three bits can represent eight things because there are eight combinations of 0 and 1 that can be made from three bits.

6. Binary & Alphanumeric Codes A binary code is a group of n bits that assume up to 2 n distinct combinations of 1’s and 0’s with each combination representing one element of the set that is being coded- i.e. permutations With two bits we can form a set of four elements With three bits we can represent 8 elements With four bits we can represent 16 elements

7. Binary Representations (Cont’d) Figure 3.4 Bit combinations

8. Binary Representations (Cont’d) In general,  bits can represent 2  things because there are 2  combinations of 0 and 1 that can be made from n bits. Note that every time we increase the number of bits by 1, we double the number of things we can represent.

9. Memory Units 1 nibble 1 byte 1 word 1 long word 1 quad word 1 octa-word 4 consecutive bits 8 consecutive bits 2 consecutive bytes 4 consecutive bytes 8 consecutive bytes 16 consecutive bytes

10. Larger Units of Memory 1 Kilobyte 1 Megabyte 1 Gigabyte 1 Terabyte 1 Petabyte 1 Exabyte 1024 bytes ~10 6 bytes ~10 9 bytes ~10 12 bytes ~10 15 bytes ~10 18 bytes 32 Mb = 32*10 3 Kb = 32 * 10 3 *1024 bytes = 32,768,000 bytes

11. Representing Data The computer knows the type of data stored in a particular location from the context in which the data are being used; i.e. individual bytes, a word, a longword, etc 01100011 01100101 01000100 01000000 Bytes: 99 (10, 101 (10, 68 (10, 64 (10 Two byte words: 24,445 (10 and 17,472 (10 Longword: 1,667,580,992 (10

12. 2 Natural Numbers Zero and any number obtained by repeatedly adding one to it. Examples: 100, 0, 45645, 32 Negative Numbers A value less than 0, with a – sign Examples: -24, -1, -45645, -32 Numbers

13. 3 Integers A natural number, a negative number, zero Examples: 249, 0, - 45645, - 32 Rational Numbers An integer or the quotient of two integers Examples: -249, -1, 0, ¼, - ½ Numbers (Cont’d)

14. 4 How many ones are there in 642? 600 + 40 + 2 ? Or is it 384 + 32 + 2 ? -- Octal Or maybe… 1536 + 64 + 2 ? -- Hexadecimal Natural Numbers

15. 5 642 is 600 + 40 + 2 in BASE 10 The base of a number determines the number of digits and the value of digit positions Natural Numbers

16. 6 Continuing with our example… 642 in base 10 positional notation is: 6 x 10² = 6 x 100 = 600 + 4 x 10¹ = 4 x 10 = 40 + 2 x 10º = 2 x 1 = 2 = 642 in base 10 This number is in base 10 The power indicates the position of the number Positional Notation

17. 7 d n * R n-1 + d n-1 * R n-2 +... + d 2 * R + d 1 As a formula: 642 is: 6 3 * 10 2 + 4 2 * 10 + 2 1 R is the base of the number n is the number of digits in the number d is the digit in the i th position in the number Positional Notation

18. 68 What if 642 has the base of 13? 642 in base 13 is equivalent to 1068 in base 10 + 6 x 13² = 6 x 169 = 1014 + 4 x 13¹ = 4 x 13 = 52 + 2 x 13º = 2 x 1 = 2 = 1068 in base 10 Positional Notation

19. 9 Decimal is base 10 and has 10 digits: 0,1,2,3,4,5,6,7,8,9 Binary is base 2 and has 2 digits: 0,1 For a number to exist in a given number system, the number system must include those digits. For example: The number 284 only exists in base 9 and higher. Binary

20. Codes Given any positive integer base (RADIX) N, there are N different individual symbols that can be used to write numbers in the system. The value of these symbols range from 0 to N-1 All systems we use in computing are positional systems 495 = 400 + 90 +5

21. Number Systems We use the DECIMAL (10 system Computers use BINARY (2 or some shorthand for it like OCTAL (8 or HEXADECIMAL (16

22. BinaryOctalDecimal 00000 00111 01022 01133 10044 10155 11066 11177 100108 1001119 10101210 16 Power of 2 Number System

23. 10 How are digits in bases higher than 10 represented? Base 16: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F Bases Higher than 10

24. Conversions

25. Decimal Equivalents Assuming the bits are unsigned, the decimal value represented by the bits of a byte can be calculated as follows: 1. Number the bits beginning on the right using superscripts beginning with 0 and increasing as you move left Note: 2 0, by definition is 1 2. Use each superscript as an exponent of a power of 2 3. Multiply the value of each bit by its corresponding power of 2 4. Add the products obtained

26. What is the decimal equivalent of the octal number 642? 6 x 8² = 6 x 64 = 384 + 4 x 8¹ = 4 x 8 = 32 + 2 x 8º = 2 x 1 = 2 = 418 in base 10 11 Converting Octal to Decimal

27. What is the decimal equivalent of the hexadecimal number DEF? D x 16² = 13 x 256 = 3328 + E x 16¹ = 14 x 16 = 224 + F x 16º = 15 x 1 = 15 = 3567 in base 10 Remember, base 16 is 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F Converting Hexadecimal to Decimal

28. What is the decimal equivalent of the binary number 1101110? 1 x 2 6 = 1 x 64 = 64 + 1 x 2 5 = 1 x 32 = 32 + 0 x 2 4 = 0 x 16 = 0 + 1 x 2 3 = 1 x 8 = 8 + 1 x 2 2 = 1 x 4 = 4 + 1 x 2 1 = 1 x 2 = 2 + 0 x 2º = 0 x 1 = 0 = 112 in base 10 13 Converting Binary to Decimal

29. Horner’s Method Another procedure to calculate the decimal equivalent of a binary number Note: This method works with any base Horner’s Method: Step 1: Start with the first digit on the left Step 2: Multiply it by the base Step 3: Add the next digit Step 4: Multiply the sum by the base Step 5: Continue the process until you add the last digit

30. Groups of Three (from right) Convert each group 10101011 10 101 011 2 5 3 10101011 is 253 in base 8 17 Converting Binary to Octal

31. Binary to Hex Step 1: Form four-bit groups beginning from the rightmost bit of the binary number If the last group (at the leftmost position) has less than four bits, add extra zeros to the left of the group to make it a four-bit group 0110011110101010100111 becomes 0001 1001 1110 1010 1010 0111 Step 2: Replace each four-bit group by its hexadecimal equivalent 19EAA7 (16

32. Converting Decimal to Other Bases Step 1: Divide the number by the base you are converting to (r) Step 2: Successively divide the quotients by (r) until a zero quotient is obtained Step 3: The decimal equivalent is obtained by writing the remainders of the successive division in the opposite order in which they were obtained Know as modulus arithmetic Step 4: Verify the result by multiplying it out

33. 222 13 0 16 3567 16 222 16 13 32 16 0 36 62 13 32 48 47 14 32 15 D E F 21 Converting Decimal to Hexadecimal

34. Addition & Subtraction Terms A + B A is the augend B is the addend C – D C is the minuend D is the subtrahend

35. Addition Rules – All Bases Addition Step 1: Add a column of numbers Step 2: Determine if there is a single symbol for the result Step 3: If so, write it and go to the next column. If not, write the accompanying number and carry the appropriate value to the next column

36. Addition of Binary Numbers Rules for adding or subtracting very similar to the ones in decimal system Limited to only two digits 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 carry 1

37. Remember: there are only 2 digits in binary: 0 and 1 Position is key, carry values are used: Carry Values 1 1 1 1 1 1 1 0 1 0 1 1 1 +1 0 0 1 0 1 1 1 0 1 0 0 0 1 0 14 Arithmetic in Binary

38. Number Overflow Overflow occurs when the value that we compute cannot fit into the number of bits we have allocated for the result. For example, if each value is stored using eight bits, adding 127 to 3 overflow: 01111111 + 00000011 10000010 Overflow is a classic example of the type of problems we encounter by mapping an infinite world onto a finite machine.

39. Subtraction Rules – All Bases Step1: Start with the rightmost column, if the column of the minuend is greater than that of the subtrahend, do the subtraction, if not… Step 2: Borrow one unit from the digit to the left of the once being processed The borrowed unit is equal to “borrowing” the radix Step 4: Decrease the column form which you borrowed by one Step 3: Subtract the subtrahend from the minuend and go to the next column

40. Remember borrowing? Apply that concept here: 1 2 2 0 2 1 0 1 0 1 1 1 - 1 1 1 0 1 1 0 0 1 1 1 0 0 15 Subtracting Binary Numbers

41. Addition & Subtraction of Hex Due to the propensity for errors in binary, it is preferable to carry out arithmetic in hexadecimal and convert back to binary If we need to borrow in hex, we borrow 16 It is convenient to think “in decimal” and then translate the results back to hex

42. Representing Real Numbers Real numbers have a whole part and a fractional part. For example 104.32, 0.999999, 357.0, and 3.14159 the digits represent values according to their position, and those position values are relative to the base. The positions to the right of the decimal point are the tenths position (10 -1 or one tenth), the hundredths position (10 -2 or one hundredth), etc.

43. Representing Real Numbers (Cont’d) In binary, the same rules apply but the base value is 2. Since we are not working in base 10, the decimal point is referred to as a radix point. The positions to the right of the radix point in binary are the halves position (2 -1 or one half), the quarters position (2 -2 or one quarter), etc.

44. Representing Real Numbers (Cont’d) A real value in base 10 can be defined by the following formula: The representation is called floating point because the number of digits is fixed but the radix point floats.

45. Representing Real Numbers (Cont’d) Likewise, a binary floating –point value is defined by the following formula: sign * mantissa * 2 exp

46. Representing Real Numbers (Cont’d) Scientific notation is a term with which you may already be familiar, so we mention it here. Scientific notation is a form of floating- point representation in which the decimal point is kept to the right of the leftmost digit. For example, 12001.32708 would be written as 1.200132708E+4 in scientific notation.

47. Representing Text To represent a text document in digital form, we simply need to be able to represent every possible character that may appear. There are finite number of characters to represent. So the general approach for representing characters is to list them all and assign each a binary string. A character set is simply a list of characters and the codes used to represent each one. By agreeing to use a particular character set, computer manufacturers have made the processing of text data easier.

48. Alphanumeric Codes American Standard Code for Information Interchange (ASCII) 7-bit code Since the unit of storage is a bit, all ASCII codes are represented by 8 bits, with a zero in the most significant digit H e l l o W o r l d 48 65 6C 6C 6F 20 57 6F 72 6C 64 Extended Binary Coded Decimal Interchange Code (EBCDIC)

49. The ASCII Character Set ASCII stands for American Standard Code for Information Interchange. The ASCII character set originally used seven bits to represent each character, allowing for 128 unique characters. Later ASCII evolved so that all eight bits were used which allows for 256 characters.

50. The ASCII Character Set (Cont’d)

51. Note that the first 32 characters in the ASCII character chart do not have a simple character representation that you could print to the screen.

52. The Unicode Character Set The extended version of the ASCII character set is not enough for international use. The Unicode character set uses 16 bits per character. Therefore, the Unicode character set can represent 216, or over 65 thousand, characters. Unicode was designed to be a superset of ASCII. That is, the first 256 characters in the Unicode character set correspond exactly to the extended ASCII character set.

53. The Unicode Character Set (Cont’d) A few characters in the Unicode character set

54. Representing Signed Numbers Remember, all numeric data is represented inside the computer as 1s and 0s Arithmetic operations, particularly subtraction raise the possibility that the result might be negative Any numerical convention needs to differentiate two basic elements of any given number, its sign and its magnitude Conventions Sign-magnitude Ten's complement Two’s complement One’s complement

55. Representing Negatives It is necessary to choose one of the bits of the “basic unit” as a sign bit Usually the leftmost bit By convention, 0 is positive and 1 is negative Positive values have the same representation in all conventions However, in order to interpret the content of any memory location correctly, it necessary to know the convention being used used for negative numbers

56. Comparing the Conventions

57. Representing Negative Values You have used the signed-magnitude representation of numbers since grade school. The sign represents the ordering, and the digits represent the magnitude of the number.

58. Representing Negative Values (Cont’d) There is a problem with the sign- magnitude representation: There are two representations of zero. There is plus zero and minus zero. Two representations of zero within a computer can cause unnecessary complexity.

59. Sign-Magnitude For a basic unit of N bits, the leftmost bit is used exclusively to represent the sign The remaining (N-1) bits are used for the magnitude The range of number represented in this convention is –2 N+1 to +2 N-1 -1

60. Sign-magnitude Operations Addition of two numbers in sign-magnitude is carried out using the usual conventions of binary arithmetic If both numbers are the same sign, we add their magnitude and copy the same sign If different signs, determine which number has the larger magnitude and subtract the other from it. The sign of the result is the sign of the operand with the larger magnitude If the result is outside the bounds of –2 n+1 to +2 n-1 –1, an overflow results

61. Other Ways of Representing Negative Values If we allow only a fixed number of values, we can represent numbers as just integer values, where half of them represent negative numbers. For example, if the maximum number of decimal digits we can represent is two, we can let 1 through 49 be the positive numbers 1 through 49 and let 50 through 99 represent the negative numbers -50 through -1.

62. Representing Negative Values (Cont’d) To perform addition within this scheme, you just add the numbers together and discard any carry.

63. Representing Negative Values (Cont’d) A-B=A+(-B). We can subtract one number from another by adding the negative of the second to the first.

64. Representing Negative Values (Cont’d) There is a formula that you can use to compute the negative representation: This representation of negative numbers is called the ten’s complement.

65. Representing Negative Values (Cont’d) Two’s Complement: To make it easier to look at long binary numbers, we make the number line vertical.

66. Representing Negative Values (Cont’d) Addition and subtraction are accomplished the same way as in 10’s complement arithmetic: -12710000001 + 100000001 -12610000010 Notice that with this representation, the leftmost bit in a negative number is always a 1.

67. Two’s Complement Convention A positive number is represented using a procedure similar to sign-magnitude To express a negative number 1. Express the absolute value of the number in binary 2. Change all the zeros to ones and all the ones to zeros (called “complementing the bits”) 3. Add one to the number obtained in Step 2 – The range of negative numbers is one larger than the range of positive numbers – Given a negative number, to find its positive counterpart, use steps 2 & 3 above

68. Two’s Complement Operations Addition: Treat the numbers as unsigned integers The sign bit is treated as any other number Ignore any carry on the leftmost position Subtraction Treat the numbers as unsigned integers If a "borrow" is necessary in the leftmost place, borrow as if there were another “invisible” one-bit to the left of the minuend

69. Overflows in Two’s Complement The range of values in two’s-complement is –2 n+1 to +2 n-1 –1 Results outside this band are overflows In all overflow conditions, the sign of the result of the operation is different than that of the operands If the operands are positive, the result is negative If the operands are negative, the result is positive

70. One’s Complement Devised to make the addition of two numbers with different signs the same as two numbers with the same sign Positive numbers are represented in the usual way For negatives STEP 1: Start with the binary representation of the absolute value STEP 2: Complement all of its bits

71. One's Complement Operations Treat the sign bit as any other bit For addition, carry out of the leftmost bit is added to the rightmost bit – end-around carry