Math in Computers A Lesson in the Math

About This Presentation This presentation is part of the “Math + Fun!” series devised

Counters and Clocks 9 Math in Computers 1 8 2 7 3 6 Nov.

A Mechanical Calculator Photo of the 1874 hand-made version Photo of production version, made

The Inside of an Odhner Calculator 197. . . 08642 + 5365 140 07

Decimal versus Binary Calculator 5 0 2 5 1000 10 1 5000 + no

Decimal versus Binary Abacus Decimal Binary If all 10 beads have moved, push them

Other Types of Abacus Each of these beads is worth 5 units Each of

Activity 1: Counting on a Binary Abacus 1. Form a binary abacus with 6

Activity 2: Adding on a Binary Abacus 1. Form a binary abacus with 6

Activity 3: Reading a Binary Clock Dark = 0 What time is it? Show

Ten-State versus Two-State Devices To remember one decimal digit, we need a wheel with

Addition Table Binary addition table 1 + 0 0 0 1 10 Write down

Secret of Mind-Reading Game Revealed 1. Think of a number between 1 and 30.

Activity 4: Binary Addition Binary addition table 1 + 0 0 0 1 10

Adding with a Checkerboard Binary Calculator 128 64 32 16 8 4 2 1

Multiplication Table Binary multiplication table 1 0 0 1 0 1 Carry over to

Activity 5: Binary Multiplication Binary multiplication table 1 0 0 1 0 1 I

Fast Addition in a Computer Forget for a moment that computers work in binary

Fast Addition in a Computer: 2 nd Try 1 st number: 2 nd number:

Next Lesson January 2006 Nov. 2005 Math in Computers 21

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Math in Computers A Lesson in the “Math + Fun!” Series Nov. 2005 Math in Computers 1

About This Presentation This presentation is part of the “Math + Fun!” series devised by Behrooz Parhami, Professor of Computer Engineering at University of California, Santa Barbara. It was first prepared for special lessons in mathematics at Goleta Family School during three school years (2003 -06). “Math + Fun!” material can be used freely in teaching and other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami Nov. 2005 Edition Released First Nov. 2005 Revised Math in Computers Revised 2

Counters and Clocks 9 Math in Computers 1 8 2 7 3 6 Nov. 2005 0 5 4 3

A Mechanical Calculator Photo of the 1874 hand-made version Photo of production version, made in Sweden (ca. 1940) Odhner calculator: invented by Willgodt T. Odhner (Russia) in 1874 Nov. 2005 Math in Computers 4

The Inside of an Odhner Calculator 197. . . 08642 + 5365 140 07 Nov. 2005 Math in Computers 5

Decimal versus Binary Calculator 5 0 2 5 1000 10 1 5000 + no hundred + 20 + 5 = Five thousand twenty-five 0 1 2 0 1 1 8 4 2 1 8 + no 4 + 2 + 1 = Eleven 3 0 4 After movement by 10 notches (one revolution), move the next wheel to the left by 1 notch. Nov. 2005 1 After movement by 2 notches (one revolution), move the next wheel to the left by 1 notch. Math in Computers 6

Decimal versus Binary Abacus Decimal Binary If all 10 beads have moved, push them back and move a bead in the next position Nov. 2005 If both beads have moved, push them back and move a bead in the next position Math in Computers 7

Other Types of Abacus Each of these beads is worth 5 units Each of these beads is worth 1 unit 3141592654 Display the digit 9 by shifting one 5 -unit bead and four 1 -unit beads 512 256 128 64 32 16 8 4 2 1 0000110110 Display the digit 1 by shifting one bead Nov. 2005 Math in Computers 8

Activity 1: Counting on a Binary Abacus 1. Form a binary abacus with 6 positions, using people as beads Leader 32 16 8 4 2 A person sits for 0, stands up for 1 1 2. The person who controls the counting stands at the right end, but is not part of the binary abacus 3. The leader sits down any time he/she wants the count to go up 4. Each person switches pose (sitting to standing, or standing to sitting) whenever the person to his/her left switches from standing to sitting 1 0 0 0 1 1 Questions: What number is shown? 32 Nov. 2005 16 8 4 2 1 Math in Computers What happens if the leader sits down? 9

Activity 2: Adding on a Binary Abacus 1. Form a binary abacus with 6 positions, using people as beads 32 16 8 4 2 A person sits for 0, stands up for 1 1 2. Show the binary number 0 1 1 0 on the abacus This number is 16 + 4 + 2 = 22 32 16 8 4 2 1 0 0 1 1 0 0 This number is 8 + 4 = 12 3. Now add the binary number 0 0 1 1 0 0 to the one shown This number is 32 + 2 = 34 32 Nov. 2005 16 8 4 2 1 Math in Computers 10

Activity 3: Reading a Binary Clock Dark = 0 What time is it? Show the time: 8 4 2 1 sec min hour 1 2: 3 4: 5 6 Each decimal digit is represented as a 4 -bit binary number. For example: 1: 6: 0 0 0 1 1 0 8 Nov. 2005 4 2 1 __ : __ 8 : 41 : 22 __ : __ 15 : 09 : 43 __ : __ 9 : 15 : 00 Light = 1 Math in Computers 11

Ten-State versus Two-State Devices To remember one decimal digit, we need a wheel with 10 notches (a ten-state device) 1 IN OU 0 T 0 1 0 A binary digit (aka bit) needs just two states Nov. 2005 Math in Computers 1 0 1 12

Addition Table Binary addition table 1 + 0 0 0 1 10 Write down in place Carry over to the left Write down in place Nov. 2005 Math in Computers 13

Secret of Mind-Reading Game Revealed 1. Think of a number between 1 and 30. 2. Tell me in which of the five lists below the number appears. List A: 1 3 5 7 List B: 2 3 6 7 List C: 4 5 6 7 List D: 8 9 10 11 List E: 16 17 18 19 9 10 12 12 20 11 11 13 13 21 13 14 14 14 22 15 15 23 17 18 20 24 24 19 19 21 25 25 21 22 22 26 26 23 23 23 27 27 25 26 28 28 28 27 27 29 29 30 30 Find the number by adding the first entries of the lists in which it appears B A E D B 0 0 0 1 1 = 3 1 1 0 = 26 16 8 4 2 1 Nov. 2005 Math in Computers 14

Activity 4: Binary Addition Binary addition table 1 + 0 0 0 1 10 Wow! Binary addition is a snap! 32 Check: Think of 5 numbers and add them Nov. 2005 Math in Computers 8 4 2 1 0 0 1 1 0 0 + 0 1 1 1 0 1 + 0 0 0 1 1 1 + 0 0 1 1 ------1 1 1 0 1 1 32 Rule: for every pair of 1 s in a column, put a 1 in the next column to the left 16 16 8 4 2 1 12 + 29 + 7 + 11 -------57 15

Adding with a Checkerboard Binary Calculator 128 64 32 16 8 4 2 1 12 + 29 + 7 + 11 59 1. Set up the binary numbers on different rows 2. Shift all beads straight down to bottom row 3. Remove pairs of beads and replace each pair with one bead in the square to the left Nov. 2005 Math in Computers 16

Multiplication Table Binary multiplication table 1 0 0 1 0 1 Carry over to the left Write down in place Nov. 2005 Math in Computers 17

Activity 5: Binary Multiplication Binary multiplication table 1 0 0 1 0 1 I ♥ this simple multiplication table! 0 1 1 0 0 1 ------0 1 1 0 0 0 0 0 ------0 0 1 1 0 16 Think of two 3 -bit binary numbers and multiply them Nov. 2005 Check: 0 1 Math in Computers 6 ---30 5 8 4 2 1 0 1 1 0 0 1 ---------1 1 0 18

Fast Addition in a Computer Forget for a moment that computers work in binary Suppose we want to add the following 12 -digit numbers Is there a way to use three people to find the sum faster? 1 st number: 2 nd number: 2 7 2 4 3 9 7 2 5 6 2 1 3 1 7 5 6 0 2 7 4 9 8 5 Idea 1: Break the 12 -digit addition into three 4 -digit additions and let each person complete one of the parts 0 0 5 8 9 9 1 9 9 0 6 This won’t work, because three groups of digits cannot be processed independently Nov. 2005 Math in Computers 19

Fast Addition in a Computer: 2 nd Try 1 st number: 2 nd number: 2 7 2 4 3 9 7 2 5 6 2 1 3 1 7 5 6 0 2 7 4 9 8 5 Idea 2: Break the 12 -digit addition into two 6 -digit additions; use two people to do the left half in two different forms 0 1 5 8 9 9 0 1 0 0 0 6 Sum 5 9 0 0 Once the carry from the right half is known, the correct left-half of the sum can be chosen quickly from the two possible values Nov. 2005 Math in Computers 20

Next Lesson January 2006 Nov. 2005 Math in Computers 21