Reciprocals of Irrational Numbers
Every number excluding zero has a reciprocal, and reciprocals of certain irrational numbers often can prove useful for reasons linked to the irrational number in question. Examples of this are the reciprocal of e which is special because no other positive number can produce a lower number when put to the power of itself, and the golden ratio's reciprocal which, being roughly 0.6180339887, is exactly one less than the golden ratio and in turn illustrates the uniqueness of the number.
There are an infinite number of irrational reciprocal pairs that differ by an integer (giving the curious effect that the pairs share their infinite mantissa). These pairs can be found by simplifying n+√(n2+1) for any integer n, and taking the reciprocal.
Read more about this topic: Multiplicative Inverse
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