# Perspectives in Mathematics and Art

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While browsing through NUS’s Undergraduate Research Opportunities Programme in Science,  I discovered an interesting project report published in year 2001. The report is titled “Perspectives in Mathematics and Art” by Kevin Heng Ser Guan (then an first year undergraduate, now a physics professor, I guess!). # Special Numbers

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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.

# Primes: popular and lonely

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ulam While doodling in class, I made a 10 x 10 grid and filled it with numbers from 1 to 100. The motivations behind 10 x 10 grid was human bias towards the number 10. Then inspired by Ulam Spiral, I started creating paths (allowing diagonal, horizontal and vertical moves) starting from the smallest number. Following paths emerged:

• 2→ 3 →13 → 23
• 2 → 11
• 7 → 17
• 19 → 29
• 31 → 41
• 37 → 47
• 43 → 53
• 61 → 71
• 73 → 83
• 79 → 89

So, longest path is of length 4 and others are of length 2.

The number 2 is special one here, since it leads to two paths. I will call such primes, with more than one paths, popular primes.

Now, 5, 59, 67 and 97 don’t have any prime number neighbour. I will call such primes, with no neighbour, lonely primes.

I hope to create other $b \times b$ grids filled with 1 to $b^2$ natural numbers written in base $b$. Then will try to identify such lonely and popular primes.

# 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”).