Real number

A real number is any number that can be used to represent a physical quantity.

Intuitively, real numbers are any number that can be found between two integers, such as \(0,\) \(1,\) \(-1,\) \(\frac{3}{2},\) \(\frac{-7}{2},\) \(\pi,\) \(e\), \(100 \cdot \sqrt{2},\) and so on. The set of real numbers is written \(\mathbb R.\) You can think of \(\mathbb R\) as \(\mathbb Q\) extended to include the irrational numbers like \(\pi\) and \(e\) which can be found between rational numbers but which cannot be completely written out in decimal notation.

Definitions of the real numbers

The most commonly used definitions of the real numbers are constructions as extensions of the rational numbers, which involve either Cauchy sequences or Dedekind cuts.

Cauchy sequences

Broadly speaking, a Cauchy sequence is a sequence where as the sequence goes on, all the elements past that point get closer and closer together. In the real numbers, every Cauchy sequence converges to a real number. However, in the set of rational numbers, not all Cauchy sequences converge to a rational number. In the set of rationals, a Cauchy sequence which does not converge to a rational number cannot really be said to “converge” at all: the set of rationals is “missing some of the points” that would be required to make every Cauchy sequence converge.

For example, the sequence of fractions of consecutive Fibonacci numbers \(1/1, ²⁄₁, ³⁄₂, ⁵⁄₃, ⁸⁄₅, \ldots\) gets closer and closer to \(\frac{1 + \sqrt{5}}{2}\), but cannot be said to converge to that number because it is not in the set of rational numbers.

For each of these non-convergent Cauchy sequences, we define a new irrational number to “fill in the gap”, and for the Cauchy sequences that do converge, we define a real number equal to that rational number.

Dedekind cuts

A Dedekind cut of a totally ordered set is a partition of that set into two sets so that every element in the first set is less than every element in the second set, and the second set has no smallest element. The latter restriction requires that the set also be a perfect set (have no isolated points), in the sense used in topology.

In the real numbers, such a partition will always have the first set having a greatest element, which is known as the least-upper-bound property. However, in the rational numbers, we might come across a partition where the first set does not have such an element.

For example, define a Dedekind cut \((A, B)\) of the rational numbers such that \(B = \{x \in \mathbb{Q} \ | \ x > 0 \wedge x^2 > 2\}\) and \(A\) is the complement of \(B\). In plainer language, \(B\) consists of all the numbers greater than \(\sqrt{2}\), but because \(\sqrt{2}\) doesn’t exist in the space of rational numbers, we can’t use that to formulate our definition. Obviously every element of \(A\) is less than every element of \(B\), but \(A\) has no greatest element either, because we can create a sequence of numbers in \(A\) that gets bigger and bigger (as it approaches \(\sqrt{2}\)) but never stops at a maximum value.

For each of these “strict cuts” where neither set has a “boundary element”, we define a new irrational number to “fill in the gap”, just like with the Cauchy sequences. For the Dedekind cuts where one of the sets does have a least or greatest element, we define a real number equal to that rational number.

This definition has the advantage that each real number is represented by a unique Dedekind cut, unlike the Cauchy sequences where multiple sequences can converge to the same number.