May 19, 2017 Friday

Bedtime Story 

Each Russian Doll Empowers Mathematics Incrementally

I shall continue quoting a little bit more on the number theory from ‘Prime Obsession’.

“Each family of members, each Russian doll, is denoted by a hollow letter (technically this kind of typeface style is either called double-struck or blackboard bold.)

ℕ is the family of all natural numbers; ℤ is the integers; ℚ the rationals; and ℝ the reals.

Each family is, in a sense, contained in the next one.

Each expands the power of math.

It lets us do something we couldn’t do with the previous doll.

For example, ℤ allows us to subtract any two numbers and get an answer, which we couldn’t do with ℕ (7 – 12 =?).

Likewise, ℚ let us divide by any number (except zero) and get an answer, which we couldn’t do in ℤ (-7 ÷ 12).

And ℝ opens the door to analysis, the mathematics of limits, because any convergent sequence of numbers in ℝ has a limit in ℝ, a thing not true in ℚ.

Consider this simple series:

 1⁄2 + 1⁄4  + 1⁄9 + 1⁄16…

This series is made up elements from the doll ℚ yet its solution is not in ℚ but in ℝ.

It was in 1735 that Euler found out that this series converges exactly to  𝜋²⁄6.

There are other categories of numbers within, or cutting across, the ℕ, ℤ, ℚ, ℝ, and ℂ schema. 

Prime numbers, to take an obvious case, are a subset of ℕ.

They are very occasionally referred to collectively as ℙ.

There is a very important subset of ℂ called the algebraic numbers, sometimes also given a hollow letter of its own, 𝔸.

An algebraic number is a number that is a zero of some polynomial with coefficients all in ℤ, for example

2x⁷ – 11x⁶ – 4x⁵ + 19x³ - 35x² + 8x – 3

Among the real numbers, every rational number–and, therefore, every integer and natural number, is algebraic; 39541⁄24565 is a zero of 24565x – 39541 (or a solution of 24565x – 39541 = 0, if you prefer the language of equations and solutions to the language of functions and zero). 

An irrational number might or might not be algebraic.

Those that are not are called transcendental.

Both 𝑒 and 𝜋 are transcendental, as proved by, respectively, Hermite in 1873 and Ferdinand von Lindemann in 1882.”

This is a most wonderful account of numbers that I have ever come across till date and hence thought of sharing with you.

It also forms a part of our story line too as we were talking about one-to-one correspondence between different types of infinities such as natural number, integers and real numbers.

(They are all infinite sets).

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:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg