What about infinity?
What about infinity times infinity?
Infinity times infinity
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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?
Hilbert's Hotel
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Is infinity equal to infinity plus one?
Is infinity plus infinity larger than infinity?
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Is infinity times infinity larger than infinity?
A former Math 210 project on large nu
Ordered pairs
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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 diferent
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
The number of rational numbers is equal to the
number of whole numbers
Countable sets
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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-toone correspondence with the whole numbers
1,2,3,….
The Hilbert hotel is a formula for such a
correspondence
Any number between 0 and 1 can be represented by an
infinite sequence of zeros and ones
Binary representation
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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
corresponding 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….?
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)
Example: Binary decomposition of N=27
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Question: Is there any relationship between
the binary decomposition of N and of 1/N?
Example: compare 3 and 1/3.
One-to-one correspondence
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Two sets are in a one-toone correspondence if
there is a mapping that
assigns to each element
of the first set a unique
element of the second
set
The numbers between 0 and 1 are
uncountable.
In search of…Georg Cantor
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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
Aleph naught
Clicker question
Cardinal numbers refer only to numbers worn
on the jerseys of St Louis Cardinals players
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A – True
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B -False
Clicker question
Cardinal numbers can be infinite (larger than
any finite number)
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A – True
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B - False
Clicker question
All infinite cardinal numbers are the same size
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A – True
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B - False
Describing numbers: some history
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The first recorded use of numbers consisted of
notches on bones.
Numbers were first use for counting
Humans used addition before recorded
history
Nowadays large numbers are used to encode
information, not to describe quantities
Nowadays we use big numbers:
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Numbers are represented by symbols:
274,207,281 − 1 with 22,338,618 digits, discovered by the GIMPS in 2016
To see:
74,207,281/3.32193=22338604.6666…
At 3000 characters per page, would take about 7500 pages to write
down its digits. The next largest known prime, discovered in 2013, only
has about 15 million digits.
Very large finite numbers are represented by descriptions. For example,
Shannon’s number is the number of chess game sequences.
VERY large numbers require increasingly abstract descriptions.
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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
Some ancient number systems
Cuneiform (Babylonians): base 60
360
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Sumerians, approx 2500 BC: 360 is
approximately the number of days in a year. It
is small enough to subdivide but large enough
Egyptians approx 1500 BC: divided day into 24
hours (length of hours varied by season. Greek
astronomers made hours equal)
Mayans: Base 20 (with zero)
Egyptians: base 10
Greeks (base 10)
Romans (base 10)
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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
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2
3
4
5
6
7
8
9
10
Scientific notation
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Scientific notation allows us to represent
numbers conveniently when only order of
magnitude matters.
Powers of 10
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Alt 1
More videos and other sources on powers of 10
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Other cosmic questions
Orders of Magnitude
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Shannon number
the number of atoms in the
observable Universe is
estimated to be between
4x10^79 and 10^81.
Bog quesitons: Are we alone?
Small questions: what is the universe m
Some orders on human scales
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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
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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
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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.
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How much do the bacteria in a typical human weigh?
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A) < 10 grams
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B) between 10 and 100 grams
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C) between 100 grams and 1 kg
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D) between 1 Kg and 10 Kg
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E) > 10 Kg
Summary:
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Changing an order of magnitude (multiplying
or dividing by 10) is the same as shifing over
one decimal place. This is an efcient way of
dealing with quantities on human scales.
Making measurements depends on properties
of matter. Our ability to make measurements
is limited by our ability to understand matter
at large or small scales.
How big is a googol?
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We can take advantage of the existence of
very large and very small numbers to use
numbers as tools to encode information. In
doing so, we are only limited by our ability to
describe large or small numbers and their
properties.
Some small numbers
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17.5 trillion: national debt
1 trillion: a partial bailout
314 million: number of Americanos
1 billion: 3 x (number of Americans) (approx)
1 trillion: 1000 x 1 billion
$ 55,700: your share of the national debt
Each month the national debt increases by the annual GDP of
New Mexico
Visualizing quantities
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How many pennies would it take to fill the
empire state building?
Your share of the national debt
Clicker question
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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
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Googol:
Googolplex:
Power towers
Power towers and large numbers
Number and Prime Numbers
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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. (
)
Division allows us to divide the things counted into equal
groups.
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
Every whole number is either prime or is
divisible by a smaller prime number.
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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.
There are infinitely many prime numbers
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Proof by contradiction.
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If there were only finitely many then we could list them all: p1,p2,…,pN
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Set Q=p1*p2*…*pN+1
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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.
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Fundamental theorem of arithmetic: Every
whole number can be writen 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.
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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….pknk and b=p1m1p2m2…pkmk
Here p1, p2,….,pk are 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.
Clicker Question
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Which of the following correctly 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:
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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
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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.
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Some things that are not known
about
prime
Goldbach’s
conjecture:
every evennumbers
number bigger than two is
the sum of two prime numbers (e.g., 8=3+5; 112=53+59; etc)
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**Twin prime conjecture: there are infinitely many primes p
such that p+2 is also a prime. In this case, p and p+2 are called
twin primes.
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E.g., (3,5), (5,7), (11,13), (29,31) etc
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However,…, Gap prime conjecture: there is a number N
(<70 million) such that there are infinitely many prime pairs of
the form (p, p+N).
How big is the largest known prime number?
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At 3000 characters per page, would take about 7500
pages to write down its digits. The
274,207,281 − 1 has 22,338,618 digits (GIMPS’16)
257,885,161-1 has 17,425,170 digits (2013)
A typical 8x10 page of text contains a maximum of
about 3500 characters (digits)
Printing out all of the digits would take about 6400
pages. That’s about 0.75 trees.
Security codes
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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.
The Euclidean algorithm
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"[The Euclidean algorithm] is the granddaddy
of all algorithms, because it is the oldest
nontrivial algorithm that has survived to the
present day."
Donald Knuth, The Art of Computer
Programming, Vol. 2: Seminumerical
Algorithms, 2nd edition (1981), p. 318.
Euclid’s algorithms: GCD
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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
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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)
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77 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.
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Clicker question: find GCD
(1234,121)
A) 1
B) 11
C) 21
D) 121