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CS 230 Introduction to Computers and Computer Systems Lecture 3 Binary Arithmetic and Twos Complement CS 230 - Spring 2020 1-1 Binary Addition Same as regular addition Add digits right to left least significant bit (LSB)

  1. CS 230 – Introduction to Computers and Computer Systems Lecture 3 – Binary Arithmetic and Two’s Complement CS 230 - Spring 2020 1-1

  2. Binary Addition  Same as regular addition  Add digits right to left  least significant bit (LSB) to most significant bit (MSB)  carry CS 230 - Spring 2020 1-2

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  23. Binary Multiplication  Same as regular multiplication  Multiply bits of lower term by upper term  LSB to MSB  Add zeros for each bit multiplied so far  Will always be zeros or copy of upper term  Add copies together to get product CS 230 - Spring 2020 1-23

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  63. Revisit Binary Addition  We’ve been doing unsigned binary  But, if you are building circuits then you have a limit to the number of bits available to represent values  Fixed width representation has overflow  An n -bit binary number has range  Min: 0  Max: 2 n -1 CS 230 - Spring 2020 1-63

  64. What about negative numbers?  On paper we can draw a negative sign  In a computer we need a bit  What does this bit mean? CS 230 - Spring 2020 1-64

  65. Signed Magnitude (BAD)  Just have MSB be sign bit  If it’s one the number is negative  If it’s zero the number is positive  Problems  Two zeros  Addition doesn’t work  Ex: 4-bit version (1 sign bit, 3 magnitude bits) 0011 + 1001 = 1100  3 + (-1) = -4  CS 230 - Spring 2020 1-65

  66. One’s Complement (BAD)  Negative numbers are inverted positive numbers  Problems  Still two zeros  Addition sort-of works  Have to re-add in the carry out at the end CS 230 - Spring 2020 1-66

  67. Two's Complement (GOOD)  Negative numbers are inverted positive numbers and then add 1  Single zero! n − 1 ...2 n − 1 − 1 − 2  Range:  Addition just works (drop carry out)  Multiplication also works  Have to extend input to 2x the bits CS 230 - Spring 2020 1-67

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