In mathematics, signed frequency (negative and positive frequency) expands upon the concept of frequency, from just an absolute value representing how often some repeating event occurs, to also have a positive or negative sign representing one of two opposing orientations for occurrences of those events. The following examples help illustrate the concept: For a rotating object, the absolute value of its frequency of rotation indicates how many rotations the object completes per unit of time, while the sign could indicate whether it is rotating clockwise or counterclockwise. Mathematically speaking, the vector has a positive frequency of +1 radian per unit of time and rotates counterclockwise around the unit circle, while the vector has a negative frequency of -1 radian per unit of time, which rotates clockwise instead. For a harmonic oscillator such as a pendulum, the absolute value of its frequency indicates how many times it swings back and forth per unit of time, while the sign could indicate in which of the two opposite directions it started moving. For a periodic function represented in a Cartesian coordinate system, the absolute value of its frequency indicates how often in its domain it repeats its values, while changing the sign of its frequency could represent a reflection around its y-axis. Sinusoids Let be a nonnegative angular frequency with units of radians per unit of time and let be a phase in radians. A function has slope When used as the argument of a sinusoid, can represent a negative frequency. Because cosine is an even function, the negative frequency sinusoid is indistinguishable from the positive frequency sinusoid Similarly, because sine is an odd function, the negative frequency sinusoid is indistinguishable from the positive frequency sinusoid or Thus any sinusoid can be represented in terms of positive frequencies only. The sign of the underlying phase slope is ambiguous. Because leads by radians (or cycle) for positive frequencies and lags by the same amount for negative frequencies, the ambiguity about the phase slope is resolved simply by observing a cosine and sine operator simultaneously and seeing which one leads the other. The sign of is also preserved in the complex-valued function: since and can be separately observed and compared. A common interpretation is that is a simpler function than either of its components, because it simplifies multiplicative trigonometric calculations, which leads to its formal description as the analytic representation of . The sum of