inverting

Inverting refers to turning something upside down, reversing its order, or reversing its sign. In mathematics, inverting denotes the process of finding an inverse: the reciprocal of x is 1/x; the inverse of a function f is f−1; the inverse of a matrix A is a matrix A−1 such that AA−1 = I. In geometry, geometric inversion about a circle maps points along lines through the circle’s center to points whose distances satisfy a constant product, and it can convert lines into circles and vice versa. In calculus and algebra, inverting a relation means solving for the dependent variable in terms of the independent variable.

In engineering and computer science, inverting a signal or a waveform means multiplying by -1, producing a phase-inverted signal; an inverting amplifier produces an output that is 180 degrees out of phase with the input. Bitwise inversion flips all bits in a binary word.

In music, inversion refers to changing the bass note of a chord without altering the other notes; this changes voicing but preserves chord quality. In chemistry, invert sugar is produced by hydrolyzing sucrose into glucose and fructose. In linguistics, inversion reverses typical word order, often for emphasis or in questions.

The term also appears in finance as inverted yield curves, and in imaging as color inversion, where colors are mapped to their opposites.

In summary, inverting encompasses mathematical, geometric, electronic, musical, chemical, linguistic, and financial uses, all centered on reversing, undoing, or reconfiguring a structure or signal.