Tag Archives: natural numbers

Prime Consequences

Standard

Most of us are aware of the following consequence of Fundamental Theorem of Arithmetic:

There are infinitely many prime numbers.

The classic proof by Euclid is easy to follow. But I wanted to share the following two analytic equivalents (infinite series and infinite products) of the above purely arithmetical statement:

  • \displaystyle{\sum_{p}\frac{1}{p}}   diverges.

For proof, refer to this discussion: https://math.stackexchange.com/q/361308/214604

  • \displaystyle{\sum_{n=1}^\infty \frac{1}{n^{s}} = \prod_p\left(1-\frac{1}{p^s}\right)^{-1}}, where s is any complex number with \text{Re}(s)>1.

The outline of proof,   when s is a real number, has been discussed here: http://mathworld.wolfram.com/EulerProduct.html

Special Numbers: update

Standard

This post is a continuation of my earlier post: Special Numbers

Four (4)

This is the only euclidean space with properties different from other n-dimensional euclidean spaces. For example, there are smooth 4-manifolds which are homeomorphic but not diffeomorphic.  Put differently, for any dimension except four there is only one differentiable structure on the space underlying the Euclidean space of that dimension. For a discussion in this direction see this article by Liviu Nicolaescu. For other special properties of 4-dimesnions read Wikipedia article on 4-manifold.
Thanks to Dr. Ritwik Mukherjee for explaining this fact about four-space.

Special Numbers

Standard

Recently I realized the special properties of following numbers:

Zero (0)

It is the only number which is both real and purely imaginary at same time.

One (1)

It is sufficient to create all the counting numbers (a.k.a. natural numbers).

Two (2)

This is the maximum exponent, n, for which x^n + y^n=z^n has solution in natural numbers. This peculiar property leads to “Fermat’s Last Theorem”.

If you also have some special numbers in mind, please do share them below as comments.

Rationals…

Standard

A few days ago I noticed some fascinating properties of so called rational numbers.

Natural Bias:

Our definition of a number being rational or irrational is very much biased. We implicitly assume our numbers to be in decimals (base 10), and then define rational numbers as those numbers which have terminating or recurring decimal representation.

But it is interesting to note that, for example, √5 is irrational in base-10 (non-terminating, non-repeating decimal representation) but if we consider “golden-ratio base“, √5 = 10.1, has terminated representation, just like rational number!!

Ability to complete themselves:

When we construct numbers following Peano’s Axioms we can “easily” create (set of) natural numbers (\mathbb{N}), and from them integers (\mathbb{Z}) and rational numbers (\mathbb{Q}). Notice that \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}.

But it is comparatively difficult to create real numbers (\mathbb{R}) from rational numbers (\mathbb{Q}) although still we want to create a set from its subset. Notice that unlike previous cases, to create \mathbb{R} we will first have to create so-called irrational numbers (\overline{\mathbb{Q}}) from \mathbb{Q}. The challenge of creating the complementary set (\overline{\mathbb{Q}}) of a given set (\mathbb{Q}) using the given set (\mathbb{Q}) itself makes it difficult to create \mathbb{R} from \mathbb{Q} . We overcome this difficulty by using specialized techniques like Dedekind cut or Cauchy sequences (the process is called “completion of rational numbers”).

 

Happy Birthday Ramanujam

Standard

Today is 128th birthday of Srinivasa Ramanujam Iyengar.

[Please note that Britishers gave wrong English spelling for his name, they called him “Ramanujan”,  but it should be “Ramanujam” which means  “The younger brother of Lord Rama”]

r

John Littlewood said (as recalled by G. H. Hardy (1921), in “Obituary Notices: Srinivasa Ramanujan”, Proceedings of the London Mathematical Society 19, p. lvii) :

Every positive integer is one of Ramanujan’s personal friends.

So, let’s talk about numbers on his birthday.

128 is 7th power of 2 and leads to 4th Mersenne Prime (2^7 - 1=127 is a prime)

128 is the largest number which is not the sum of distinct squares. [https://oeis.org/A001422]

and here is my present for him:

RAMANUJAM’S MAGIC SQUARE

magic

When I was in High School, I learned making such “Special Date Magic Squares” form P.K. Srinivasan’s (1924-2005) book regarding Ramanujam’s work. [see: https://nrich.maths.org/1380 ]

Magic sum of this square is 69. Thus all rows, columns and diagonals add up to the same total of 69 and today’s date has been placed in first row.

69 is a value of n where n^2 and n^3 together contain each digit once. Since, 69^2=4761 and 69^3 = 328509

http://www2.stetson.edu/~efriedma/numbers.html]

Ramanujam also studied “Partition Function”, this function gives the number of ways of writing the integer n as a sum of positive integers, where the order of addends is not considered significant. Observe that, in the magic square above we created 10 distinct 4-partitions of 69. (though 2376 distinct 4-partitions of 69 are possible!!)

Also, the total sum of numbers in our magic square is 69\cdot 4 = 276 = 1^5 + 2^5 + 3^5

Once more, Happy Birthday Ramanujam!

 

Let’s construct Natural Numbers…

Standard

Everybody is introduced to mathematics by “remembering” natural numbers. Today we will see, “How to construct natural numbers”.

Without much delay let’s construct “Natural Numbers”, as in “plane geometry” (you must have studied it in high school), we define “Euclid’s Axioms”, here we will define “Peano’s Axioms”, the axioms are:

  • Axiom One: 1 is a natural number.
  • Axiom Two: Every natural number has a successor.
  • Axiom Three: 1 is not the successor of any natural number.
  • Axiom Four: If the successor of x equals the successor of y, then x equals y.
  • Axiom Five (Principle of mathematical induction): If a statement is true of 1, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.

Some texts, which swap meaning of natural numbers and whole numbers, consider natural numbers to include the number zero! In that case, we will start our construction with zero instead of one (see: http://mathworld.wolfram.com/PeanosAxioms.html). This supports a very criticized philosophical idea from Indian civilization:

Everything started from nothing.

Once we are able to construct natural numbers, we can construct all kinds of number systems (with motivation of solving certain algebraic equations):

  • whole numbers: add ‘0’ to the list of natural numbers (I don’t know why, but some mathematicians use whole numbers and natural numbers interchangeable)
  • integers: add negative natural numbers to the list of whole numbers
  • rational numbers: make fractions from integers, where denominator is not allowed to be zero.
  • irrational numbers: not all numbers can be represented as fractions
  • real numbers: define “Dedekind cut”, and construct real numbers out of rational numbers (lengthy task!)
  • complex numbers: any polynomial equation must have a solution

There is another classification of “real numbers” , called “algebraic numbers” and “transcendental numbers“, but it is altogether a different topic of discussion which I will discuss in some other blog post.