CS 230 Introduction to Computers and Computer Systems Lecture 3 - - PowerPoint PPT Presentation

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CS 230 Introduction to Computers and Computer Systems Lecture 3 - - PowerPoint PPT Presentation

Dec 15, 2022 557 likes •1.61k views

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)

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CS 230 – Introduction to Computers and Computer Systems Lecture 3 – Binary Arithmetic and Two’s Complement

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Binary Addition

 Same as regular addition  Add digits right to left

 least significant bit (LSB) to most significant bit (MSB)  carry

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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

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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: 2n-1

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What about negative numbers?

 On paper we can draw a negative sign  In a computer we need a bit

 What does this bit mean?

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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

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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

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Two's Complement (GOOD)

 Negative numbers are inverted positive

numbers and then add 1

 Single zero!  Range:  Addition just works (drop carry out)  Multiplication also works

 Have to extend input to 2x the bits

− 2

n− 1...2 n− 1− 1

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