Editing Multiplicative inverse (section)

==Reciprocals of irrational numbers==
Every real or complex number excluding zero has a reciprocal, and reciprocals of certain [[irrational number]]s can have important special properties. Examples include the reciprocal of ''[[e (mathematical constant)|e]]'' (≈&nbsp;0.367879) and the [[Golden ratio#Golden ratio conjugate and powers|golden ratio's reciprocal]] (≈&nbsp;0.618034). The first reciprocal is special because no other positive number can produce a lower number when put to the power of itself; <math>f(1/e)</math> is the [[global optimum|global minimum]] of <math>f(x)=x^x</math>. The second number is the only positive number that is equal to its reciprocal plus one:<math>\varphi = 1/\varphi + 1</math>. Its [[additive inverse]] is the only negative number that is equal to its reciprocal minus one:<math>-\varphi = -1/\varphi - 1</math>.

The function <math display="inline">f(n) = n + \sqrt{ n^2+1 }, n \in \N, n>0</math> gives an infinite number of irrational numbers that differ with their reciprocal by an integer. For example, <math>f(2)</math> is the irrational <math>2+\sqrt 5</math>. Its reciprocal <math>1 / (2 + \sqrt 5)</math> is <math>-2 + \sqrt 5</math>, exactly <math>4</math> 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 [[simple continued fraction]]s, which have a number of remarkable properties relating to the representation of (both rational and) irrational numbers.