Tag Archives: rational numbers

Huntington’s Red-Blue Set

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While reading Lillian Lieber’s book on infinity, I came across an astonishing example of infinite set (on pp. 207). Let’s call the property of existence of a rational number between given two rational number to be “beauty” (a random word introduced by me to make arguments clearer).

The set of rational numbers between 0 and 1 are arranged in ascending order of magnitude, and all of them are coloured blue. This is clearly a beautiful set. Then another another set of rational numbers between 0 and 1 is taken and arrange in ascending order of magnitude, but all of them are coloured red. This is also a beautiful set. Now, put these two sets together in such a way that each blue number is immediately followed by the corresponding red number. For example, 1/2 is immediately followed by 1/2 etc.  It appears that if we interlace two beautiful sets, the resulting set should be even more beautiful. But since each blue number has an immediate successor, namely the corresponding red number, so that between these two we can’t find even a single other rational number, red or blue, the resulting set is NOT beautiful.

The set created above is called Huntington’s Red-Blue set. It is an ingenious invention, where two beautiful sets combined together lead to loss of beauty. For more details, read the original paper:

Huntington, Edward V. “The Continuum as A Type of Order: An Exposition of the Modern Theory.” Annals of Mathematics, Second Series, 7, no. 1 (1905): 15-43. doi:10.2307/1967192.

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”).