Boolean Algebra Binary Arithmetic August 28, 2001 Ellen Spertus MCS 111

2 What you will learn this semester Given devices that implement simple boolean functions… Understand how a computer works –Build an actual computer that you will program –Study the MIPS architecture Write assembly language programs Learn the implementation of a MIPS subset

3 How you will learn Readings Lectures and discussion Homework assignments to review lecture material  Lab assignments to apply ideas in depth (every 2-3 weeks)

4 Boolean algebra George Boole ( ) What is an algebra? Boolean values Boolean operations Boolean functions

5 What is an algebra? Term from Hisab al-jabr w'al-muqabala by al-Khwarizmi (~ , Baghdad) Consists of –a set of values (e.g., the integers) –a set of functions multiplication addition

6 Expressing functions What’s a function? What are these functions? Why is this a bad way of expressing a function over the integers?

7 Boolean values false 0 low true 1 high

8 Boolean operations identity not or and xor (exclusive or) nand (not and) nor (not or)

9 Identity 0 Note: Bebop calls this “buf” (for “buffer”).

10 Not 0 Note: Some people write x’ instead of x. The “bubble” (or “bobble”) means “not”.

11 And Notes: Some people write a^b or a&b. The gate is shaped like a “D” as in “anD”.

12 Or Note: Some people write a  b or a|b.

13 Exclusive-or (xor)

14 Nand

15 Nor

16 What you will learn Given devices that implement simple boolean functions… Understand how a computer works

17 Base-10 (decimal) arithmetic Uses the ten numbers from 0 to 9 Each column represents a power of 10

18 Base-2 (binary) arithmetic Uses the two numbers from 0 to 1 Every column represents a power of 2

19 Converting from base-2 to base-10

20 Converting from base-10 to base-2

21 Addition Base-10Base-2 Note: A binary digit is called a bit. The bit with the lowest exponent (2 0 ) is called the least significant bit (LSB). The bit with the highest exponent is the most significant bit (MSB).

22 Practice binary arithmetic

23 Count on your fingers! How high can you count with –5 fingers? –10 fingers?

24 Looking ahead For you to do –Complete and turn in information form –Read syllabus and policies –Homework, due Tuesday, September 4 Coming up –Thursday: Digital computing –Next week: First lab sessions