4. What about infinity?

5. What about infinity times infinity?

6. Infinity times infinity
Are all infinities the same?
Is infinity plus one larger than infinity?
Is infinity plus infinity larger than infinity?
Is infinity times infinity larger than infinity?

7. Hilbert's Hotel
Is infinity equal to infinity plus one?

8. Is infinity plus infinity larger than infinity?
Is infinity times infinity larger than infinity?

9. A former Math 210 project on large numbers

10. Ordered pairs
An ordered pair of numbers is simply two numbers, one listed before the other:
(3,2), (3.14,2.71), (m,n)
An ordered pair of elements of a set is simply two elements of the set, one listed before the other. For example, if the set is the alphabet then (a,b) is an ordered pair. (b,a) is a different ordered pair.
Given any two sets A and B, the collection of all ordered pairs of elements, one from A then one from B, defines another set called the Cartesian product, denoted AxB

11. The number of rational numbers is equal to the number of whole numbers

12. Countable sets
A set is countable if its elements can be enumerated using the whole numbers.
A set is countable if it can be put in a one-to-one correspondence with the whole numbers 1,2,3,….
The Hilbert hotel is a formula for such a correspondence

13. Any number between 0 and 1 can be represented by an infinite sequence of zeros and ones

14. Binary representation

15. Binary representation (cont)
The nth bit of x can be thought of as a microscope that zooms into the (dyadic) interval of length containing x
And decides whether x is in the left half or right half of this interval.
The number corresponds to the left endpoint of the interval of length containing x

16. The sequence where is an increasing sequence of numbers in [0,1] that lie to the left of x and get closer and closer to x (or eventually equal x).
since both numbers lie in the same interval of length
We write meaning, literally, that x is a limit of the terminating numbers

17. Any number between zero and 1 also has a decimal representation
In this case each digit takes the value between 0 and 9.
One divides [0,1] into 10 equal bins and assigns the digit corresponding to which bin contains x,
If x is not an endpoint then one repeats the process on
and so on.
Example:
Note: in this case.
What about ….?

18. Binary representation of whole numbers
Here Algorithm: Step 1: Find the largest power of 2 less than or equal to N. This is k. Step 2: If then done . Otherwise, subtract from N. Apply step 1 to stop when either the remainder is a power of two (possibly equal to one)

19. Example: Binary decomposition of N=27

20. Clicker question The binary decomposition of the number 8 is: A) 0.1

21. Question: Is there any relationship between the binary decomposition of N and of 1/N?
Example: compare 3 and 1/3.

22. One-to-one correspondence
Two sets are in a one-to-one correspondence if there is a mapping that assigns to each element of the first set a unique element of the second set

23. The numbers between 0 and 1 are uncountable.

24. In search of…Georg Cantor

25. 2^N: number of subsets of a set of N elements
Ordinal number: 0,1,2, etc
Cardinal number:
2^N: number of subsets of a set of N elements
Number of subsets of the natural numbers
The “Continuum hypothesis”
Aleph naught

26. Clicker question
Cardinal numbers refer only to numbers worn on the jerseys of St Louis Cardinals players
A – True
B -False

27. Clicker question
Cardinal numbers can be infinite (larger than any finite number)
A – True
B - False

28. Clicker question All infinite cardinal numbers are the same size
A – True
B - False

29. Describing numbers: some history
The first recorded use of numbers consisted of notches on bones.
Numbers were first use for counting
Humans used addition before recorded history
Nowadays large numbers are used to encode information, not to describe quantities

30. Nowadays we use big numbers:
Numbers are represented by symbols:
274,207,281 − 1 with 22,338,618 digits, discovered by the GIMPS in 2016
To see:
74,207,281/ = …
At 3000 characters per page, would take about 7500 pages to write down its digits. The next largest known prime, discovered in 2013, only has about 15 million digits.
Very large finite numbers are represented by descriptions. For example, Shannon’s number is the number of chess game sequences.
VERY large numbers require increasingly abstract descriptions.

32. We use symbols to represent mathematical concepts such as numbers
Some number systems facilitate calculations and handling large magnitudes better than others
The symbols 0,1,2,3,4,5,6,7,8,9 are known as the Hindu arabic numerals

34. Some ancient number systems

35. Cuneiform (Babylonians): base 60

36. 360
Sumerians, approx 2500 BC: 360 is approximately the number of days in a year. It is small enough to subdivide but large enough
Egyptians approx 1500 BC: divided day into 24 hours (length of hours varied by season. Greek astronomers made hours equal)

37. Mayans: Base 20 (with zero)

38. Egyptians: base 10

39. Greeks (base 10)

40. Romans (base 10)

41. Only the Mayan’s had a “zero”
Babylonians: base 60 inherited today in angle measures. Used for divisibility.
No placeholder: the idea of a “power” of 10 is present, but a new symbol had to be introduced for each new power of 10.
Decimal notation was discovered several times historically, notably by Archimedes, but not popularized until the mid 14th cent.
Numbers have names

42. Base 10 2 3 4 5 6 7 8 9 10

43. Scientific notation
Scientific notation allows us to represent numbers conveniently when only order of magnitude matters.

44. Powers of 10
Alt 1
More videos and other sources on powers of 10

45. Other cosmic questions

46. Just for fun (11/03/16) Eamus Catuli means:
A) May the force be with you
B) Live long and prosper
C) Don’t tread on me
D) Let’s go Cubs

47. Orders of Magnitude Shannon number
the number of atoms in the observable Universe is estimated to be between 4x10^79 and 10^81.

48. Big questions: Are we alone?

49. Small questions: what is the universe made of?

50. Powers of 10: Charles and Ray Eames
In metric units, there are 40 orders of magnitude in the observable universe;
10^24 meters: 100 million light years: deep space
10^-16:10^-6 angstroms: subatomic particles
(10^-14: a single proton)

51. Some orders on human scales
Human scale I: things that humans can sense directly (e.g., a bug, the moon, etc)
Human scales II: things that humans can sense with light, sound etc amplification (e.g., bacteria, a man on the moon, etc)
Large and small scales: things that require specialized instruments to detect or sense indirectly
Indirect scales: things that cannot possibly be sensed directly: subatomic particles, black holes

52. These are a few of my least favorite things

53. Viruses vary in shape from simple helical and icosahedral shapes, to more complex structures. They are about 100 times smaller than bacteria
Bacterial cells are about one tenth the size of eukaryotic cells and are typically 0.5–5.0 micrometres in length
There are approximately five nonillion (.5×10^30) bacteria on Earth, forming much of the world's biomass.

55. Clicker question
If the average weight of a bacterium is a picogram (10^12 or 1 trillion per gram).
The average human is estimated to have about 50 trillion human cells, and it is estimated that the number of bacteria in a human is ten times the number of human cells.
How much do the bacteria in a typical human weigh?
A) < 10 grams
B) between 10 and 100 grams
C) between 100 grams and 1 kg
D) between 1 Kg and 10 Kg
E) > 10 Kg

56. Summary:
Changing an order of magnitude (multiplying or dividing by 10) is the same as shifting over one decimal place. This is an efficient way of dealing with quantities on human scales.
Making measurements depends on properties of matter. Our ability to make measurements is limited by our ability to understand matter at large or small scales.

57. How big is a googol?
We can take advantage of the existence of very large and very small numbers to use numbers as tools to encode information. In doing so, we are only limited by our ability to describe large or small numbers and their properties.

58. Some small numbers 19.8 trillion: national debt
1 trillion: a partial bailout
318.9 million: number of Americanos
1 billion: 3 x (number of Americans) (approx)
1 trillion: 1000 x 1 billion
$ 62000: your approximate share of the national debt
Each month the national debt increases by the annual GDP of New Mexico

59. Trump tweet Feb 25: national debt went down 12 billion vs 200 billion increase under Obama
Congressional Budget office estimate: $559 billion increase in 2017, about 7 times NM GDP of Billion

60. Visualizing quantities
How many pennies would it take to fill the empire state building?
Your share of the national debt

61. Clicker question
If one cubic foot of pennies is worth $491.52, your share of the national debt, in pennies, would fill a cube closest to the following dimensions:
A) 1x1x1 foot (one cubic foot)
B) 3x3x3 (27 cubic feet)
C) 5x5x5 feet (125 cubic feet)
D) 100x100x100 (1 million cubic feet)
E) 1000x1000x1000 (1 billion cubic feet)

62. big numbers
Small Numbers have names

63. How to make bigger numbers faster
Googol:
Googolplex:

64. Power towers

65. Power towers and large numbers

66. Bigfoot Bigfoot, the number
Note: explaining all of this in a sensible way would make a good project
FOOT: a function that takes a large number and returns a much larger number.
Example: Exponentiation:
Iteration:
And so on.
FOOT is a process that makes big numbers faster, such as busy beaver numbers
More history of big numbers

67. Googology is the study and nomenclature of large numbers
Little Bigeddon is considered the current holder of the title of largest valid googologism
Finitism: infinite objects (e.g. transfinite ordinals) do not exist
Ultrafinitism: natural numbers end because after some point they cannot be computed: they are circularly defined

68. Number and Prime Numbers
Natural numbers: 0,1,2,3,… allow us to count things.
Divisible: p is divisible by q if some whole number multiple of q is equal to p. ( )
Division allows us to divide the things counted into equal groups.
Remainder: if p>q but p is not divisible by q then there is a largest m such that mq<p and we write p=mq+r where 0<=r<q
p is prime if its only divisors are p and itself.

69. Some facts about prime numbers

70. Every whole number is either prime or is divisible by a smaller prime number.
Proof: If Q is not prime then we can write Q=ab for whole numbers a, b where a>1 (and hence b<Q)
Suppose that a is the smallest whole number, larger than one, that divides into Q. Then a is prime since, otherwise, we could write a=cd where c>1 (and hence d<a). But then d is a smaller number than a that divides into Q, which contradicts our choice of a.

71. There are infinitely many prime numbers
Proof by contradiction.
If there were only finitely many then we could list them all: p1,p2,…,pN
Set Q=p1*p2*…*pN+1
Claim: Q is not divisible by any of the numbers in the list. Otherwise, Q=Pm for some integer m and P in the list, say P=p1 (the same argument applies or the other pi’s) Then
p1*(p2*…*pN)+1 =p1*m or p1*(m-p2*…*pN)=1
But this is impossible because if the product of two whole numbers a and b is 1, i.e., a*b=1, then a=1 and b=1. But p1 is not equal to one.
This contradiction proves that Q is not divisible by any prime number on the list so either Q itself is a prime number not on the list or it is divisible by a prime number not on the list.

72. Fundamental theorem of arithmetic: Every whole number can be written uniquely as a product of prime powers.
We use the principal of mathematical induction: if the statement is true for n=1 and if its being true for all numbers smaller than n implies that it is true for n, then it is true for all whole numbers.

73. If n is itself prime then we are done (why?)
Otherwise n is composite, ie, n=ab where a,b are whole numbers smaller than 1. The induction hypothesis is that a and b can be written uniquely as products of prime powers, that is,
a=p1n1p2n2….pknk and b=p1m1p2m2…pkmk
Here p1, p2,….,pk are all primes smaller than n and the exponents could equal zero.
Then n=ab=p1n1+m1p2n2+m2….pknk+mk
The exponents are unique since changing any of them would change the product.

74. Clicker Question
Which of the following correctly expresses as a product of prime factors:
A) =2*3*3*3*3*769*991
B) =29*
C) =3*3*3607*3803
D) =2*2*7*13*17*71*281

75. What this means:
There is a code (the prime numbers) for generating any whole number via the code
Given the code, it is simple to check the code (by multiplying)
Given the answer, it is not easy, necessarily, to find the code.

76. Large prime numbers Euclid: there are infinitely many prime numbers
Proof: given a list of prime numbers, multiply all of them together and add one.
Either the new number is prime or there is a smaller prime not in the list.

77. Some things that are not known about prime numbers
Goldbach’s conjecture: every even number bigger than two is the sum of two prime numbers (e.g., 8=3+5; 112=53+59; etc)
**Twin prime conjecture: there are infinitely many primes p such that p+2 is also a prime. In this case, p and p+2 are called twin primes.
E.g., (3,5), (5,7), (11,13), (29,31) etc
However,…, Gap prime conjecture: there is a number N
(<70 million) such that there are infinitely many prime pairs of the form (p, p+N).

78. How big is the largest known prime number?
At 3000 characters per page, would take about 7500 pages to write down its digits. The
274,207,281 − 1 has 22,338,618 digits (GIMPS’16)
257,885,161-1 has 17,425,170 digits (2013)
A typical 8x10 page of text contains a maximum of about 3500 characters (digits)
Printing out all of the digits would take about 6400 pages. That’s about 0.75 trees.

79. Security codes
Later we might discuss RSA encryption, which is based on prime number pairs, M=E*D where E,D are prime numbers. Standard 2048 bit encryption uses numbers M that have about 617 digits. In principle we have to check divisibility by prime numbers up to about 300 digits.

80. The Euclidean algorithm
"[The Euclidean algorithm] is the granddaddy of all algorithms, because it is the oldest nontrivial algorithm that has survived to the present day."
Donald Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 2nd edition (1981), p. 318.

81. Euclid’s algorithms: GCD
The greatest common divisor of M and N is the largest whole number that divides evenly into both M and N
GCD (6 , 15 ) = 3
If GCD (M, N) = 1 then M and N are called relatively prime.
Euclid’s algorithm is a method to find GCD (M,N)

82. Euclid’s algorithm M and N whole numbers.
Suppose M<N. If N is divisible by M then GCD(M,N) = M
Otherwise, subtract from N the biggest multiple of M that is smaller than N. Call the remainder R.
Claim: GCD(M,N) = GCD (M,R).
Repeat until R divides into previous.

83. Example: GCD (105, 77) 77 does not divide 105.
Subtract 1*77 from 105. Get R=28
28 does not divide into 77. Subtract 2*28 from 77. Get R=77-56=21
Subtract 21 from 28. Get 7.
7 divides into 21. Done.
GCD (105, 77) = 7.

84. Clicker question: find GCD (1234,121)
A) 1
B) 11
C) 21
D) 121