Special Numbers
A Lesson in the “Math + Fun!” Series
May 2005
Special Numbers
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 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
May 2005
Edition
Released
First
May 2005
Revised
Special Numbers
Revised
Slide 2
What is
Special
About
These
Numbers?
Numbers in
purple squares?
Numbers in
green squares?
Circled
numbers?
May 2005
Special Numbers
Slide 3
Atoms in the Universe of Numbers
H2O
Prime
number
(atom)
Composite
number
(molecule)
Two hydrogen atoms
and one oxygen atom
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
May 2005
Special Numbers
Are the following numbers
atoms or molecules?
For molecules, write down
the list of atoms:
12 = 22  3
13 = 13
14 = 2  7
15 = 3  5
19 = 19
27 = 33
30 = 2  3  5
32 = 25
47 = 47
50 = 2  52
70 = 2  5  7
Molecule
Atom
Molecule
Molecule
Atom
Molecule
Molecule
Molecule
Atom
Molecule
Molecule
Slide 4
Is There a Pattern to Prime Numbers?
Primes
appear to
be randomly
distributed
in this list
that goes up
to 620.
Primes
become
rarer as we
go higher,
but there
are always
more
primes, no
matter how
high we go.
May 2005
Special Numbers
Slide 5
Ulam’s Discovery
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
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
May 2005
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
Special Numbers
Slide 6
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.
May 2005
Special Numbers
Slide 7
Explaining Ulam’s Rose
Table of numbers that is 6 columns
wide shows that primes, except for
2 and 3, all fall in 2 columns
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
May 2005
The two columns whose
numbers are potentially
prime form this pattern
when drawn in a spiral
Special Numbers
6k – 1
6k + 1
Pattern
Slide 8
Activity 1: More Number Patterns
Color all boxes that contain
multiples of 5 and explain
the pattern that you see.
Color all boxes that contain
multiples of 7 and explain
the pattern that you see.
2
3
4
5
6
7
2
3
4
5
6
7
8
9
10
11
12
13
8
9
10
11
12
13
14
15
16
17
18
19
14
15
16
17
18
19
20
21
22
23
24
25
20
21
22
23
24
25
26
27
28
29
30
31
26
27
28
29
30
31
32
33
34
35
36
37
32
33
34
35
36
37
38
39
40
41
42
43
38
39
40
41
42
43
44
45
46
47
48
49
44
45
46
47
48
49
50
51
52
53
54
55
50
51
52
53
54
55
56
57
58
59
60
61
56
57
58
59
60
61
62
63
64
65
66
67
62
63
64
65
66
67
68
69
70
71
72
73
68
69
70
71
72
73
74
75
76
77
78
79
74
75
76
77
78
79
80
81
82
83
84
85
80
81
82
83
84
85
86
87
88
89
90
91
86
87
88
89
90
91
92
93
94
95
96
97
92
93
94
95
96
97
May 2005
Special Numbers
Slide 9
Activity 2: Number Patterns in a Spiral
73
74
75
76
77
78
79
80
81
73
74
75
76
77
78
79
80
81
72
43
44
45
46
47
48
49
50
72
43
44
45
46
47
48
49
50
71
42
21
22
23
24
25
26
51
71
42
21
22
23
24
25
26
51
70
41
20
7
8
9
10
27
52
70
41
20
7
8
9
10
27
52
69
40
19
6
1
2
11
28
53
69
40
19
6
1
2
11
28
53
68
39
18
5
4
3
12
29
54
68
39
18
5
4
3
12
29
54
67
38
17
16
15
14
13
30
55
67
38
17
16
15
14
13
30
55
66
37
36
35
34
33
32
31
56
66
37
36
35
34
33
32
31
56
65
64
63
62
61
60
59
58
57
65
64
63
62
61
60
59
58
57
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).
May 2005
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.
Special Numbers
Slide 10
Perfect Numbers
A perfect number equals the sum of its divisors, except itself
6:
28:
496:
1+2+3=6
1 + 2 + 4 + 7 + 14 = 28
1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496
An abundant number has a sum of divisors that is larger than itself
36:
60:
100:
1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55 > 36
1 + 2 + 3 + 4 + 5 + 6 + 10 + 15 + 20 + 30 = 96 > 60
1 + 2 + 4 + 5 + 10 + 20 + 25 + 50 = 117 > 100
A deficient number has a sum of divisors that is smaller than itself
9:
23:
128:
May 2005
1+3=4<9
1 < 23
1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 < 128
Special Numbers
Slide 11
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.
Number Divisors (other than the number itself)
Sum of divisors
Type
12
18
28
30
45
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?
May 2005
Special Numbers
Slide 12
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
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)
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
May 2005
Special Numbers
Slide 13
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.
May 2005
Special Numbers
Slide 14
Special Numbers and Patterns
Why is the number 37 special?
3  37 = 111
6  37 = 222
9  37 = 333
12  37 = 444
and
and
and
and
Here is an amazing pattern:
12 = 1
112 = 121
1112 = 12321
11112 = 1234321
111112 = 123454321
1+1+1=3
2+2+2=6
3+3+3=9
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:
Playing around with a number
and its digits:
1 1/2  3 = 1 1/2 + 3
198 = 11 + 99 + 88
153 = 13 + 53 + 33
1634 = 14 + 64 + 34 + 44
1 1/3  4 = 1 1/3 + 4
1 1/4  5 = 1 1/4 + 5
May 2005
Special Numbers
Slide 15
Activity 4: More Special Number Patterns
Continue these patterns and find out
what makes them special.
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
May 2005
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
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
Special Numbers
Slide 16
Activity 5: Special or Surprising Answers
What is special about 9?
Can you find something special
in each of the following groups?
1  9 + 2 = ___
12  9 + 3 = ____
123  9 + 4 = _____
What’s special about the following?
12  483 = 5796
27  198 = 5346
39  186 = 7254
42  138 = 5796
What is special about 327?
327  1 = _____
327  2 = _____
327  3 = _____
Do the following multiplications:
4  1738 = _______
4  1963 = _______
18  297 = _______
28  157 = _______
48  159 = _______
May 2005
Do the following multiplications:
3  51249876 = ____________
9  16583742 = ____________
6  32547891 = ____________
Special Numbers
Slide 17
Numbers as Words
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
0
1
2
3
4
5
6
7
8
9
10
Zero
One
Two
Three
Four
Five
Six
Seven
Eight
Nine
Ten
May 2005
Eight
Five
Four
Nine
One
Seven
Six
Ten
Three
Two
Zero
Three
Nine
One
Five
Ten
Seven
Zero
Two
Four
Eight
Six
Special Numbers
One
Two
Six
Ten
Zero
Four
Five
Nine
Three
Seven
Eight
Eight
Four
Six
Ten
Two
Zero
Five
Nine
One
Seven
Three
Slide 18
Activity 6: Numbers as Words
Alpha order
0
1
2
3
4
5
6
7
8
9
10
Zero
One
Two
Three
Four
Five
Six
Seven
Eight
Nine
Ten
Eight
Five
Four
Nine
One
Seven
Six
Ten
Three
Two
Zero
Alpha order,
from the end
Three
Nine
One
Five
Ten
Seven
Zero
Two
Four
Eight
Six
Length order
Evens and odds
(in alpha order)
One
Two
Six
Ten
Zero
Four
Five
Nine
Three
Seven
Eight
Eight
Four
Six
Ten
Two
Zero
Five
Nine
One
Seven
Three
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”?
May 2005
Special Numbers
Slide 19
Activity 7: Sorting the Letters in Numbers
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!
0 eorz
1 eno
2 otw
3 eehrt
4 foru
5 efiv
6 isx
7 eensv
8 eghit
9 einn
May 2005
10 ent
11 eeelnv
12
13
14
15
16
17
18
19
20 enttwy
21 eennottwy
22
23
24
25
26
27
28
29
Special Numbers
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
Slide 20
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?
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.
May 2005
Special Numbers
Slide 21