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

Nov. 2005 Math in Computers Slide 1

Math in ComputersA Lesson in the “Math + Fun!” Series


About This Presentation

Edition Released Revised Revised

First Nov. 2005

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


Counters and Clocks

5

0

3

9

4

1

2

7

8

6


A Mechanical Calculator

Odhner calculator: invented by Willgodt T. Odhner (Russia) in 1874

Photo of production version, made in Sweden (ca. 1940)

Photo of the 1874 hand-made version


The Inside of an Odhner Calculator

. . . 0 8 6 4 2

70

7

0

9

4

1

1

+ 5 3 6 5


Decimal versus Binary Calculator

After movement by 10 notches (one revolution), move the next wheel to the left by 1 notch.

0

1

2

3

4

After movement by 2 notches (one revolution), move the next wheel to the left by 1 notch.

0

5 0 2 5 1000 100 10 1

5000 + no hundred + 20 + 5= Five thousand twenty-five

1 0 1 1 8 4 2 1

8 + no 4 + 2 + 1 = Eleven


Decimal versus Binary Abacus

If all 10 beads have moved, push them back and move a bead in the next position

If both beads have moved, push them back and move a bead in the next position

Decimal Binary


Other Types of Abacus

3 1 4 1 5 9 2 6 5 4Each of these beads is worth 5 units

Each of these beads is worth 1 unit Display the digit 9 by

shifting one 5-unit bead and four 1-unit beads

0 0 0 0 1 1 0 1 1 0

512 256 128 64 32 16 8 4 2 1

Display the digit 1 by shifting one bead


Activity 1: Counting on a Binary Abacus1. Form a binary abacus with 6 positions, using people as beads

32 16 8 4 2 1

2. The person who controls the counting stands at the right end, but is not part of the binary abacus

A person sits for 0, stands up for 1

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

Questions:

What number is shown?

What happens if the leader sits down?

Leader

1 0 0 0 1 1

32 16 8 4 2 1


Activity 2: Adding on a Binary Abacus1. Form a binary abacus with 6 positions, using people as beads

This number is16 + 4 + 2 = 22

32 16 8 4 2 1

32 16 8 4 2 1

3. Now add the binary number 0 0 1 1 0 0 to the one shown0 0 1 1 0 0 This number is

8 + 4 = 12

32 16 8 4 2 1

This number is32 + 2 = 34

A person sits for 0, stands up for 1

2. Show the binary number 0 1 0 1 1 0 on the abacus


hour min sec1

2

4

8

Activity 3: Reading a Binary Clock

1 2 : 3 4 : 5 6Each decimal digit is represented as a 4-bit binary number.For example:

1: 0 0 0 1 6: 0 1 1 0

8 4 2 1

__ :__ :__

__ :__ :__

What time is it?

__ :__ :__

Show the time:

8 :41 :22

15 :09 :43

9 :15 :00

Dark = 0

Light = 1


IN

OUT

Ten-State versus Two-State DevicesTo remember one decimal digit,we need a wheel with 10 notches(a ten-state device)

A binary digit (aka bit) needs just two states

01

01

0 1

0 10

1


Addition Table

+ 0 1

0

1

1

1

0

10

Binary additiontable

Write downin place

Carry overto the left

Write downin place



Secret of Mind-Reading Game Revealed1. 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 9 11 13 15 17 19 21 23 25 27 29

List B: 2 3 6 7 10 11 14 15 18 19 22 23 26 27 30

List C: 4 5 6 7 12 13 14 15 20 21 22 23 28 29 30

List D: 8 9 10 11 12 13 14 15 24 25 26 27 28 29 30

List E: 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Find the number by adding the first entries of the lists in which it appears

0 0 0 1 1 = 316 8 4 2 1

AB

1 1 0 1 0 = 2616 8 4 2 1

BDE


Activity 4: Binary Addition

Check: 1 2+ 2 9+ 7+ 1 1-------- 5 7

+ 0 1

0

1

1

1

0

10

Binary addition

table

Wow! Binary addition is a snap! 0 0 1 1 0 0

+ 0 1 1 1 0 1+ 0 0 0 1 1 1+ 0 0 1 0 1 1------------- 1 1 1 0 1 1

32 16 8 4 2 1

32 16 8 4 2 1

Rule: for every pair of 1s in a column, put a 1 in the next column to the left

Think of 5 numbers and add them


128 64 32 16 8 4 2 1

Adding with a Checkerboard Binary Calculator128 64 32 16 8 4 2 1

12

+ 29

+ 7

+ 11

59

32 16 8 2 1

1. Set up the binary numbers on different rows2. Shift all beads straight down to bottom row3. Remove pairs of beads and replace each pair with one bead in the square to the left


Multiplication Table

0 1

0

1

0

0

0

1

Binary multiplication

table

Write downin place



Activity 5: Binary Multiplication

0 1 1 0 0 1 0 1 ------- 0 1 1 0 0 0 0 0 0 1 1 00 0 0 0-------------0 0 1 1 1 1 0

Check: 6 0 1 1 0 5

0 1 0 1---- ----

---------------30 1 1 1 1

0

16 8 4 2 1

0 1

0

1

0

0

0

1

Binary multiplication

table

I ♥ this simple multiplication

table!

Think of two 3-bit binary numbers and multiply them


Idea 1: Break the 12-digit addition into three 4-digit additions

and let each person complete one of the parts

3 9 7 26 0 2 7

2 7 2 4 3 1 7 5

5 6 2 14 9 8 5

2 7 2 4 3 1 7 5

3 9 7 26 0 2 7

5 6 2 14 9 8 5

Fast Addition in a ComputerForget for a moment that computers work in binarySuppose we want to add the following 12-digit numbersIs there a way to use three people to find the sum faster?

1st number: 2nd number: 1st number:2nd number:

This won’t work, because the three groups of digits cannot be processed independently

9 9 9 9

0

0 6 0 6

1

5 8 9 9

0


Idea 2: Break the 12-digit addition into two 6-digit additions;

use two people to do the left half in two different forms

2 7 2 4 3 9 3 1 7 5 6 0

7 2 5 6 2 12 7 4 9 8 5

2 7 2 4 3 9 3 1 7 5 6 0

7 2 5 6 2 12 7 4 9 8 5

Fast Addition in a Computer: 2nd Try

1st number: 2nd number: 1st number:2nd number:

Once the carry from the right half is known, the correct left-halfof the sum can be chosen quickly from the two possible values

0 0 0 6 0 6

1

5 9 0 0 0 0

0

2 7 2 4 3 9 3 1 7 5 6 0

5 8 9 9 9 9

0

1

Sum


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