big numbers
Large numbers: it's the law
• Given a random variable with a fnite expected
value, if its values are repeatedly sampled, as
the number of these observatons increases,
their mean will tend to approach and stay
close to the expected value.
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 tmes smaller than bacteria
• Bacterial cells are about one tenth the size of
eukaryotc cells and are typically 0.5–5.0
micrometres in length
• There are approximately fve nonillion (5×10^30)
bacteria on Earth, forming much of the world's
biomass.
YouTube - From Quarks to Outer Space
• Quark: 100 atometers
• Milky way: 100,000 light years
Orders of Magnitude
• Shannon number
• the number of atoms in the
observable Universe is
estmated to be between
4x10^79 and 10^81.
How big is a googol?
• Numbers have names
Some small numbers
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10 trillion: natonal debt
1 trillion: a partal 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 natonal debt
Visualizing quanttes
• How many pennies would it take to fll the
empire state building?
How to make bigger numbers faster
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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 infnitely many prime
numbers
• Proof: given a list of prime numbers, multply
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
relatvely prime.
• Euclid’s algorithm is a method to fnd 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
multple of M that is smaller than N. Call the
remainder R.
• Claim: GCD(M,N) = GCD (M,R).
• Repeat untl 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: fnd GCD (1234,121)