April 06, 2017 Thursday
Bedtime Story
The Transcendental Nature of 𝝅 Sealed the Fate of Circling the Square
It was known for some time that it would be impossible to carry
out the task of circling the square if it was proven that pi is transcendental.
Yet it was not proven so until 1882.
A transcendental number is a real or complex number that is not
algebraic.
A transcendental number is not a root of any non-zero polynomial having
rational coefficients.
An algebraic number is any complex number that is a root of a
non-zero polynomial in one variable with rational coefficients (or with integer
coefficients if denominator has been cleared).
This is the reason why the square root of 2 and the golden ratio,
though being irrational are not transcendental.
The golden ratio is denoted by the Greek letter phi
.
Any two numbers or quantities are said to exist in a golden ratio
if their ratio is same as the ratio of their sum to the larger quantities.
Algebraically, let a > b > 0.
Then
a/b = (a+b)/a
The golden ratio or 𝜑
This number continues and if you are interested, you can follow
this unending sequence at The On-Line Encyclopedia of Integer Sequences or the
OEIS.
According to the popular mathematics writer Ian Stewart, this
is the most irrational of
all the irrational numbers.
Yet, just like the root of 2, it is not a transcendental number.
The root of 2 is a solution to the polynomial equation:
x2 – 2 = 0
Similarly, golden ratio or
is a solution to the
polynomial equation:
x2 - x - 1 = 0
By the way, it was in the prolific nineteenth century that the
transcendental numbers were first shown to exist.
The credit goes to the French mathematician Joseph Liouville.
In number theory, the numbers that he constructed go by the name
of Liouville numbers and in 1844, the man proved that all Liouville numbers are
transcendental.
Two German mathematicians, Ferdinand von Lindemann and Karl
Weierstrass in 1882 published a theorem that goes by the name of
Lindemann-Weierstrass theorem.
This theorem is very useful is proving the transcendence of a
number.
The theorem is a landmark feat in human intellect.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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