QUIZ: Generations of computer technology Hardware: 1. 2. 3. 4. 5. 1
QUIZ: Generations of computer technology Software: 1. 2. 3. 4. 5. 6. 2
Chapter 2 Binary Values and Number Systems
Numbers Natural numbers, a.k.a. positive integers 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 4 2
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, 3/7, -2/5 Real numbers In general cannot be represented as the quotient of any two integers. They have an infinite # of fractional digits. Example: Pi = 3.14159265 5 3
2.2 Positional notation How many ones (units) are there in 642? 600 + 40 + 2 6 x 100 + 4 x 10 + 2 x 1 6 x 10 2 + 4 x 10 1 + 2 x 10 0 10 is called the base This is called the polynomial expansion of the number We also write 642 10 6 4
QUIZ Write the polynomial expansion of 642 9 (642 in base nine), and convert it to decimal. Use the previous expansion as example: 600 + 40 + 2 6 x 100 + 4 x 10 + 2 x 1 6 x 10 2 + 4 x 10 1 + 2 x 10 0 7 4
QUIZ How many ones (units) are there in 642 9 (642 in base nine)? 6 x 9 2 + 4 x 9 1 + 2 x 9 0 = 524 10 642 9 = 524 10 8 4
Positional Notation The base of a number determines how many digits are used and the value of each digit s position. To be specific: In base R, there are R digits, from 0 to R-1 The positions have for values the powers of R, from right to left: R 0, R 1, R 2, 9 5
Positional Notation Formula: R is the base of the number d n * R n-1 + d n-1 * R n-2 +... + d 2 * R + d 1 n is the number of digits in the number d is the digit in the i th position in the number 10 7
Positional Notation reloaded The text shows the digits numbered like this: d n * R n-1 + d n-1 * R n-2 +... + d 2 * R + d 1 but, in CS, the digits are numbered from zero, to match the power of the base: d n-1 * R n-1 + d n-2 * R n-2 +... + d 1 * R 1 + d 0 * R 0 11 7
QUIZ What is 642 in base 13? 12 68
QUIZ What is 642 in base 13? + 6 x 13 2 = 6 x 169 = 1014 + 4 x 13 1 = 4 x 13 = 52 + 2 x 13º = 2 x 1 = 2 = 1068 in base 10 642 13 = 1068 10 13 68
Nota bene! In a given base R, the digits range from 0 up to R 1 R itself cannot be a digit in base R Trick problem: Convert the number 473 from base 6 to base 10 14
Binary 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 15 9
QUIZ: Converting Binary to Decimal What is the decimal equivalent of the binary number 110 1110? 110 1110 2 =??? 10 16 13
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 = 110 in base 10 = 110 10 17 13
Base 8 = octal system What is the decimal equivalent of the octal number 642? 642 8 =??? 10 18 11
Converting Octal to Decimal What is the decimal equivalent of the octal number 642? 6 x 8 2 = 6 x 64 = 384 + 4 x 8 1 = 4 x 8 = 32 + 2 x 8º = 2 x 1 = 2 Add the above = 418 in base 10 = 418 10 19 11
Bases Higher than 10 How are digits in bases higher than 10 represented? Base 16 (hexadecimal, a.k.a. hex) has 16 digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, F 20 10
Converting Hexadecimal to Decimal What is the decimal equivalent of the hexadecimal number DEF? D x 16 2 = 13 x 256 = 3328 + E x 16 1 = 14 x 16 = 224 + F x 16º = 15 x 1 = 15 = 3567 in base 10 21
QUIZ: 2AF 16 =??? 10 22
Are there any non-positional number systems? Hint: Why did the Roman civilization have no contributions to mathematics? 23
QUIZ: Convert to decimal 0001 0011 2 = C7 16 = 42 6 = 71 8 = 24
Today we ve covered pp.33-39 of text (stopped before Arithmetic in Other Bases) Solve in notebook as individual work for next class: 1, 2, 3, 4, 5, 20, 21 25
QUIZ: Convert to decimal 1101 0011 2 = AB7 16 = 513 7 = 692 8 = 26
The inverse problem: Converting Base 10 to Other Bases Algorithm for converting a number in base 10 to any other base R: While (the quotient is not zero) Divide the decimal number by R Make the remainder the next digit to the left in the answer Replace the original decimal number with the quotient Known as repeated division (by the base) 27 19
Converting Decimal to Binary Example: Convert 179 10 to binary 179 2 = 89 rem. 1 2 = 44 rem. 1 2 = 22 rem. 0 2 = 11 rem. 0 MSB LSB 2 = 5 rem. 1 2 = 2 rem. 1 2 = 1 rem. 0 179 10 = 10110011 2 2 = 0 rem. 1 Notes: The first bit obtained is the rightmost (a.k.a. LSB) The algorithm stops when the quotient (not the remainder!) becomes zero 28 19
Repeated division QUIZ Convert 42 10 to binary 42 2 = rem. 42 10 = 2 29 19
The repeated division algorithm can be used to convert from any base into any other base (but normally we use it only for 10 2) For next time: Read text example on p.43: Converting Decimal to Hex using repeated division 30
Addition in Binary Remember that there are only 2 digits in binary, 0 and 1 1 + 1 is 0 with a carry 0 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 Carry Values 31 14
Addition QUIZ 1 0 1 0 1 1 0 +1 0 0 0 0 1 1 Carry values go here Check in base ten! 32 14
Putting it all together! Computer hardware Python (or another high-level language) Convert decimal to binary Compute in binary 1 1001 + 101 1110 Convert binary to decimal 33
Direct conversions between bases that are powers of 2 binary hexadecimal octal 34
Converting Binary to Octal Mark groups of three (from right) Convert each group 10101011 10 101 011 2 5 3 10101011 is 253 in base 8 35 17
Converting Binary to Hexadecimal Mark groups of four (from right) Convert each group 10101011 1010 1011 A B 10101011 is AB in base 16 36 18
Extra-credit QUIZ: 37
Converting Octal to Hexadecimal End-of-chapter ex. 25: Explain how base 8 and base 16 are related 10 101 011 1010 1011 2 5 3 A B 253 in base 8 = AB in base 16 38 18
End-of-chapter ex.37 Perform the following octal additions: a. 770 + 665 b. 101 + 707 39 14
Read and take notes: Binary SUBTRACTION (with borrow bits) p.40 of text 40
Today we ve covered pp.36-43 of text (stopped before Binary Values and Computers) Solve in notebook as individual work for next class: 6 through 11 41
Addition QUIZ 1 1 1 0 1 1 0 +1 0 0 0 1 1 1 Carry values go here Check in base ten! 42 14
End-of-chapter ex.37 Perform the following octal additions: c. 202 + 667 43 14
Counting 44
Basic skill: counting (in any base!) 0, 1, 2, 3, 4, 5, 0, 1, 10, 11, 100, 101, 45
Hex 46
Conclusion: In order to represent any octal digit, we need at most bits In order to represent any hex digit, we need at most bits 47
Binary and Computers Word = group of bits that the computer processes at a time The number of bits in a word determines the word length of the computer. It is usually a multiple of 8. 1 Byte = 8 bits 8, 16, 32, 64-bit computers 128? 256? 48 23
6-bit computers: an Evolutionary dead-end? Motivated by the 6-bit codes for printable graphic patterns created by the U.S. Army and Navy 6, 18, 24, 36, 48-bit words Some history: 18-Bit Computers from DEC 36-bit Wikipedia Edged out of the market by the need for floatingpoint numbers IBM System/360 (1965) 8-bit microprocessors (1970s) Not in text 49
6-bit computers: an Evolutionary dead-end? Unisys is still successful with their 36- and 48-bit machines : Clearpath Dorado line of 36-bit CISC high-end servers Clearpath Libra line of 48-bit mainframes Not in text although they are being transitioned to Intel Xeon chips (64-bit): see article 50
Grace Murray Hopper Ph.D. in mathematics Wrote A-0, the world s first compiler, in 1952! Co-invented COBOL Rear-admiral of the US Navy Nanosecond wires 51
Grace also liked telling this story Harvard University Mark II Aiken Relay Calculator 52
Ethical Issues Tenth Strand What do the following acronyms stand for: ACM? IEEE? What is the tenth strand? Why the tenth? 53
Why the tenth? 54 A: In the 1989 ACM report ( Computing as a Discipline p.12), the following 9 areas (strands) of CS were defined: Algorithms and data structures Programming languages Computer Architecture Numerical and symbolic computations Operating systems (OS) Software engineering Databases Artificial intelligence (AI) Human-computer interaction
Ethical Issues Tenth Strand The latest official release of the IEEE/ACM CS Curriculum was in 2001: 3 levels of organization: areas knowledge units topics There are now 14 areas, and the tenth strand is listed in position 12 What does SP stand for? How many knowledge units does the SP area have? Name two of these units! 55
Chapter Review questions Describe positional notation (polynomial in the base) Convert numbers in other bases to base 10 Convert base-10 numbers to numbers in other bases Add and subtract in binary Convert between bases 2, 8, and 16 using groups of digits Count in binary Explain the importance to computing of bases that are powers of 2 56 24 6
Chapter Review questions IEEE/ACM CS Curriculum and the Tenth Strand 57 24 6
Homework for Ch.2 End-of-ch. 23, 24, 25, 28, 29, 30, 33, 38 Due next Wednesday, Sep.9 The latest homework assigned is always available on the course webpage 58
FYI: Subtraction in Binary Remember borrowing? 1 2 0 2 0 2 1 0 1 0 1 1 1-1 1 1 0 1 1 0 0 1 1 1 0 0 Borrow values Check in base ten! 59 15
Subtraction QUIZ 1 0 1 0 0 0 0-1 0 0 1 0 1 Borrow values go here Check in base ten! 60 15
Subtraction QUIZ 1 1 1 0 1 0 0-1 1 0 1 1 1 Borrow values go here Check in base ten! 61 15
Another subtraction QUIZ 1 0 1 0 0 0 1-1 0 0 1 1 1 Borrow values go here Check in base ten! 62 15