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The term amplitude refers to the height of a wave, e.g., for a wave described by the sine function, the distance from the function's mean value to its peak value, which is the number multiplied by the sine function. The term is used in descriptions of sound waves, of the waves carried by electrical circuits and can be used for electromagnetic radiation though often other measures are used that carry the same information. For a basic-shaped (sine) wave:
f(t) = m sin t
The term complex amplitude is for a variant that is a complex number whose magnitude is the (non-complex) amplitude. The wave's phase can also be extracted (by other means); complex numbers consist of pairs of scalars, sufficient information for two separate such values. With the complex amplitudes of the components of some wave, the exact wave can be (re)constructed whereas knowing only their non-complex amplitudes is insufficient.
The peak-to-peak amplitude is the difference between the high point and the low point (e.g., of a sine function over its period), and the RMS amplitude is the RMS of the function values (e.g., of the sine function over its period). The term peak amplitude is used for the "ordinary" amplitude to distinguish it.
Amplitude is a commonly-used term within quantum mechanics; it is literally the an amplitude (as above) of the waves of wave mechanics, which is one version of quantum mechanics. This wave-amplitude is the square-root of the probability of some phenomenon occurring, and in some cited calculations you see the term amplitude defined as the square root of the probability, even in text that makes no mention of the quantum mechanical waves. This amplitude is of high interest because the quantum-mechanics calculation of a situation's future probabilities depend not merely on a current probability but on which of the probability's two possible (complex) square roots corresponds to the associated wave's amplitude.