What about infinity?

2. What about infinity?

4. What about infinity times infinity?

5. 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?

6. A former Math 210 project on large numbers

7. 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

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

9. 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,…. • Paradox: the Hilbert hotel

10. Any number between 0 and 1 can be represented by a sequence of zeros and ones

12. 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 correponding to which bin contains x, • If x is not an endpoint then one repeats the process on 10(x -and so on. • Example: • Note: in this case. • What about 0.99999….?

13. 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)

14. Example: Binary decomposition of N=27

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

16. The numbers between 0 and 1 are uncountable.

17. In search of…Georg Cantor

18. 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

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

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

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

22. Counting: some history • The first recorded use of numbers consisted of notches on bones. • Humans used addition before recorded history

23. Nowadays we use big numbers: • Numbers are represented by symbols: • 257,885,161-1 has 17,425,170 digits • To see: • 57,885,161/3.32193=17,425,169.7… • At 3000 characters per page, would take about 5000 pages to write down its digits. • Very large numbers are represented by descriptions. For example, Shannon’s number is the number of chess game sequences. • Very very large numbers are represented by increasingly abstract descriptions.

24. 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

26. Some ancient number systems

27. Cuneiform (Babylonians): base 60

28. Mayans: Base 20 (with zero)

29. Egyptians: base 10

30. Greeks (base 10)

31. Romans (base 10)

32. 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

33. Base 10 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10

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

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

36. Other cosmic questions

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

38. 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

39. These are a few of my least favorite things

40. 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.

42. 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

43. How big is a googol?

44. Some small numbers • 17 trillion: national debt • 1 trillion: a partial bailout • 314 million: number of americans • 1 billion: 3 x (number of americans) (approx) • 1 trillion: 1000 x 1 billion • $ 54,134: your share of the national debt • Each month the national debt increases by the annual GDP of New Mexico

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

46. 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)

47. big numbers Small Numbers have names

48. How to make bigger numbers faster • Googol: • Googolplex:

49. Power towers

50. Power towers and large numbers