structural:expansion
Reciprocals of irrational numbers

Every real or complex number excluding zero has a reciprocal, and reciprocals of certain irrational numbers can have important special properties. Examples include the reciprocal of e (≈ 0.367879) and the golden ratio's reciprocal (≈ 0.618034). The first reciprocal is special because no other positive number can produce a lower number when put to the power of itself; f (1 / e) {\displaystyle f(1/e)} is the global minimum of f (x) = x x {\displaystyle f(x)=x^{x}}. The second number is the only positive number that is equal to its reciprocal plus one: φ = 1 / φ + 1 {\displaystyle \varphi =1/\varphi +1}. Its additive inverse is the only negative number that is equal to its reciprocal minus one: − φ = − 1 / φ − 1 {\displaystyle -\varphi =-1/\varphi -1}.

The function f (n) = n + n 2 + 1, n ∈ N, n > 0 {\textstyle f(n)=n+{\sqrt {n^{2}+1}},n\in \mathbb {N},n>0} gives an infinite number of irrational numbers that differ with their reciprocal by an integer. For example, f (2) {\displaystyle f(2)} is the irrational 2 + 5 {\displaystyle 2+{\sqrt {5}}}. Its reciprocal 1 / (2 + 5) {\displaystyle 1/(2+{\sqrt {5}})} is − 2 + 5 {\displaystyle -2+{\sqrt {5}}}, exactly 4 {\displaystyle 4} less. Such irrational numbers share an evident property: they have the same fractional part as their reciprocal, since these numbers differ by an integer.

The reciprocal function plays an important role in continued fractions, which have a number of remarkable properties relating to the representation of (both rational and) irrational numbers.