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

May 2005 Special Numbers Slide 1

Special NumbersA Lesson in the “Math + Fun!” Series


About This Presentation

Edition Released Revised Revised

First May 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


What is Special About These

Numbers?

Numbers in purple squares?

Numbers in green squares?

Circled numbers?


Atoms in the Universe of Numbers

Two hydrogen atoms and one oxygen atom

H2O

2 3 4 5 6

7 8 9 10 11

12 13 14 15 16

17 18 19 20 21

22 23 24 25

13 Atom

3 5 Molecule2 7 Molecule

Are the following numbers atoms or molecules? For molecules, write down the list of atoms:

12 = 22 3 Molecule13 =14 =15 =19 =27 =30 =32 =47 =50 =70 =

19 Atom33 Molecule2 3 5 Molecule25 Molecule47 Atom2 52 Molecule2 5 7 Molecule

Prime number(atom)

Composite number(molecule)


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.


Ulam’s Discovery73 74 75 76 77 78 79 80 81

72 43 44 45 46 47 48 49 50

71 42 21 22 23 24 25 26 51

70 41 20 7 8 9 10 27 52

69 40 19 6 1 2 11 28 53

68 39 18 5 4 3 12 29 54

67 38 17 16 15 14 13 30 55

66 37 36 35 34 33 32 31 56

65 64 63 62 61 60 59 58 57

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


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.


Explaining Ulam’s Rose

2 3 4 5 6 7

8 9 10 11 12 13

14 15 16 17 18 19

20 21 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

50 51 52 53 54 55

56 57 58 59 60 61

62 63 64 65 66 67

68 69 70 71 72 73

74 75 76 77 78 79

80 81 82 83 84 85

86 87 88 89 90 91

92 93 94 95 96 97

Table of numbers that is 6 columns wide shows that primes, except for 2 and 3, all fall in 2 columns

6k – 1 6k + 1Pattern

The two columns whose numbers are potentially prime form this pattern when drawn in a spiral


Activity 1: More Number Patterns

2 3 4 5 6 7

8 9 10 11 12 13

14 15 16 17 18 19

20 21 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

50 51 52 53 54 55

56 57 58 59 60 61

62 63 64 65 66 67

68 69 70 71 72 73

74 75 76 77 78 79

80 81 82 83 84 85

86 87 88 89 90 91

92 93 94 95 96 97

Color all boxes that contain multiples of 5 and explain the pattern that you see.

2 3 4 5 6 7

8 9 10 11 12 13

14 15 16 17 18 19

20 21 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

50 51 52 53 54 55

56 57 58 59 60 61

62 63 64 65 66 67

68 69 70 71 72 73

74 75 76 77 78 79

80 81 82 83 84 85

86 87 88 89 90 91

92 93 94 95 96 97

Color all boxes that contain multiples of 7 and explain the pattern that you see.


Activity 2: Number Patterns in a Spiral

73 74 75 76 77 78 79 80 81

72 43 44 45 46 47 48 49 50

71 42 21 22 23 24 25 26 51

70 41 20 7 8 9 10 27 52

69 40 19 6 1 2 11 28 53

68 39 18 5 4 3 12 29 54

67 38 17 16 15 14 13 30 55

66 37 36 35 34 33 32 31 56

65 64 63 62 61 60 59 58 57

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).

73 74 75 76 77 78 79 80 81

72 43 44 45 46 47 48 49 50

71 42 21 22 23 24 25 26 51

70 41 20 7 8 9 10 27 52

69 40 19 6 1 2 11 28 53

68 39 18 5 4 3 12 29 54

67 38 17 16 15 14 13 30 55

66 37 36 35 34 33 32 31 56

65 64 63 62 61 60 59 58 57


Perfect NumbersA perfect number equals the sum of its divisors, except itself

6: 1 + 2 + 3 = 6 28: 1 + 2 + 4 + 7 + 14 = 28496: 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 > 60100: 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 < 23128: 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 < 128


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 (32 = 9, 72 = 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?

Number Divisors (other than the number itself) Sum of divisors Type

12

18

28

30

45


Why Perfect Numbers Are Special

Some things we know about perfect numbers

There are only about a dozen perfect numbers up to 10160

All even perfect numbers end in 6 or 8

10160 = 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000

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 10300)


1089: A Very Special NumberFollow 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.


Special Numbers and Patterns

Why is the number 37 special?

3 37 = 111 and 1 + 1 + 1 = 36 37 = 222 and 2 + 2 + 2 = 69 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 = 13 + 53 + 33

1634 = 14 + 64 + 34 + 44

Here is an amazing pattern:

12 = 1112 = 121

1112 = 1232111112 = 1234321

111112 = 123454321


Activity 4: More Special Number Patterns

11 + 3

1 + 3 + 51 + 3 + 5 + 7

1 + 3 + 5 + 7 + 91 + 3 + 5 + 7 + 9 + 11

1 + 3 + 5 + 7 + 9 + 11 + 13

13 + 5

7 + 9 + 1113 + 15 + 17 + 19

21 + 23 + 25 + 27 + 2931 + 33 + 35 + 37 + 39 + 41

43 + 45 + 47 + 49 + 51 + 53 + 55

1 7 + 3 = 1014 7 + 2 = 100

142 7 + 6 = 10001428 7 + 4 = 10000

14285 7 + 5 = 100000142857 7 + 1 = 1000000

1428571 7 + 3 = 1000000014285714 7 + 2 = 100000000

142857142 7 + 6 = 10000000001428571428 7 + 4 = 10000000000

11 + 2 + 1

1 + 2 + 3 + 2 + 11 + 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.


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 = 579627 198 = 534639 186 = 725442 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 = _____


Numbers as Words

0 Zero 1 One 2 Two 3 Three 4 Four 5 Five 6 Six 7 Seven 8 Eight 9 Nine10 Ten

We can write any number as words. Here are some examples:12 Twelve 21 Twenty-one 80 Eighty3547 Three thousand five hundred forty-seven

EightFiveFourNineOneSevenSixTenThreeTwoZero

ThreeNineOneFiveTen

SevenZeroTwoFourEight

Six

OneTwoSixTenZeroFourFiveNineThreeSevenEight

EightFourSixTenTwoZeroFiveNineOneSevenThree


Activity 6: Numbers as Words

0 Zero 1 One 2 Two 3 Three 4 Four 5 Five 6 Six 7 Seven 8 Eight 9 Nine10 Ten

Alpha order

EightFiveFourNineOneSevenSixTenThreeTwoZero

ThreeNineOneFiveTen

SevenZeroTwoFourEight

Six

OneTwoSixTenZeroFourFiveNineThreeSevenEight

EightFourSixTenTwoZeroFiveNineOneSevenThree

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”?


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 ent11 eeelnv1213141516171819

20 enttwy21 eennottwy2223242526272829

30 313233343536373839

40414243444546474849


Next LessonNot 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?

000 001 010 011 100 101 110 111

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.

Documents

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