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Learn about 2’s complement representation, ranging from basic concepts to exercises and practical applications in computer systems engineering. Understand the significance of signed integers, overflow detection, and arithmetic operations.
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Topic 3aTwo’s Complement Representation Introduction to Computer Systems Engineering (CPEG 323) cpeg323-05F\Topic3a-323
2’s Complement • Number and its representation are not the same thing! • Representation is a convention defined for expressing the number. • A number may have different representations, e.g. BCD, hexadecimal, 2’s complement, etc. • 2’s complement: a kind of representation defined as follows cpeg323-05F\Topic3a-323
Excercises • What is 2’s complement representation of +3 ? Answer:+3 = 0011 => + 3 = 0x(-2^3) + 0011 = 011 How about 2’s complement representation of -3 ? Answer: -3 = -(0011) => - 3 => 1x(-2^3) + 0011 = -5 => 1101 cpeg323-05F\Topic3a-323
2’s Complement bn-1 bn-2 ……b1b0 cpeg323-05F\Topic3a-323
Signed Integers(2’s Complement) (For 4 bits) Reference: Katz: Contemporary Logic Design, p243 -1 +0 +1 -2 1111 0000 1110 0001 -3 +2 1101 0010 -4 +3 1100 0011 Why 1111 and 0000 are adjacent? What is the meaning of bit 3? What is the sign of 0? What is the smallest number? What is the biggest number? -5 1011 +4 0100 1010 0101 -6 +5 1001 0110 +6 -7 1000 0111 +7 -8 cpeg323-05F\Topic3a-323
2’s Complement (Cont.) • Representing a negative number –x • From the representation of x to that of –x Invert each bi, then add 1 • Proof of correctness: • Shortcut: Invert every bit and add 1 to the least significant bit Example: - (0100) = 1011 + 1 = 1100 Example: 10010 = 01100100 10011100 = -10010 cpeg323-05F\Topic3a-323
Why –X can be calculated from X by bit-wise complemeting X and add 1 ? • Hint : using x = +4 as an example • How is +4 represented ? • How to do -4 brute force from +4’s representation ? • More cleverly ? (hint: negation of the most significant bit of +4 is what (in binary) ?) • Try yourself ! cpeg323-05F\Topic3a-323
Fit a shorter number to longer bits • Why is it necessary? • Compare an integer with a long integer: typecasting • Load a byte to a word • To add an immediate field to a 32-bit number • MIPS ALU only works with values in 32-bit registers • How to deal with smaller sizes? • What about larger sizes? cpeg323-05F\Topic3a-323
Fit a shorter number to longer bits Example(2-bit): 10= 1*(-21)+0*20=-210 Representing -210 in 4-bit word: Can you do that? What is the intuition? • copy the sign bit into the other bits(sign extension) 0010 -> 0000 0010 1010 -> 1111 1010 • Proof? (Hint ? 1x(2^(-7) + 111 1010 = ?) cpeg323-05F\Topic3a-323
Addition & Subtraction • Addition: Just like in grade school 0110 + 0001 • Substraction: a-b=a+(-b) • Two's complement operations easy 0111-0110= 0111 +1010 01 1 1 0 0 1 • So 0111-0110= 0001 • Have you specially handled the sign bits? cpeg323-05F\Topic3a-323
Addition & Subtraction (Cont.) • Wait! There is one extra bit lost! 0111-0110= 0111 +1010 10001 Why the 1 can be omitted? • Because 1 1 is But treated in the addition as 0*23 + 1*(-23) 1*23 0 0*23 + 1*23 1*23 10 • 1-1=1+1=0 from the viewpoint of 1-bit. • So as long as you do not consider the carry brought by 1+1 (the green 1), you are right! cpeg323-05F\Topic3a-323
Addition & Subtraction (Cont.) • Questions: • Do you have an adder and a “subtractor”? • What about unsigned numbers? (Another adder?) • Advantage of using 2’s complement • Sub can share the same logic as add • Sign bit can be treated as a normal number bit in addition. • These are the clever points! cpeg323-05F\Topic3a-323
Overflow • Addition: 0111 1111 + 0110 +1000 What is the sign of the input? What is the sign of the output? • Can overflow occur in adding a positive and a negative number? • Can overflow occur with 0? • Consider the operations A + B, and A – B cpeg323-05F\Topic3a-323
Overflow • Addition: 0111 1111 + 0110 + 1000 1101 10111 What is the sign of the input? What is the sign of the output? cpeg323-05F\Topic3a-323
Detecting Overflow • Overflow occurs when: • add two positives yields a negative • add two negatives gives a positive • or, subtract a negative from a positive and get a negative • or, subtract a positive from a negative and get a positive • What about addition and subtraction of unsigned numbers? cpeg323-05F\Topic3a-323
What should CPU do about overflow? • Ignore it? • Don't always want to detect overflow • Addu, addiu, subu (MIPS: do not generate a overflow) • Generate a trap so that the programmer can try to deal with it? • An exception (interrupt) occurs • Control jumps to predefined address for exception • Interrupted address is saved for possible resumption • Add, sub cpeg323-05F\Topic3a-323
Instructions • Comparison • Unsigned numbers sltu: set on less than unsigned sltiu: set on less than immediate unsigned • Signed numbers slt: set on less than slti: set on less than immediate Example: What are the values of $t0 and $t1? $s0= 1111 1111 1111 1111 1111 1111 1111 1111 $s1= 0000 0000 0000 0000 0000 0000 0000 0001 (1) slt $t0, $s0, $s1 (2) sltu $t1, $s0, $s1 cpeg323-05F\Topic3a-323
Example (cont’d) • What are the values of $s0 and $s1? • $s0= 1111 1111 1111 1111 1111 1111 1111 1111 • $s1= 0000 0000 0000 0000 0000 0000 0000 0001 Answer: slt $t0, $s0, $s1 -- the result of $t0 is 1 if both are signed sltu $t1, $s0, $s1 -- the result of $t1 is 0 if both are unsigned cpeg323-05F\Topic3a-323
Instructions (Cont.) • Load/Store • lb: load byte • lbu: load byte unsigned • Example: lb $s1, 100($s2) What is the values of $s1 • When memory[$s2+100] = 0000 0000? • When memory[$s2+100] = 0000 0001? • When memory[$s2+100] = 1111 1111? • Example: lbu $s1, 100($s2) What is the values of $s1 • when memory[$s2+100] = 1111 1111? cpeg323-05F\Topic3a-323
