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

2. What is Special About These Numbers?
Numbers in purple squares?
Numbers in green squares?
Circled numbers?
Special Numbers

3. Atoms in the Universe of Numbers
Are the following numbers atoms or molecules?
For molecules, write down the list of atoms:
12 = 22  3 Molecule
13 =
14 =
15 =
19 =
27 =
30 =
32 =
47 =
50 =
70 =
Prime number
(atom)
H2O
Composite number
(molecule)
Two hydrogen atoms and one oxygen atom
13 Atom
2  7 Molecule
3  5 Molecule
19 Atom
33 Molecule
2  3  5 Molecule
25 Molecule
47 Atom
2  Molecule
2  5  7 Molecule
Special Numbers

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.
Special Numbers

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
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
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
Special Numbers

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.
Special Numbers

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
6k – 1
6k + 1
Pattern
The two columns whose numbers are potentially prime form this pattern when drawn in a spiral 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
Special Numbers

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 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 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
Special Numbers

9. Activity 2: Number Patterns in a Spiral73 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 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 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).
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

10. Perfect Numbers
A perfect number equals the sum of its divisors, except itself
6: = 6
28: = 28
496: = 496
An abundant number has a sum of divisors that is larger than itself
36: = 55 > 36
60: = 96 > 60
100: = 117 > 100
[Crub97] “Looking for Perfect Numbers” by M. Crubellier and J. Sip, Chapter 15 in [Week97].
See also, [Wago85] Wagon, “Perfect Numbers,” The Mathematical Intelligencer, Vol. 7, No. 2, pp , 1985.
A deficient number has a sum of divisors that is smaller than itself
9: = 4 < 9
23: 1 < 23
128: = 127 < 128
Special Numbers

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?
Special Numbers

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)
Sum of inverses of divisors of a perfect number is 1 (exclude 1/1; else 2). If there are odd perfect numbers, they must be greater than 10^300 and have at least 37 prime factors ([GUYR04] Unsolved Problems in Number Theory). Even perfect numbers are of the form 2^n(2^(n+1) – 1), with the second term prime. The largest known perfect number is 2^216,090(2^216,091 – 1).
10160 =
Special Numbers

13. The answer is always 1089. 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: = 1089)
The answer is always 1089.
Special Numbers

14. Special Numbers and Patterns
Here is an amazing pattern:
12 = 1
112 = 121
1112 = 12321
11112 = =
Why is the number 37 special?
3  37 = 111 and = 3
6  37 = 222 and = 6
9  37 = 333 and = 9
12  37 = 444 and = 12
When adding or multiplying does not make a difference.
You know that 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 =
153 =
1634 =
Special Numbers

15. Activity 4: More Special Number Patterns
Continue these patterns and find out what makes them special. 1 1
1 + 3
1  = 10
14  = 100
142  = 1000
1428  = 10000
14285  =
 =
 =
 =
 =
 = 1
3 + 5
Special Numbers

16. Activity 5: Special or Surprising Answers
What is special about 9?
1  = ___
12  = ____
123  = _____
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
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 = _______
Do the following multiplications:
3  = ____________
9  = ____________
6  = ____________
Special Numbers

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

18. Activity 6: Numbers as Words
Alpha order, from the end
Evens and odds (in alpha order)
Alpha order
Length order
0 Zero
1 One
2 Two
3 Three
4 Four
5 Five
6 Six
7 Seven
8 Eight
9 Nine
10 Ten
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
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”?
Special Numbers

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
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
Special Numbers

20. Next Lesson Math and Computers
Not definite, at this point: Thursday, June 9, 2005
Math and Computers
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?
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.
       
Special Numbers