1. May 2005Special NumbersSlide 1 Special Numbers A Lesson in the “Math + Fun!” Series

2. May 2005Special NumbersSlide 2 About This Presentation EditionReleasedRevised FirstMay 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 the 2003-04 and 2004-05 school years. The slides can be used freely in teaching and in other educational settings. Unauthorized uses are strictly prohibited. © Behrooz Parhami

3. May 2005Special NumbersSlide 3 What is Special About These Numbers? Numbers in purple squares? Numbers in green squares? Circled numbers?

4. May 2005Special NumbersSlide 4 Atoms in the Universe of Numbers Two hydrogen atoms and one oxygen atom H2OH2O 2 3 4 5 6 7 8 91011 1213141516 1718192021 22232425 13 Atom 3  5 Molecule 2  7 Molecule Are the following numbers atoms or molecules? For molecules, write down the list of atoms: 12 = 2 2  3Molecule 13 = 14 = 15 = 19 = 27 = 30 = 32 = 47 = 50 = 70 = 19 Atom 3 3 Molecule 2  3  5 Molecule 2 5 Molecule 47 Atom 2  5 2 Molecule 2  5  7 Molecule Prime number (atom) Composite number (molecule)

5. May 2005Special NumbersSlide 5 Is There a Pattern to Prime Numbers? Primes become rarer as we go higher, but there are always more primes, no matter how high we go. Primes appear to be randomly distributed in this list that goes up to 620.

6. May 2005Special NumbersSlide 6 Ulam’s Discovery 737475767778798081 724344454647484950 714221222324252651 704120789102752 694019612112853 683918543122954 673817161514133055 663736353433323156 656463626160595857 Stanislaw Ulam was in a boring meeting, so he started writing numbers in a spiral and discovered that prime numbers bunch together along diagonal lines Primes pattern for numbers up to about 60,000; notice that primes bunch together along diagonal lines and they thin out as we move further out

7. May 2005Special NumbersSlide 7 Ulam’s Rose Primes pattern for numbers up to 262,144. Just as water molecules bunch together to make a snowflake, prime numbers bunch together to produce Ulam’s rose.

8. May 2005Special NumbersSlide 8 Explaining Ulam’s Rose 234567 8910111213 141516171819 202122232425 262728293031 323334353637 383940414243 444546474849 505152535455 565758596061 626364656667 686970717273 747576777879 808182838485 868788899091 929394959697 Table of numbers that is 6 columns wide shows that primes, except for 2 and 3, all fall in 2 columns 6k – 1 6k + 1 Pattern The two columns whose numbers are potentially prime form this pattern when drawn in a spiral

9. May 2005Special NumbersSlide 9 Activity 1: More Number Patterns 234567 8910111213 141516171819 202122232425 262728293031 323334353637 383940414243 444546474849 505152535455 565758596061 626364656667 686970717273 747576777879 808182838485 868788899091 929394959697 Color all boxes that contain multiples of 5 and explain the pattern that you see. 234567 8910111213 141516171819 202122232425 262728293031 323334353637 383940414243 444546474849 505152535455 565758596061 626364656667 686970717273 747576777879 808182838485 868788899091 929394959697 Color all boxes that contain multiples of 7 and explain the pattern that you see.

10. May 2005Special NumbersSlide 10 Activity 2: Number Patterns in a Spiral 737475767778798081 724344454647484950 714221222324252651 704120789102752 694019612112853 683918543122954 673817161514133055 663736353433323156 656463626160595857 Color all the even numbers that are not multiples of 3 or 5. For example, 4 and 14 should be colored, but not 10 or 12. Color the multiples of 3. Use two different colors for odd multiples (such as 9 or 15) and for even multiples (such as 6 or 24). 737475767778798081 724344454647484950 714221222324252651 704120789102752 694019612112853 683918543122954 673817161514133055 663736353433323156 656463626160595857

11. May 2005Special NumbersSlide 11 Perfect Numbers A perfect number equals the sum of its divisors, except itself 6:1 + 2 + 3 = 6 28:1 + 2 + 4 + 7 + 14 = 28 496:1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496 An abundant number has a sum of divisors that is larger than itself 36:1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55 > 36 60:1 + 2 + 3 + 4 + 5 + 6 + 10 + 15 + 20 + 30 = 96 > 60 100:1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 = 117 > 100 A deficient number has a sum of divisors that is smaller than itself 9:1 + 3 = 4 < 9 23:1 < 23 128:1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 < 128

12. May 2005Special NumbersSlide 12 Activity 3: Abundant, Deficient, or Perfect? For each of the numbers below, write down its divisors, add them up, and decide whether the number is deficient, abundant, or perfect. Challenge questions: Are prime numbers (for example, 2, 3, 7, 13,... ) abundant or deficient? Are squares of prime numbers (3 2 = 9, 7 2 = 49,... ) abundant or deficient? You can find powers of 2 by starting with 2 and doubling in each step. It is easy to see that 4 (divisible by 1 and 2), 8 (divisible by 1, 2, 4), and 16 (divisible by 1, 2, 4, 8) are deficient. Are all powers of 2 deficient? NumberDivisors (other than the number itself)Sum of divisorsType 12 18 28 30 45

13. May 2005Special NumbersSlide 13 Why Perfect Numbers Are Special Some things we know about perfect numbers There are only about a dozen perfect numbers up to 10 160 All even perfect numbers end in 6 or 8 10 160 = 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 Some open questions about perfect numbers Are there an infinite set of perfect numbers? (The largest, discovered in 1997, has 120,000 digits) Are there any odd perfect numbers? (Not up to 10 300 )

14. May 2005Special NumbersSlide 14 1089: A Very Special Number Follow these instructions: 1. Take any three digit number in which the first and last digits differ by 2 or more; e.g., 335 would be okay, but not 333 or 332. 2. Reverse the number you chose in step 1. (Example: 533) 3. You now have two numbers. Subtract the smaller number from the larger one. (Example: 533 – 335 = 198) 4. Add the answer in step 3 to the reverse of the same number. (Example: 198 + 891 = 1089) The answer is always 1089.

15. May 2005Special NumbersSlide 15 Special Numbers and Patterns Why is the number 37 special? 3  37 = 111 and 1 + 1 + 1 = 3 6  37 = 222 and 2 + 2 + 2 = 6 9  37 = 333 and 3 + 3 + 3 = 9 12  37 = 444 and 4 + 4 + 4 = 12 When adding or multiplying does not make a difference. You know that 2  2 = 2 + 2. But, these may be new to you: 1 1 / 2  3 = 1 1 / 2 + 3 1 1 / 3  4 = 1 1 / 3 + 4 1 1 / 4  5 = 1 1 / 4 + 5 Playing around with a number and its digits: 198 = 11 + 99 + 88 153 = 1 3 + 5 3 + 3 3 1634 = 1 4 + 6 4 + 3 4 + 4 4 Here is an amazing pattern: 1 2 = 1 11 2 = 121 111 2 = 12321 1111 2 = 1234321 11111 2 = 123454321

16. May 2005Special NumbersSlide 16 Activity 4: More Special Number Patterns 1 1 + 3 1 + 3 + 5 1 + 3 + 5 + 7 1 + 3 + 5 + 7 + 9 1 + 3 + 5 + 7 + 9 + 11 1 + 3 + 5 + 7 + 9 + 11 + 13 1 3 + 5 7 + 9 + 11 13 + 15 + 17 + 19 21 + 23 + 25 + 27 + 29 31 + 33 + 35 + 37 + 39 + 41 43 + 45 + 47 + 49 + 51 + 53 + 55 1  7 + 3 = 10 14  7 + 2 = 100 142  7 + 6 = 1000 1428  7 + 4 = 10000 14285  7 + 5 = 100000 142857  7 + 1 = 1000000 1428571  7 + 3 = 10000000 14285714  7 + 2 = 100000000 142857142  7 + 6 = 1000000000 1428571428  7 + 4 = 10000000000 1 1 + 2 + 1 1 + 2 + 3 + 2 + 1 1 + 2 + 3 + 4 + 3 + 2 + 1 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 Continue these patterns and find out what makes them special.

17. May 2005Special NumbersSlide 17 Activity 5: Special or Surprising Answers Can you find something special in each of the following groups? What’s special about the following? 12  483 = 5796 27  198 = 5346 39  186 = 7254 42  138 = 5796 Do the following multiplications: 4  1738 = _______ 4  1963 = _______ 18  297 = _______ 28  157 = _______ 48  159 = _______ Do the following multiplications: 3  51249876 = ____________ 9  16583742 = ____________ 6  32547891 = ____________ What is special about 327? 327  1 = _____ 327  2 = _____ 327  3 = _____ What is special about 9? 1  9 + 2 = ___ 12  9 + 3 = ____ 123  9 + 4 = _____

18. May 2005Special NumbersSlide 18 Numbers as Words 0 Zero 1 One 2 Two 3 Three 4 Four 5 Five 6 Six 7 Seven 8 Eight 9 Nine 10 Ten We can write any number as words. Here are some examples: 12 Twelve 21 Twenty-one 80 Eighty 3547 Three thousand five hundred forty-seven Eight Five Four Nine One Seven Six Ten Three Two Zero Three Nine One Five Ten Seven Zero Two Four Eight Six One Two Six Ten Zero Four Five Nine Three Seven Eight Four Six Ten Two Zero Five Nine One Seven Three

19. May 2005Special NumbersSlide 19 Activity 6: Numbers as Words 0 Zero 1 One 2 Two 3 Three 4 Four 5 Five 6 Six 7 Seven 8 Eight 9 Nine 10 Ten Alpha order Eight Five Four Nine One Seven Six Ten Three Two Zero Three Nine One Five Ten Seven Zero Two Four Eight Six One Two Six Ten Zero Four Five Nine Three Seven Eight Four Six Ten Two Zero Five Nine One Seven Three Alpha order, from the end Length order Evens and odds (in alpha order) If we wrote these four lists from “zero” to “one thousand,” which number would appear first/last in each list? Why? What about to “one million”?

20. May 2005Special NumbersSlide 20 Activity 7: Sorting the Letters in Numbers 0 eorz 1 eno 2 otw 3 eehrt 4 foru 5 efiv 6 isx 7 eensv 8 eghit 9 einn Spell out each number and put its letters in alphabetical order (ignore hyphens and spaces). You will discover that 40 is a very special number! 10 ent 11 eeelnv 12 13 14 15 16 17 18 19 20 enttwy 21 eennottwy 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

21. May 2005Special NumbersSlide 21 Next Lesson Not definite, at this point: Thursday, June 9, 2005 It is believed that we use decimal (base-10) numbers because humans have 10 fingers. How would we count if we had one finger on each hand? 000001010011100101110111 Computers do math in base 2, because the two digits 0 and 1 that are needed are easy to represent with electronic signals or on/off switches.  