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What does the lowercase j actually mean in expressions of impedance such as 4 j7 and 45 j83 This was briefly discussed earlier in this book, but what is this thing, really Mathematicians use the lowercase letter i to represent j. (Mathematicians and physicists/engineers often differ in notation as well as in philosophy.) This imaginary number is the square root of 1. It is the number that, when multiplied by itself, gives 1. So i j, and j j 1. The entire set of imaginary numbers derives from this single unit. The square of an imaginary number is negative always. No real number has this property. Whether a real number is positive, zero, or negative, its square can never be negative never. The notion of j (or i, if you re a mathematician) came about simply because some mathematicians wondered what the square root of 1 would behave like, if there were such a thing. So the mathematicians imagined the existence of this animal, and found that it had certain properties. Eventually, the number i was granted a place among the 264

dB = 10 log (Q/P)

At the same time, the business world found more and more applications for data, and the need to move that data from place to place became a major contributor to the growth in data traffic on the world s telephone networks In those heady, early days, data did not represent a problem for the bandwidth-limited local loop The digital information created by a computer and intended for transmission through the telephone network was received by a modem, converted into a modulated analog waveform that fell within the 4-KHz voice band, and fed to the network without incident As we mentioned in a previous chapter, the modem s job was (and is) quite simple Invoke the Wizard of Oz protocol: when a computer is doing the talking, the modem must make the network think it is talking to a telephone.

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Complex numbers 265 realm of numbers. Mathematically, it s as real as the real numbers. But the original term imaginary stuck, so that this number carries with it a mysterious aura. It s not important, in this context, to debate the reality of the abstract, but to reassure you that imaginary numbers are not particularly special, and are not intended or reserved for just a few eccentric geniuses. Imaginary numbers are as real as the real ones. And just as unreal, in that neither kind are concrete; you can hold neither type of number in your hand, nor eat them, nor throw them in a wastebasket. The unit imaginary number j can be multiplied by any real number, getting an infinitude of imaginary numbers forming an imaginary number line (Fig. 15-1). This is a duplicate of the real number line you learned about in school. It must be at a right angle to the real number line when you think of real and imaginary numbers at the same time.

As an example, suppose a speaker emits 1 W of sound, and then you turn up the volume so that it emits 2 W of sound power. Then P = 1 and Q = 2, and dB = 10 log (2/1) = 10 log 2 = 10 0.3 = 3 dB. This is the minimum detectable level of volume change if you aren t expecting it: doubling of the actual sound power! If you turn the volume level back down again, then P/Q = 1/2 = 0.5, and you can calculate dB = 10 log 0.5 = 10 0.3 = 3 dB. A gain or loss of 10 dB (that is, a change of +10 dB or 10 dB, often shortened to 10 dB) represents a 10-fold increase or decrease in sound power. A change of 20 dB represents a 100-fold increase or decrease in sound power. It is not unusual to encounter sounds that vary in intensity over ranges of 60 dB, which represents a 1,000,000-fold increase or decrease in sound power!

When you add a real number and an imaginary number, such as 4 j7 or 45 j83, you have a complex number. This term doesn t mean complicated; it would better be called composite. But again, the original name stuck, even if it wasn t the best possible thing to call it. Real numbers are one dimensional. They can be depicted on a line. Imaginary numbers are also one dimensional for the same reason. But complex numbers need two dimensions to be completely defined.

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