4/8/17. Admin. Assignment 5 BINARY. David Kauchak CS 52 Spring 2017

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1 4/8/17 Admin! Assignment 5 BINARY David Kauchak CS 52 Spring 2017 Diving into your computer Normal computer user 1

2 After intro CS After 5 weeks of cs52 What now One last note on CS52 memory address binary representation of code How do we get this instructions (assembly code) 2

3 Encoding assembly instructions Binary numbers revisited What number does 1001 represent in binary opcode rx ry rz Depends! Is it a signed number or If signed, what convention are we using Twos complement Twos complement For a number with n digits the high order bit represents -2 n-1 What number is it signed (twos complement) signed (twos complement)

4 Twos complement Twos complement What number is it What number is it signed (twos complement) signed (twos complement) Twos complement How many numbers can we represent with each approach using 4 bits 16 (2 4 ) numbers, 0000, 0001,., 1111 Doesn t matter the representation! Twos complement How many numbers can we represent with each approach using 32 bits billion numbers signed (twos complement) signed (twos complement)

5 Twos complement What is the range of numbers that we can represent for each approach with 4 bits : 0, 1, 15 signed: -8, -7,, 7 signed (twos complement) binary representation binary representation twos complement binary representation twos complement

6 binary representation twos complement binary representation twos complement binary representation twos complement How can you tell if a number is negative binary representation twos complement High order bit! 6

7 A two s complement trick You can also calculate the value of a negative number represented as twos complement as follows:! Flip all of the bits (0 " 1 and 1" 0)! Add 1! The resulting number is the magnitude of the original negative number A two s complement trick You can also calculate the value of a negative number represented as twos complement as follows:! Flip all of the bits (0 " 1 and 1" 0)! Add 1! The resulting number is the magnitude of the original negative number flip the bits add flip the bits add Shifting Shifting 37 >> 2 number to be shifted right shift number of positions to shift 7

8 Shifting Shifting 37 >> 2 37 >> shift right two positions decimal form Shifting Shifting with fixed bit representations In real computers, we generally have a fixed number of bits we use to represent a number (e.g. 8-bits, 16-bits, 32-bits) 37 >> shift right three positions decimal form 8

9 Shifting 8-bit numbers 37 pad with 0s 37 >> 2 What is 37 as an 8-bit binary number Shifting 8-bit numbers >> 2 How do we fill in the leftmost bits shift right two positions Shifting 8-bit numbers 37 >> 2 Shifting 8-bit numbers 37 >> 2 How do we fill in the leftmost bits shift right two positions shift right two positions (discard away bits shifting off) decimal form 9

10 Shifting 8-bit numbers 15 << 2 Shifting 8-bit numbers 15 << 2 15 Shifting 8-bit numbers 15 << 2 Shifting 8-bit numbers 15 << shift left two positions 10

11 Shifting 8-bit numbers 15 << 2 Shifting 8-bit numbers 15 << shift left two positions shift left two positions Shifting 8-bit numbers 15 << 2 Shifting mathematically What does left shifting by one position do mathematically 0 A B C shift left two positions decimal form 11

12 Shifting mathematically What does left shifting by one position do mathematically Shifting mathematically What does left shifting by one position do mathematically 0 A B C 0 A B C = A * B * C * 2 0 A B C 0 A B C 0 = A * B * C * 2 1 = 2 *(A * B * C * 2 0 ) Shifting mathematically What does left shifting by one position do mathematically Shifting mathematically What does left shifting by n positions do mathematically 0 A B C = A * B * C * 2 0 Doubles the number! Multiply by 2 n (double n times) A B C 0 = A * B * C * 2 1 = 2 *(A * B * C * 2 0 ) 12

13 Shifting mathematically What does right shifting by one position do mathematically 0 A B C Shifting mathematically What does right shifting by one position do mathematically 0 A B C 0 0 A B Shifting mathematically What does right shifting by one position do mathematically Shifting mathematically What does right shifting by one position do mathematically 0 A B C = A * B * C * A B C = A * B * C * 2 0 Integer divide by A B = A * B * 2 0 = (A * B * C * 2 0 ) div A B = A * B * 2 0 = (A * B * C * 2 0 ) div 2 13

14 Shifting mathematically What does right shifting by n positions do mathematically Integer division by 2 n (halve n times) >> 1 >> 1 What is as a 4-bit binary number >> 1 How do we fill in the leftmost bit shift right one position 14

15 Two types of right shifts: - arithmetic shift: shift in the same as the highorder bit Two types of right shifts: - arithmetic shift: shift in the same as the highorder bit shift right one position shift right one position Two types of right shifts: - arithmetic shift: shift in the same as the highorder bit Two types of right shifts: - arithmetic shift: shift in the same as the highorder bit >> shift right one position shift right one position decimal form 15

16 Two types of right shifts: - arithmetic shift: shift in the same as the highorder bit >> 1 >> shift right one position decimal form Two types of right shifts: - arithmetic shift: shift in the same as the highorder bit >> 2 Arithmetic shifting mathematically What does right arithmetic shifting by n positions do mathematically for signed numbers Integer division by 2 n (halve n times) Same thing!! shift right two positions decimal form 16

17 Two types of right shifts: - arithmetic shift: shift in the same as the highorder bit Two types of right shifts: - arithmetic shift: shift in the same as the highorder bit shift right one position shift right one position Two types of right shifts: - arithmetic shift: shift in the same as the highorder bit >>> 1 Two types of right shifts: - arithmetic shift: shift in the same as the highorder bit >>> shift right one position decimal form shift right one position decimal form 17

18 Left shifts Two types of left shifts - arithmetic shift: Left shifts Two types of left shifts - arithmetic shift: arithmetic -3 << 1 arithmetic -3 << (double the number) logical -3 <<< 1 logical -3 <<< Left shifts Two types of left shifts - arithmetic shift: arithmetic logical -3 << 1-3 <<< Only one type of left shift (double the number) Shifting summarized Arithmetic shift:! Right shift n # shift n bits to the right # discard right n bits # left n bits match high-order bits of original number # Effect: Integer division by 2 n (halve n times)! Left shift # shift n bits to the left # discard left n bits # right n bits are 0s # Effect: multiply by 2 n (double n times) Logical shift right:! left n bits are 0s (no mathematical guarantees for negative numbers) 18

19 Adding numbers base 10 Adding numbers base 10 Add: 456 and Adding numbers base 10 Adding numbers base Add: and

20 Adding numbers base 5 Adding numbers base Adding numbers base 5 Adding numbers base Add: and

21 Adding numbers base 2 Adding numbers base Adding numbers base 2 Addition with 4-bit twos complement numbers

22 Addition with 4-bit twos complement numbers Addition with 4-bit twos complement numbers (Note: I m going to stop writing the base 2 ) Addition with 4-bit twos complement numbers Addition with 4-bit twos complement numbers (11 ) Overflow! We cannot represent this number (it s too large) 22

23 Addition with 4-bit twos complement numbers Addition with 4-bit twos complement numbers Addition with 4-bit twos complement numbers Subtraction ignore the last carry Ideas 23

24 Subtraction Negate the 2 nd number (flip the bits and add 1) Add them! Midterm Average: 28 (77%) Q1: 24.9 (70%) Median: 28.5 (81%) Q3: 32 (91%) 24

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