Identity written as a power or quotient

Read the equation

Raising an ordinary real z to the first power, or dividing it by one, returns z. These alternate expressions do not add a new transformation. They remain identity operators inside the illustrative composition.

Terms & definitions

How it works

Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.

Assumptions & limits

Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.

Original context

f1z=z+1,  f2z=z−1,  f3z=z.

f4z=z,  f5z=1,  f6z=z,  f7z=1/z.

Here f4z=z1, f5z=1z, and f6z=z/1. The reciprocal requires a nonzero input. An ordered composite can nevertheless be defined at an initial zero when a preceding operator supplies a nonzero reciprocal input.

6.1 Proposition 1: the sequential loop is a constant map

For every finite real initial value in the ordinary interpretation,

f7∘f6∘f5∘f4∘f3∘f2∘f1z=1.

{f}_{4}\left(z\right)={z}^{1}