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a /b = c /d You can multiply the dividend on the left-hand side of this equation (here, that s a) by the divisor on the right-hand side (d ), and get the same result as when you multiply the divisor on the left-hand side (b) by the dividend on the right-hand side (c) In formal terms, If a /b = c /d, then ad = bc This rule works in the reverse sense, too For any four integers a, b, c, and d, where b is not equal to 0 (written b 0) and d 0, If ad = bc, then a /b = c /d When a rule works in both logical directions, mathematicians use the expression if and only if and abbreviate it as iff Now we know that for any four integers a, b, c, and d, where b 0 and d 0, ad = bc iff a /b = c /d

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Suppose you see an expression where you have to divide repeatedly, with no parentheses telling you which division to do first Here is an example: 200/2/5/4 In a case like this, proceed from left to right Here, that means you should take 200 and divide it by 2, getting 100 Then divide 100 by 5, getting 20 Finally, divide 20 by 4, getting 5

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Radiation Resistance At fo and all of the odd harmonics, the antenna behaves like a series-resonant RLC circuit with a fairly low resistance At all even harmonics, the antenna acts like a parallel-resonant RLC circuit with a high resistance Does this confuse you There s no resistor in Fig 17-14! Where, you ask, does the resistance come from in the half-wave dipole The answer to this is rather esoteric, and it brings to light an interesting property that all antennas have It is called radiation resistance, and is a crucial factor in the design and construction of all RF antenna systems When electromagnetic energy is fed into an antenna, power is radiated into space in the form of radio waves This is a manifestation of true power, just as the dissipation of power in a pure resistance is a manifestation of true power Although there is no physical resistor in Fig 17-14, the radiation of radio waves is like power dissipation in a pure resistance In fact, if a half-wave dipole antenna were replaced with a 73- nonreactive resistor that could dissipate enough power without burning out, a radio transmitter connected to the opposite end of the line wouldn t know the difference (But a receiver would!) Problem 17-14 How many feet long is a quarter-wave section of transmission line at 705 MHz, if the velocity factor is 0800

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Here s a challenge!

Under what circumstances can we divide an integer a by an integer b, and get the same quotient (or ratio) as when we divide b by a Assume that a 0 and b 0

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Another signal that traders look for is the shape of the bands For example, if the upper and lower bands suddenly contract (squeeze together), this is a clue that volatility is low Don t be fooled by a quiet and low volatile market When Bollinger Bands are very narrow, it is a signal the market could suddenly explode in the opposite direction Put another way, periods of low volatility are often followed by periods of high volatility It works the other way too If the bands are expanding, it means that the market is extremely volatile There are no guaran-

We have two integers a and b, and we are told that a /b = b /a This equation is always true when a and b are the same In that case, we can substitute a for b and get a /a = a /a which simplifies to1 = 1 That s trivial! Now suppose that a and b are additive inverses Therefore, a = b Here, we can substitute a for b in the original equation and get a /( a) = ( a)/a which simplifies to 1 = 1 We know this because it is an application of one of those two facts we re supposed to have homogenized earlier in this chapter What s the verdict If we have two nonzero integers a and b whose absolute values are the same, meaning that they re either identical or are additive inverses of each other, then a/b = b/a In symbols, we can write If |a| = |b|, then a /b = b /a

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