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

2. Nov. 2005Math in ComputersSlide 2 About This Presentation EditionReleasedRevised FirstNov. 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

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

4. Nov. 2005Math in ComputersSlide 4 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

5. Nov. 2005Math in ComputersSlide 5 The Inside of an Odhner Calculator... 0 8 6 4 2 70 7 0 9 4 1 1 + 5 3 6 5

6. Nov. 2005Math in ComputersSlide 6 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 25 1000 100 10 1 5000 + no hundred + 20 + 5 = Five thousand twenty-five 1011 8 4 2 1 8 + no 4 + 2 + 1 = Eleven

7. Nov. 2005Math in ComputersSlide 7 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 DecimalBinary

8. Nov. 2005Math in ComputersSlide 8 Other Types of Abacus 3 1 4 1 5 9 2 6 5 4 Each 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

9. Nov. 2005Math in ComputersSlide 9 Activity 1: Counting on a Binary Abacus 1. Form a binary abacus with 6 positions, using people as beads 3216 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 100011 3216 8 4 2 1

10. Nov. 2005Math in ComputersSlide 10 Activity 2: Adding on a Binary Abacus 1. Form a binary abacus with 6 positions, using people as beads This number is 16 + 4 + 2 = 22 3216 8 4 2 1 3216 8 4 2 1 3. Now add the binary number 0 0 1 1 0 0 to the one shown 001100 This number is 8 + 4 = 12 3216 8 4 2 1 This number is 32 + 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

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

12. Nov. 2005Math in ComputersSlide 12 IN OUT Ten-State versus Two-State Devices To 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 01 01 0 1

13. Nov. 2005Math in ComputersSlide 13 Addition Table + 01 0 1 1 1 0 10 Binary addition table Write down in place Carry over to the left Write down in place Carry over to the left

14. Nov. 2005Math in ComputersSlide 14 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 00011 = 3 16 8 4 2 1 AB 11010 = 26 16 8 4 2 1 BDE

15. Nov. 2005Math in ComputersSlide 15 Activity 4: Binary Addition Check: 1 2 + 2 9 + 7 + 1 1 -------- 5 7 + 01 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 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

16. Nov. 2005Math in ComputersSlide 16 1286432168421 Adding with a Checkerboard Binary Calculator 1286432168421 12 + 29 + 7 + 11 59 3216821 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

17. Nov. 2005Math in ComputersSlide 17  Multiplication Table  01 0 1 0 0 0 1 Binary multiplication table Write down in place Carry over to the left

18. Nov. 2005Math in ComputersSlide 18 Activity 5: Binary Multiplication 0 1 1 0 0 1 0 1 ------- 0 1 1 0 0 0 0 0 0 1 1 0 0 0 ------------- 0 0 1 1 1 1 0  Check: 60 1 1 0  5  0 1 0 1 ---- ------------------- 30 11 1 1 0 16 8 4 2 1  01 0 1 0 0 0 1 Binary multiplication table I ♥ this simple multiplication table! Think of two 3-bit binary numbers and multiply them

19. Nov. 2005Math in ComputersSlide 19 Idea 1:Break the 12-digit addition into three 4-digit additions and let each person complete one of the parts 3 9 7 2 6 0 2 7 2 7 2 4 3 1 7 5 5 6 2 1 4 9 8 5 2 7 2 4 3 1 7 5 3 9 7 2 6 0 2 7 5 6 2 1 4 9 8 5 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? 1st number: 2nd number: 1st number: 2nd number: This won’t work, because the three groups of digits cannot be processed independently 9 9 0 0 6 1 5 8 9 9 0

20. Nov. 2005Math in ComputersSlide 20 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 1 2 7 4 9 8 5 2 7 2 4 3 9 3 1 7 5 6 0 7 2 5 6 2 1 2 7 4 9 8 5 Fast Addition in a Computer: 2 nd Try 1st number: 2nd number: 1st number: 2nd number: 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 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

21. Nov. 2005Math in ComputersSlide 21 Next Lesson January 2006