A Lesson in the “Math + Fun!” Series
Nov. 2005 Math in Computers Slide 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
Edition
First
Released Revised
Nov. 2005
Revised
Nov. 2005 Math in Computers Slide 2

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

The Inside of an Odhner Calculator
1 9 7
. . . 0 8 6 4 2
+ 5 3 6 5
1 4 0 0 7
Nov. 2005 Math in Computers Slide 5

Decimal versus Binary Calculator
0
1
2
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
0
4
3
After movement by 10 notches
(one revolution), move the next wheel to the left by 1 notch.
After movement by 2 notches
(one revolution), move the next wheel to the left by 1 notch.
Nov. 2005 Math in Computers Slide 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 Math in Computers
If both beads have moved, push them back and move a bead in the next position
Slide 7

Each of these beads is worth 5 units
Other Types of Abacus
3 1 4 1 5 9 2 6 5 4
Each of these beads is worth 1 unit 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
0 0 0 0 1 1 0 1 1 0
Display the digit 1 by shifting one bead
Math in Computers Slide 8 Nov. 2005

Activity 1: Counting on a Binary Abacus
1. Form a binary abacus with 6 positions, using people as beads
Leader
A person sits for 0, stands up for 1
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
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
32
Nov. 2005
16
0
8
0
4
1 1
2 1
Math in Computers
Questions:
What number is shown?
What happens if the leader sits down?
Slide 9

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

Dark = 0
Activity 3: Reading a Binary Clock
What time is it?
Show the time:
8
4
2
1 sec hour min
1 2 : 3 4 : 5 6
Each 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
Light = 1
__:__:__
__:__:__
__:__:__
8:41:22
15:09:43
9:15:00
Nov. 2005 Math in Computers Slide 11

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

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

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 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
B A E D B
0 0 0 1 1 = 3
16 8 4 2 1
1 1 0 1 0 = 26
16 8 4 2 1
Nov. 2005 Math in Computers Slide 14

Binary addition table
+ 0 1
0 0 1
1 1
Activity 4: Binary Addition
10
Wow! Binary addition is a snap!
32 16 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 0 1 1
-------------
1 1 1 0 1 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
Check: 1 2
+ 2 9
+ 7
+ 1 1
--------
5 7
Nov. 2005 Math in Computers Slide 15

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

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

Activity 5: Binary Multiplication
Binary multiplication table

0 1
0 0 0
1 0 1
I
♥ this simple multiplication table!

0 1 1 0
0 1 0 1
-------
0 1 1 0
0 0 0 0
0 1 1 0
0 0 0 0
-------------
0 0 1 1 1 1 0
Think of two 3-bit binary numbers and multiply them
Check: 6

5
----
30
16 8 4 2 1
0 1 1 0

0 1 0 1
-------------------
1 1 1 1 0
Nov. 2005 Math in Computers Slide 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?
Idea 1: Break the 12-digit addition into three 4-digit additions and let each person complete one of the parts
0 0 1
5 8 9 9 9 9 9 9 0 6 0 6
This won’t work, because the three groups of digits cannot be processed independently
Nov. 2005 Math in Computers Slide 19

Fast Addition in a Computer: 2 nd Try
2 7 2 4 3 9 7 2 5 6 2 1
3 1 7 5 6 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
0
1
5 8 9 9 9 9
0
1
0 0 0 6 0 6
Sum
5 9 0 0 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 Slide 20

January 2006
Nov. 2005 Math in Computers Slide 21