Arithmetic Operations

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There are only 4 binary operations which we call “arithmetic operations”. These are:

• Subtractions (-)
• Multiplication (×)
• Division (÷)

Reading this fact, an obvious question is:

Why only four out of the infinitely many possible binary operations are said to be arithmetical?

Before presenting my attempt to answer this question, I would like to remind you that these are the operations you were taught when you learnt about numbers i.e. arithmetic.

In high school when $\sqrt{2}$ is introduced, we are told that real numbers are of two types: “rational” and “irrational”. Then in college when $\sqrt{-1}$ is introduced, we should be told that complex numbers are of two types: “algebraic” and “transcendental“.

As I have commented before, there are various number systems. And for each number system we have some valid arithmetical operations leading to a valid algebraic structure. So, only these 4 operations are entitled to be arithmetic operations because only these operations lead to valid algebraic numbers when operated on algebraic numbers.

Now this leads to another obvious question:

Why so much concerned about algebraic numbers?

To answer this question, we will have to look into the motivation for construction of various number systems like integers, rational, irrationals, complex numbers… The construction of these number systems has been motivated by our need to be able to solve polynomials of various degree (linear, quadratic, cubic…). And the Fundamental Theorem of Algebra says:

Every polynomial with rational coefficients and of degree n in variable $x$ has n solutions in  complex number system.

But, here is a catch. The number of complex numbers which can’t satisfy any polynomial (called transcendental numbers) is much more than the number of complex numbers which can satisfy a polynomial equation (called algebraic numbers). And we wish to find solutions of a polynomial equation (ie.e algebraic numbers) in terms of sum, difference, product, division or $m^{th}$ root of rational numbers (since coefficients were rational numbers). Therefore, sum, difference, product and division are only 4 possible arithmetic operations.

My previous statement may lead to a doubt that:

Why taking $m^{th}$ root isn’t an arithmetic operation?

This is because it isn’t a binary operation to start with, since we have fixed $m$. Also, taking $m^{th}$ root is allowed because of the multiplication property.

CAUTION: The reverse of $m^{th}$ root is multiplying a number with itself m times and it is obviously allowed. But, this doesn’t make the binary operation of taking exponents, $\alpha^{\beta}$ where $\alpha$ and $\beta$ are algebraic numbers, an arithmetic operation. For example, $2^{\sqrt{2}}$ is transcendental (called Gelfond–Schneider constant or Hilbert number) even though 2 and $\sqrt{2}$ are algebraic.

Special Numbers: update

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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.

Rationals…

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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”).

Indeed, 0.999…= 1

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There is a video by Vi Hart titled “Why Every Proof that .999… = 1 is Wrong”

It is true that few real numbers (with cardinality same as that of rational numbers) can’t have unique decimal representation, like, 0.493499999… = 0.4935000…. So, the point made in this video is that you can’t prove two different representations of a number to be equal.

But why so? As pointed out in my earlier post, this has something to do with the way we construct numbers. Such ambiguity in representation holds irrespective of representation system (binary or decimal)

In decimal representation of real numbers  we subdivide intervals into ten equal subintervals. Thus, given $x\in [0,1]$, if we subdivide $[0,1]$ into ten equal subintervals, then $x$ belongs to a subinterval $\left[\frac{b_1}{10}, \frac{b_1 + 1}{10}\right]$ for some integer $b_1$ in $\{0,1,\ldots ,9\}$. We obtain a sequence $\{b_n\}_n$ of integers with $0 \leq b_n \leq 9$ for all $n \in \mathbb{N}$ such that $x$ satisfies $\frac{b_1}{10}+ \frac{b_2}{10^2}+ \ldots + \frac{b_n}{10^n}\leq x \leq \frac{b_1}{10} + \frac{b_2}{10^2} + \ldots + \frac{b_n+1}{10^n}$

In this case we say that $x$ has a decimal representation given by $x = 0.b_1 b_2\ldots b_n \ldots$

The decimal representation of $x\in [0,1]$ is unique except when $x$  is a subdivision point at some stage, which is when $x=\frac{m}{10^n}$ for some $m,n \in \mathbb{N}; 1 \leq m \leq 10^n$. We may also assume that $m$ is not divisible by 10.

When $x$ is a subdivision point at the $n^{th}$ stage, one choice for $b_n$ corresponds to selecting the left subinterval, which causes all subsequent digits to be 9, and the other choice corresponds to selecting the right subinterval, which causes all subsequent digits to be 0.

For example, $x = \frac{1}{2} = \frac{5}{10} = 0.4999\ldots = 0.5000\ldots$ unlike $x = \frac{1}{3} = 0.333...$

A nice exposition is available on Wikipedia: https://en.wikipedia.org/wiki/0.999…

Happy Birthday Ramanujam

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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”] 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 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 $69^3 = 328509$

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!

Fibonacci, Chebyshev and Ramsey

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Pascal’s triangle has long and celebrated history, see this TedEd video:

What makes it more interesting is its relations with various domains of Mathematics (if you don’t understand the three relations discussed, refer Wikipedia). Here I will point few such connections:

1. Fibonacci Numbers: I believe that this is the most celebrated observation from pascal’s triangle. To see the jungle of Equations involved, visit http://www.maplesoft.com/applications/view.aspx?SID=3617&view=html

2. Chebyshev Polynomial: We can find coefficients of Chebyshev Polynomial using pascal’s triangle. See- http://mathpages.com/home/kmath304.htm

3. Ramsey Number: Upper bound of Ramsey Number can be found using Pascal’s triangle, for more details refer : https://plus.maths.org/content/friends-and-strangers