### Exploration of Real numbers - The Robert Noyce Scholarship Program

```Welcome to the Workshop
Presentation
Robert Noyce Teacher Scholarship
Program Conference
Renaissance Washington DC Hotel
July 8, 2011
Workshop Session IV
Meeting Room # 6
10:35 am to 11:50 am
Exploring the World of
Irrational Numbers
Dr. Viji K. Sundar
Professor of Mathematics
California State University, Stanislaus
[email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */
the rest is the work of man.
Mathematics is Beautiful.
Mathematics is Beautiful?
Mathematics is Beautiful!!
Pre-requisites
 Basic Set Theory
Equal sets and Equivalent Sets
 One to one correspondence
 Finite and Infinite Sets
 Countable and Uncountable Sets
 A set is infinite it it is NOT finite.
 Infinite set equivalent to one of its
subsets.
More Ideas to Grow on …
Let us recall what we know
 Let N be the set of Natural Numbers
 Let E be the set of Even Numbers
 Let O be the set of odd Numbers N = E U O
 Which set is ‘bigger?’ N or E N or O … Justify.
 Let Z be the set of Integers
 Let Q be the set of Rational Numbers
 Let Ir be the set of Irrational Numbers
 Let R be the set of Real Numbers
 Which is ’ bigger or larger’ has ‘more’ members ?
6
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Real Numbers
Cardinality of Numbers
 Cardinality - the number of members in a set.
 Cardinality can be finite or infinite.
 Cardinality of ‘number of heads in this room - 30’?
 Cardinality of Z is denumerably infinite. Z ~ Q
 Cardinality of Q is denumerably infinite. Z ~ Q
 Cardinality of total grains of sand in beaches is ?
Z equivalent to Q Z ~ Q
Density of Rational Numbers
that between any two rational numbers we can
insert a rational number.
Can you find a rational number between 2 and 3?
Can you find 5 rational numbers between 2 and 3?
This means, between any two rational numbers there
exist infinite number of rational numbers.
Isn't it amazing?
can also be written as 0.5
can also be written as 0.666666666...
can also be written as 0.38181818...
can also be written as 0.62855421687…
the decimals will repeat after 41 digits
Be careful when using your calculator to determine if a decimal number is
irrational. The calculator may not be displaying enough digits to show you
the repeating decimals, as was seen in the last example above.
What is an irrational number?
What is a rational number?
A number that can be expressed as a ratio of two
integers where the denominator Is not zero…
Can be expressed as a fraction a/b where b is not zero.
I want to know… What is an irrational
number?
A number that can not be expressed as a/b.
A Real Number That is not a rational number is
an irrational number !!!!!
Irrational Numbers
Every Real Number has a decimal
representation
For Rational numbers this will
Either terminate or
Non-terminating repeating
When it is
Non-terminating Non repeating …
it is Irrational
Where ARE All of Those Irrationals?
Pi
 is defined as the (constant) ratio of the
circumference to the diameter in any circle.
 the circumference and diameter of every
circle are known to be related by ….
 is bigger than 3 but less than 4
Hoax or Truth - Do your own search π = 3
Indiana Bill is the popular name for bill
#246 of the 1897 π = 3.2
 Oklahoma legislators had passed a law
making pi equal to 3.0.
 Congresswoman Martha Roby (R-Ala.) is
Simplification Act, declaring the Euclidean
mathematical constant of pi to be
precisely 3.
 Motivation …. Squaring a Circle?
Historically speaking .. Check this out
 The Bible says Pi = 3.
 A little known verse of the Bible reads:
 " And he made a molten sea, ten cubits from
the one brim to the other: it was round all
about, and his height was five cubits: and a
line of thirty cubits did compass it about.”
(I Kings 7, 23)
 C = 30 cubits
height = radius = 5 30 cubits
Lives intimately with the circle
 In recent years, the computation of the
expansion of pi has assumed the role of a
standard test of computer integrity.
 The number has been the subject of a great
deal of mathematical (and popular) folklore.
It's been worshipped, maligned, and
misunderstood. Overestimated,
underestimated, and legislated.
The Pi video
 How To Transform The Number Pi Into A Song
 by Michael John Blake
 To celebrate Pi Day — the 14th day of the third month —
Canadian musician Michael Blake has set the mathematical
constant to a tune." The idea just hit me one day — what
would happen if I found some kind of equation or formula
that I could transfer to music?" he told news.com.au."I
thought about the Fibonacci sequence, because I liked the
sound of that, but it didn't really work in my mind so the next
one I tried was Pi."Blake's music video on YouTube explains
how each of the eight notes in any major scale can be
assigned a numerical value — C being one, D two and so on.

Euler's number e
The number e is irrational.
Furthermore, it is transcendental
(All rational numbers are algebraic)
Euler's number That is it is not a root of a
polynomial with rational coefficients
.
All transcendental numbers are irrational.
Only few are known as it is difficult to prove
that a number is transcendental.
So you have a lot of problems to solve!
e … base of the Natural Logarithms
10 … base of common algorithm
e
Pi is a famous irrational number. People have calculated
Pi to over one million decimal places … no pattern.
3.1415926535897932384626433832795 (and more ...)
The number e (Euler's Number) is another famous
irrational number. The first few digits look like this:
2.7182818284590452353602874713527 (and more ...)
Phi the Golden Ratio is an irrational number.
1.61803398874989484820... (and more ...)
Many square roots, cube roots, etc are also
irrational numbers.
√3 1.7320508075688772935274463415…( more ...)
Euler's number e
 e … base of the Natural Logarithms
 10 … base of common algorithm
 The irrational e is :
 a number for algebra 2 students
 defines trig functions for Pre Calculus
students
 a limit by calculus students
But ‘e’ has a role in financial calculus and
population growth.
The compound-interest problem
 Jacob Bernoulli discovered this constant
by studying a question about compound
interest.
 Play this video
The number e and The Rule of 70
 The Rule of 70 is useful for financial as well as
demographic analysis.
 It states that to find the doubling time of a
quantity growing at a given annual percentage
rate, divide the percentage number into 70 to
obtain the approximate number of years
required to double.
 For example, at a 10% annual growth rate,
doubling time is 70 / 10 = 7 years.
The number e and The Rule of 70
Similarly, to get the annual growth rate,
divide 70 by the doubling time. For
example, 70 / 14 years doubling time = 5,
or a 5% annual growth rate.
Where is the Math?
 Mathematics Behind the Rule of 70
 The use of natural logs arises from
integrating the basic differential equation for
exponential growth:
 dN/dt = rN, over the period from t=0 to t =
the time period in question, where N is the
quantity growing and r is the growth rate.
 The integral of that equation is:
The integral of that equation is:
N(t) = N(0) x ert
 where N(t) quantity after t intervals have
elapsed,
 N(0) is the initial value of the quantity,
 r is the average growth rate over the interval
in question,
 t is the number of intervals.
The golden section is a line segment
divided according to the golden ratio:
The total length a + b is to the length
of the longer segment a as the length
of a is to the length of the shorter
segment b.
Video for the Golden Ratio
Movie Time
Pi is a famous irrational number. People have calculated Pi to over
one million decimal places and still there is no pattern.
3.1415926535897932384626433832795 (and more ...)
The number e (Euler's Number) is another famous irrational
number. The first few digits look like this:
2.7182818284590452353602874713527 (and more ...)
Phi the Golden Ratio is an irrational number.
1.61803398874989484820... (and more ...)
Many square roots, cube roots, etc are also irrational numbers.
√3 1.7320508075688772935274463415059 (etc)
√99 9.9498743710661995473447982100121 (etc)
That is all folks!
That is the end of the tour for now!!
This is just the beginning of your journey
into the wondrous world of
Irrational Numbers!
There are innumerable Irrational Numbers
waiting to be found and classified!
May your future be rich with discoveries!!!
```