Fundamentals of Programming Session 2

Transcription

1 Fundamentals of Programming Session 2 Instructor: Reza Entezari-Maleki entezari@ce.sharif.edu 1 Fall 2013 Sharif University of Technology

2 Outlines Programming Language Binary numbers Addition Subtraction Division Multiplication One s and Two s Complement 2

3 Programming Language A programming language is an artificial language designed to express computations that can be performed by a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine, to express algorithms precisely, or as a mode of human communication. Many programming languages have some form of written specification of their syntax (form) and semantics (meaning). Some languages are defined by a specification document. For example, the C programming language is specified by an ISO Standard. Other languages, such as Perl, have a dominant implementation that is used as a reference. 3

4 Programming Language Evolution of Programming Languages: First Generation: Machine languages Strings of numbers giving machine specific instructions Example: Computer only understands machine language instructions. Second Generation: Assembly languages English-like abbreviations representing elementary computer operations (translated via assemblers) 4

5 Programming Language Example: LOAD BASEPAY ADD OVERPAY STORE GROSSPAY Third Generation : High-level languages Codes similar to everyday English Use mathematical notations (translated via compilers) Example: grosspay = basepay + overpay 5

6 6 Programming Language

7 Common Software Operating System Assemblers Compilers Interpreters 7

8 Binary numbers 8 Binary numbers Why binary? Computers are built using digital circuits Inputs and outputs can have only two values True (high voltage) or false (low voltage) Converting base 10 to base 2 Octal and hexadecimal Integers Unsigned integers Integer addition Signed integers C bit operators And, or, not, and xor Shift-left and shift-right

9 Base 10 and Base 2 Base 10 Each digit represents a power of = 4* * * *10 0 Base 2 Each bit represents a power of = 1* * * * *2 0 = 22 Question: What is the binary representation of number 12? Response:

10 Base 8 Octal (base 8) Digits 0, 1,, 7 Thus the 12 bit binary number converted to Oct is: = = = = = = = = 7 10

11 Base 8 Question: What is the octal representation of number 118 (in base 10)? Response: 166 Question: What is the octal representation of number (in base 2)? Response: 1305 Question: What is the binary representation of number 1472 (in base 8)? Response:

12 Base 16 Hexadecimal (base 16) Digits 0, 1,, 9, A, B, C, D, E, F Thus the 16-bit binary number converted to Hex is: B2A = = = = = = A 0011 = = B 0100 = = C 0101 = = D 0110 = = E 0111 = = F 12

13 Base 16 Question: What is the hexadecimal representation of number 375 (in base 10)? Response: 177 Question: What is the hexadecimal representation of number (in base 2)? Response: 2C5 Question: What is the binary representation of number 6A4D2 (in base 16)? Response:

14 Integers Fixed number of bits in memory Short: usually 16 bits Int: 16 or 32 bits Long: 32 bits Unsigned integer No sign bit Always positive or 0 14 Example of unsigned int

15 Decimal Addition Add 3758 to 4657: ) Add = 15 Write down 5, carry 1 2) Add = 11 Write down 1, carry 1 3) Add = 14 Write down 4, carry ) Add = 8 Write down 8 15

16 Binary Addition Rules: = = = 1 (just like in decimal) = 2 10 = 10 2 = 0 with 1 to carry = 3 10 = 11 2 = 1 with 1 to carry

17 Binary Addition 17 Example 1: Add binary to Col 1) Add = 1 Write 1 Col 2) Add = 1 Write 1 Col 3) Add = 2 (10 in binary) Write 0, carry 1 Col 4) Add = 2 Write 0, carry 1 Col 5) Add = 3 (11 in binary) Write 1, carry 1 Col 6) Add = 2 Write 0, carry 1 Col 7) Bring down the carried 1 Write 1

18 Binary Addition You can always check your answer by converting the figures to decimal, doing the addition, and comparing the answers Verification = =

19 Decimal Subtraction Subtract 4657 from 8025: ) Try to subtract 5 7 can t. Must borrow 10 from next column. Add the borrowed 10 to the original 5. Then subtract 15 7 = 8. 2) Try to subtract 1 5 can t. Must borrow 10 from next column. But next column is 0, so must go to column after next to borrow. Add the borrowed 10 to the original 0. Now you can borrow 10 from this column. Add the borrowed 10 to the original 1.. Then subtract 11 5 = 6 3) Subtract 9 6 = 3 4) Subtract 7 4 = 3

20 Decimal Subtraction So when you cannot subtract, you borrow from the column to the left. The amount borrowed is 1 base unit, which in decimal is 10. The 10 is added to the original column value, so you will be able to subtract. 20

21 Binary Subtraction In binary, the base unit is 2 So when you cannot subtract, you borrow from the column to the left. The amount borrowed is 2. The 2 is added to the original column value, so you will be able to subtract. 21

22 Binary Subtraction Example 1: Subtract binary from Col 1) Subtract 1 0 = 1 Col 2) Subtract 1 0 = 1 Col 3) Try to subtract 0 1 can t. Must borrow 2 from next column. But next column is 0, so must go to column after next to borrow. Add the borrowed 2 to the 0 on the right. Now you can borrow from this column (leaving 1 remaining). Add the borrowed 2 to the original 0. Then subtract 2 1 = 1 Col 4) Subtract 1 1 = 0 Col 5) Try to subtract 0 1 can t. Must borrow from next column. Add the borrowed 2 to the remaining 0. Then subtract 2 1 = 1 Col 6) Remaining leading 0 can be ignored.

23 Binary Subtraction Subtract binary from : Verification = =

24 One s and Two s Complement Consider only numbers in a range E.g., five-digit car odometer: 0, 1,, E.g., eight-bit numbers 0, 1,, 255 Roll-over when you run out of space E.g., car odometer goes from to 0, 1, E.g., eight-bit number goes from 255 to 0, 1, 24

25 One s and Two s Complement One s complement: flip every bit E.g., b is (i.e., 69 in base 10) One s complement is That s simply Subtracting from is easy (no carry needed!) b one s complement 25 Two s complement Add 1 to the one s complement E.g., (255 69)

26 One s and Two s Complement 26 Computing a b for unsigned integers Same as a b Same as a + (255 b) + 1 Same as a + onecomplement(b) + 1 Same as a + twocomplement(b) Example: The original number 69: One s complement of 69: Two s complement of 69: Add to the number 172: The sum comes to: Equals: 103 in base

27 One s and Two s Complement 27 Sign-magnitude representation Use one bit to store the sign Zero for positive number One for negative number Examples E.g., E.g., Hard to do arithmetic this way, so it is rarely used Complement representation One s complement Flip every bit E.g., Two s complement Flip every bit, then add 1 E.g.,