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

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

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

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

  2. 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

  3. Any number between 0 and 1 can be represented by a sequence d1,d2,d3,… of zeros and ones • x1=x • d1=0 if 0<=x1<1/2; d1=1 if ½<=x1<1 • x2=2x1-d1 • d2=0 if 0<=x2<1/2; d2=1 if ½<=x2<1 • x3=2x2-d2, etc…. • Ex: x=0, d1=d2=…=0 • x=1/2, d1=1, d2=d3=….=0 • x=3/8: d1=0, d2=d2=1, d3=d4=…=0

  4. The numbers between 0 and 1 are uncountable.

  5. In search of…Georg Cantor

  6. 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

  7. Back down to Earth: • Numbers are represented by symbols: • 257,885,161-1 has 17,425,170 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.

  8. We use symbols to represent mathematical concepts such as numbers • The symbols 0,1,2,3,4,5,6,7,8,9 are known as the Hindu arabic numerals

  9. Some ancient number systems

  10. Cuneiform (Babylonians): base 60

  11. Mayans: Base 20 (with zero)

  12. Egyptians: base 10

  13. Greeks (base 10)

  14. Romans (base 10)

  15. 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

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

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

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

  19. 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

  20. These are a few of my least favorite things

  21. 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.

  22. 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

  23. How big is a googol?

  24. Some small numbers • 10 trillion: national debt • 1 trillion: a partial bailout • 300 million: number of americans • 1 billion: 3 x (number of americans) (approx) • 1 trillion: 1000 x 1 billion • $ 30,000: your share of the national debt

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

  26. 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)

  27. big numbers Small Numbers have names

  28. How to make bigger numbers faster • There is no biggest number • N+1 > N • 2*N>N • N^2>N if N>1 • Googol: 10^100 • Googolplex: 10^googol • “10^big = very big”

  29. Power towers

  30. Power towers and large numbers

  31. 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. • 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.

  32. Some facts about prime numbers

  33. 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.

  34. 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.

  35. 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.

  36. 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….pknkand b=p1m1p2m2…pkmk • Here p1,p2,….,pkare 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.

  37. Clicker Question • Which of the following correctly expresses expresses 123456789 as a product of prime factors: • A) 123456789=2*3*3*3*3*769*991 • B) 123456789=29*4257131 • C) 123456789=3*3*3607*3803 • D) 123456789=2*2*7*13*17*71*281

  38. 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.

  39. 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.

  40. How big is the largest known prime number? • 257,885,161-1 has 17,425,170 digits. • A typical 8x10 page of text contains a maximum of about 3500 characters (digits) • Printing out all of the digits would take about 5000 pages. That’s a full carton of standard copier paper. That’s about 0.6 trees.

  41. 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.

  42. 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)

  43. 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.

  44. Example: GCD (105, 77) • 49 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.

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

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