f38 Special Numbers

8/6/2019 f38 Special Numbers

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May 2005 Special Numbers Slide 1

Special NumbersA Lesson in the Math + Fun! Series


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About This Presentation

Edition Released Revised Revised

First May 2005

This presentation is part of the Math + Fun! series devisedby 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 educationalsettings. Unauthorized uses are strictly prohibited.

Behrooz Parhami


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What isSpecial

AboutTheseNumbers?

Numbers inpurple squares?

Numbers in

green squares?

Circled

numbers?


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Atoms in the Universe ofNumbers

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 Molecule

2

3

5Molecule25 Molecule

47 Atom2 52 Molecule2 5 7Molecule

Prime

number(atom)

Composite

number

(molecule)


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Is There a Pattern to PrimeNumbers?

Primes

become

rarer as we

go higher,

but there

are alwaysmore

primes, no

matter how

high we go.

Primes

appear to

be randomlydistributed

in this list

that goes up

to 620.


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Ulams 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 spiraland discovered that prime numbersbunch together along diagonal lines

Primes pattern for numbers up to about60,000; notice that primes bunchtogether along diagonal lines and theythin out as we move further out


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UlamsRose

Primes patternfor numbers upto 262,144.

Just as watermoleculesbunch togetherto make asnowflake,

prime numbersbunch togetherto produceUlams rose.


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Explaining Ulams 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 columnswide shows that primes, except for

2 and 3, all fall in 2 columns

6k 1

6k+ 1Pattern

The two columns whosenumbers are potentially

prime form this patternwhen drawn in a spiral


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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 3738 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 containmultiples of 5 and explainthe 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 3738 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 containmultiples of 7 and explainthe pattern that you see.


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Activity 2: Number Patterns in aSpiral

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 notmultiples of 3 or 5. For example, 4 and 14should be colored, but not 10 or 12.

Color the multiples of 3. Use two differentcolors 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


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Perfect Numbers

A perfect numberequals 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 numberhas 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 numberhas a sum of divisors that is smaller than itself9: 1 + 3 = 4 < 9

23: 1 < 23

128: 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 < 128

A ti it 3 Ab d t D fi i t


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Activity 3: Abundant, Deficient, orPerfect?

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

3045


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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 )


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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.


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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 = 1

112 = 121

1112 = 12321

11112 = 1234321

111112 = 123454321


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Activity 4: More Special NumberPatterns

11 + 3

1 + 3 + 51 + 3 + 5 + 7

1 + 3 + 5 + 7 + 9

1 + 3 + 5 + 7 + 9 + 111 + 3 + 5 + 7 + 9 + 11 + 13

13 + 5

7 + 9 + 1113 + 15 + 17 + 1921 + 23 + 25 + 27 + 29

31 + 33 + 35 + 37 + 39 + 4143 + 45 + 47 + 49 + 51 + 53 + 55

1

7 + 3 = 1014 7 + 2 = 100142 7 + 6 = 1000

1428 7 + 4 = 1000014285 7 + 5 = 100000

142857 7 + 1 = 10000001428571 7 + 3 = 10000000

14285714 7 + 2 = 100000000142857142 7 + 6 = 1000000000

1428571428 7 + 4 = 10000000000

1

1 + 2 + 11 + 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.


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Activity 5: Special or Surprising Answers

Can you find something special

in each of the following groups?

Whats 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 = _____


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Numbers as Words

0 Zero

1 One

2 Two

3 Three4 Four

5 Five

6 Six

7 Seven8 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

NineOne

Seven

Six

TenThree

Two

Zero

Three

Nine

One

FiveTen

Seven

Zero

TwoFour

Eight

Six

One

Two

Six

TenZero

Four

Five

NineThree

Seven

Eight

Eight

Four

Six

TenTwo

Zero

Five

NineOne

Seven

Three


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Activity 6: Numbers as Words

0 Zero1 One

2 Two

3 Three

4 Four

5 Five

6 Six

7 Seven

8 Eight

9 Nine10 Ten

Alpha order

EightFive

Four

Nine

One

Seven

Six

Ten

Three

TwoZero

ThreeNine

One

Five

Ten

Seven

Zero

Two

Four

EightSix

OneTwo

Six

Ten

Zero

Four

Five

Nine

Three

SevenEight

EightFour

Six

Ten

Two

Zero

Five

Nine

One

SevenThree

Alpha order,from the end Length order

Evens and odds(in alpha order)

If we wrote these four lists from zero to one thousand, whichnumber would appear first/last in each list? Why? What about to onemillion?


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Activity 7: Sorting the Letters in Numbers

0 eorz

1 eno

2 otw3 eehrt

4 foru

5 efiv

6 isx7 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

1213

14

15

1617

18

19

20 enttwy

21 eennottwy

2223

24

25

2627

28

29

30

31

3233

34

35

3637

38

39

40

41

4243

44

45

4647

48

49


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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 111Computers 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.

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f38 Special Numbers