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The concept of negative and positive frequency can be as simple as a wheel rotating one way or the other way. A ''signed value'' of frequency can indicate both the rate and direction of rotation. The rate is expressed in units such as revolutions (aka ''cycles'') per second (hertz) or radian/second (where 1 cycle corresponds to 2π radians). ==Sinusoids== Let ω be a parameter > 0 with units of radians/sec. Then the angular function (angle vs time), -ωt+θ, has slope -ω, which is called a negative frequency. But when the function is used as the argument of a cosine operator, the result is indistinguishable from cos(ωt-θ). Similarly, sin(-ωt+θ) is indistinguishable from sin(ωt-θ+π). Thus all unique sinusoids can be represented in terms of just positive frequencies. The sign of the underlying phase slope is ambiguous. But the ambiguity is resolved when both the cosine and sine operators can be observed, because cos(ωt+θ) leads sin(ωt+θ) by ¼ cycle (= π/2 radians) when ω>0, and lags by ¼ cycle when ω<0. Similarly, a vector, (cos t, sin t), rotates counter-clockwise at 1 radian/sec, and completes a circle every 2π seconds, and the vector (cos -t, sin -t) rotates in the other direction. The sign of ω is also preserved in the complex-valued function: since R(t) and I(t) can be separately extracted and compared. Although clearly contains more information than either of its components, a common interpretation is that it's a simpler function, because: *It simplifies many important trigonometric calculations, which leads to its formal description as the analytic representation of 〔See Euler's_formula#Relationship_to_trigonometry and Phasor#Addition for examples of calculations simplified by the complex representation.〕 *A corollary of is:
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