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 n1 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. 8bits, 16bits, 32bits) 37 >> shift right three positions decimal form 8
9 Shifting 8bit numbers 37 pad with 0s 37 >> 2 What is 37 as an 8bit binary number Shifting 8bit numbers >> 2 How do we fill in the leftmost bits shift right two positions Shifting 8bit numbers 37 >> 2 Shifting 8bit 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 8bit numbers 15 << 2 Shifting 8bit numbers 15 << 2 15 Shifting 8bit numbers 15 << 2 Shifting 8bit numbers 15 << shift left two positions 10
11 Shifting 8bit numbers 15 << 2 Shifting 8bit numbers 15 << shift left two positions shift left two positions Shifting 8bit 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 4bit 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 << 13 <<< 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 highorder 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 4bit twos complement numbers
22 Addition with 4bit twos complement numbers Addition with 4bit twos complement numbers (Note: I m going to stop writing the base 2 ) Addition with 4bit twos complement numbers Addition with 4bit twos complement numbers (11 ) Overflow! We cannot represent this number (it s too large) 22
23 Addition with 4bit twos complement numbers Addition with 4bit twos complement numbers Addition with 4bit 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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