whose polar coordinates are magnitude and angle is written The angle may be stated in degrees with an implied conversion from degrees to radians. For example would be assumed to be which is the vector or the number Definition A real-valued sinusoid with constant amplitude, frequency, and phase has the form: where only parameter is time-variant. The inclusion of an imaginary component: gives it, in accordance with Euler's formula, the factoring property described in the lede paragraph: whose real part is the original sinusoid. The benefit of the complex representation is that linear operations with other complex representations produces a complex result whose real part reflects the same linear operations with the real parts of the other complex sinusoids. Furthermore, all the mathematics can be done with just the phasors and the common factor is reinserted prior to the real part of the result. The function is called the analytic representation of Figure 2 depicts it as a rotating vector in the complex plane. It is sometimes convenient to refer to the entire function as a phasor, as we do in the next section. But the term phasor usually implies just the static complex number Arithmetic Multiplication by a constant (scalar) Multiplication of the phasor by a complex constant, , produces another phasor. That means its only effect is to change the amplitude and phase of the underlying sinusoid: In electronics, would represent an impedance, which is independent of time. In particular it is not the shorthand notation for another phasor. Multiplying a phasor current by an impedance produces a phasor voltage. But the product of two phasors (or squaring a phasor) would represent the product of two sinusoids, which is a non-linear operation that produces new frequency components. Phasor notation can only represent systems with one frequency, such as a linear system stimulated by a sinusoid. Addition The sum of multiple phasors produces another phasor. That is because the sum of sinusoids with the same frequency is also a sinusoid with that frequency: where: and, if we take , then is: if with the signum function; if ; if . or, via the law of cosines on the complex plane (or the trigonometric identity for angle differences): where A key point is that A3 and θ3 do not depend on ω or t, which is what makes phasor notation possible. The time and frequency dependence can be suppressed and