• shailesh_k

But what is a Real Number?

The genesis of real numbers does not lie so much in modern mathematics as it does in the picture of a line(a real number line). The Greeks and the Indians were at least some of the civilization mortals who in their time attained some insight into the nature of the numbers they encountered. A classic case is that of √2; that is the number which when squared gives 2.

A Constructive approach


Though the modern(not quite) axiomatic approach using the Peano axioms and further defining operations of addition & multiplication gives a neat construction of the natural numbers and the integers; such numbers were always known to us and had been in good use without the knowledge of such axioms. Nevertheless, the Peano axioms were a check of the consistency and completeness of the natural arithmetic and have presented a systematic construction of the numbers treating them merely as any objects. Also, the knowledge of rational numbers was existent but the idea of what real numbers are wasn’t on a firm footing till mathematicians such as Cantor and Dedekind. With ample examples of numbers such as √2, e, π and enough irrational roots of polynomials it was clear that the real number line constituted of more clutter than has been known of.


Richard Dedekind came up with a constructive way to concertize the meaning of what real numbers are and he did it in a way that made real numbers into just any objects derived from the known set of rational numbers(we denote it by ). Here, let us take a look at the idea of the construction with an example:


Consider the sets

L={ x ∈ | x²<2} &

R={ x ∈ | x²>2}.


The number √2 will be denoted by (L, R) known as a cut; a Dedekind cut which are certain subsets of ℚ. √2 could well have been defined alone by L or R.


In fact members of real numbers; ℝ, are certain subsets of called cuts. A cut is by definition, any set α ⊂ with the following properties:

(i) α is not empty & α≠ℚ.

(ii) If p ∈ α, q ∈ , and q<p, then q ∈ α.

(iii) If p ∈ α, then p<r for some r∈α.


Remarks: (iii) implies that α has no largest number. (ii) implies that: — If p ∈ α and q ∉ α then p<q & — if r ∉ α and r<s then s ∉ α.


Using this definition of a cut we can define a real number as α and thus define order properties and algebraic operations on it. The most important property that is then proved is the least upper bound property of real numbers, i.e. Let S ⊆ be a nonempty set that is bounded above. Then S has a least upper bound in ℝ. Note that this property is not in , for example, the set L={ x ∈ | x²<2} has no least upper bound in ℚ, but it has a least upper bound in ℝ.


So, what was supposed to be a number has now due to the nature of the construction become a set but that does not shy away from the fact that it can still be treated like a number as it can be proved using the definition of a cut that our arithmetic still holds well with the real numbers in place.


RESOURCES

1. Principles of Mathematical Analysis, Walter Rudin.

2. A Course of Pure Mathematics, G H Hardy.

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