big numbers

• Given a random variable with a finite expected value, if its values are repeatedly sampled, as the number of these observations increases, their mean will tend to approach and stay close to the expected value.

Large numbers: it's the law

These are a few of my 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.

YouTube - From Quarks to Outer Space

• Quark: 100 attometers• Milky way: 100,000 light years

Orders of Magnitude

• Shannon number• the number of atoms in the

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

How big is a googol?

• Numbers have names

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?

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

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.

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.

Exercise: find GCD (1234,121)