30CHAPTER 2. COMPLEX NUMBERS012345. . .. . .-1-2-3-4-5Figure 2.6: “Number line” shows both positive and negative numbers.being able to represent something more complex like the distance and direction between two cities,or the amplitude and phase of an AC waveform. To represent these kinds of quantities, we needmultidimensional representations. In other words, we need a number line that can point in diﬀerentdirections, and that’s exactly what a vector is.• REVIEW:• A scalar number is the type of mathematical object that people are used to using in everydaylife: a one-dimensional quantity like temperature, length, weight, etc.• A complex number is a mathematical quantity representing two dimensions of magnitude anddirection.• A vector is a graphical representation of a complex number. It looks like an arrow, with astarting point, a tip, a deﬁnite length, and a deﬁnite direction. Sometimes the word phasoris used in electrical applications where the angle of the vector represents phase shift betweenwaveforms.2.2Vectors and AC waveformsOK, so how exactly can we represent AC quantities of voltage or current in the form of a vector? Thelength of the vector represents the magnitude (or amplitude) of the waveform, like this: (Figure 40,2.7)The greater the amplitude of the waveform, the greater the length of its corresponding vector.The angle of the vector, however, represents the phase shift in degrees between the waveform inquestion and another waveform acting as a “reference” in time. Usually, when the phase of awaveform in a circuit is expressed, it is referenced to the power supply voltage waveform (arbitrarilystated to be “at” 0o). Remember that phase is always a relative measurement between two waveformsrather than an absolute property. (Figure 40,2.8) (Figure 41,2.9)The greater the phase shift in degrees between two waveforms, the greater the angle diﬀerencebetween the corresponding vectors. Being a relative measurement, like voltage, phase shift (vectorangle) only has meaning in reference to some standard waveform. Generally this “reference” wave-form is the main AC power supply voltage in the circuit. If there is more than one AC voltage source,then one of those sources is arbitrarily chosen to be the phase reference for all other measurementsin the circuit.This concept of a reference point is not unlike that of the “ground” point in a circuit for thebeneﬁt of voltage reference. With a clearly deﬁned point in the circuit declared to be “ground,” itbecomes possible to talk about voltage “on” or “at” single points in a circuit, being understood thatthose voltages (always relative between two points) are referenced to “ground.” Correspondingly,with a clearly deﬁned point of reference for phase it becomes possible to speak of voltages and