June 22, 2018 Friday
Bedtime Story
Defining Real Numbers
Last night we were talking about
mathematical fields and the field of real numbers is one example of such a
field.
The real numbers can be descriptively termed
as infinite decimal representations.
The truth is that defining real numbers is
a cumbersome task as the way I have defined it above completely lacks the rigor
necessary in modern mathematics.
In fact, the need to have a mathematical rigorous
definition for real numbers was one of the important developments of the nineteenth
century mathematics.
Just to show you how seriously
mathematicians take their definitions, I am going to write down - just write
down without any explanation - the modern formal definition of Real Number.
Real numbers form the unique Dedekind-complete
ordered field (R; +; . ; <), up to an isomorphism.
This is the current axiomatic definition of
real numbers and you can very well see that almost every word of this short
definition can fill pages of mathematical storytelling.
So let us not digress too much and return
back to the vector and scalar quantities.
A scalar quantity is a physical quantity
that can be described by a single element of a number field such as a real
number accompanied with a unit of measurement.
In a more simple language, a scalar
quantity is such a physical entity that has only magnitude and no other
component.
This is in contrast to vectors and tensors which
are often described by several numbers that signify the magnitude, direction
and other parameters.
We all were introduced to scalars through
physics though fundamentally being scalar is a mathematical property.
In mathematics, a scalar is stated formally
to be something that remains unchanged by co ordinate system transformations.
The beauty is that whether it is applied to
physics or pure mathematics, the essential concept of the scalars remains the
same.
A scalar will be preserved in a
mathematical operation such as rotation or reflection or in mathematical
physical operations such as space-time transformations or Lorentz
transformations.
I used the word Lorentz transformations
even though some nights back I promised to stick to classical mechanics of
Lagrange (and Hamilton) and not to fiddle even lightly with relativistic
physics; But then I may be forgiven for small transgressions even though I say
this to myself.
One of the simplest examples of scalar
quantity is the temperature.
This is so because a temperature at a given
point is just a single number followed by unit.
We shall continue with our stories on scalars
and vectors in the nights to come.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
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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