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

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

The numbers between 0 and 1 are uncountable.

In search of…Georg Cantor

• 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

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.

• 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

Some ancient number systems

Cuneiform (Babylonians): base 60

Mayans: Base 20 (with zero)

Egyptians: base 10

Greeks (base 10)

Romans (base 10)

• 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

Base 10• 2• 3• 4• 5• 6• 7• 8• 9• 10

Powers of 10

• Alt 1• More videos and other sources on powers of

10

Orders of Magnitude

• Shannon number• the number of atoms in the

observable Universe is estimated to be between 4x10^79 and 10^81.

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

These are a few of my least favorite things

• 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.

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

How big is a googol?

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

Visualizing quantities

• How many pennies would it take to fill the empire state building?

• Your share of the national debt

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)

big numbers

Small Numbers have names

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”

Power towers

Power towers and large numbers

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.

Some facts about prime numbers

• 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.

Every whole number is either prime or is divisible by a smaller prime number.

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.

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.

• 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=p1n1p2

n2….pknk and b=p1

m1p2m2…pk

mk

• Here p1, p2,….,pk are all primes smaller than n and the

exponents could equal zero.• Then n=ab=p1

n1+m1p2n2+m2….pk

nk+mk

• The exponents are unique since changing any of them would change the product.

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

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.

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.

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.

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.

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)

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.

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.

Clicker question: find GCD (1234,121)

• A) 1• B) 11• C) 21• D) 121