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sỉnce ở = ¡2 = —1. Therefore all the numbers that now belong in the rules (22.1)
have this mathematical form.
NÑow you say, “This can go on foreverl We have defined powers of imaginaries
and all the rest, and when we are all fñnished, somebody else will come along with
another equation which cannot be solved, like øŠ + 3z2 = —2. Then we have to
generalize all over again!” But it turns out that œU#th thás one more inueniion, just
the square root of —1, cuer algebraic cquation can be solued! 'This 1s a fantastic
fact, which we must leave to the Mathematics Department to prove. The proofs
are very beautiful and very interesting, but certainly not self-evident. In fact,
the most obvious supposition is that we are goïing to have to invent again and
again and again. But the greatest miracle of all is that we do not. 'Phis is the
last invention. After this invention of complex numbers, we fnd that the rules
still work with complex numbers, and we are fñnished inventing new things. We
can fnd the complex power of any complex number, we can solve any equation
that is written algebraically, in terms of a ñnite number of those symbols. We
do not fnd any new numbers. 'Phe square root oŸ ¿, for instance, has a definite
result, it is not something new; and ?' is something. We will điscuss that now.
W© have already discussed multiplication, and addition is also easy; if we add
©wo cormplex numbers, (p + 7g) + (r + 7s), the answer is (p + r) + ¿(q + s). Now
we can add and multiply complex numbers. But the real problem, of course, 1s
to compute cơomplÌez pouers oƑ complez numnbers. It turns out that the problem
1s actually no more dificult than computing complex powers of real numbers. So
let us concentrate now on the problem of calculating 10 to a complex power, not
just an irrational power, but 10†?%), Of course, we must at all tỉmes use our
rules (22.1) and (22.2). Thus
10ŒT7%) = 10710!%, (22.5)
But 10” we already know how to compute, and we can always multiply anything
by anything else; therefore the problem is to compute only 107%. Let us call it
some complex number, # + 2. Problem: given s, ñnd z, ñnd . Now ïf
108 =z +,
--- Trang 401 ---
then the complex conjugate of this equation must also be true, so that
10”? =„— 1g.
(Thus we see that we can deduce a number oŸ things without actually computing
anything, by using our rules.) We deduce another interesting thing by multiplying
these together:
108108 = 10 =1= (+ i9)( — iu) = z? + Ÿ. (22.6)
'Thus if we fnd z, we have + also.
Now the problem is ho to compute 10 to an imaginary power. What guide
1s there? We may work over our rules until we can go no further, but here is a
reasonable guide: if we can compute it for any particular s, we can get it for all
the rest. IÝ we know 10”° for any one s and then we want it for twice that s, we
can square the number, and so on. But how can we fnd 10/5 for even one special
value of øs? 'To do so we shall make one additional assumption, which is not quite
in the category of all the other rules, but which leads to reasonable results and
permits us to make progress: when the power is small, we shall suppose that the
“law” 10° = 1+ 2.3025c is right, as c gets very small, not only for real c, bu for
comjplex as uell. Therefore, we begin with the supposition that this law is true
in general, and that tells us that 10° = 1 + 2.3095 - is, for s —> 0. So we assume
that 1Í s is very small, say one part in 1024, we have a rather good approximation
to 107%.
Now we make a table by which we can compute ø/! the Imaginary DOW©rS
of 10, that is, compute + and . It ¡is done as follows. "The first power we start
with is the 1/1024 power, which we presume is very nearly 1 + 2.3025//1024.
'Thus we start with
107/192 — 1.00000 -+ 0.0022486¿, (22.7)
and ïfƒ we keep multiplying the number by itself, we can get to a higher imaginary
power. In fact, we may just reverse the procedure we used in making our logarithm
table, and calculate the square, 4th power, 8th power, etc., oŸ (22.7), and thus
buïld up the values shown in Table 22-3. We notice an interesting thing, that
the ø numbers are positive at frst, but then swing negative. We shall look into
that a little bit more in a moment. But first we may be curious to fnd for what
number s the real part of 10/5 is zero. The -value would be 1, and so we would
have 103 = 1¿, or js = logjg7. As an example of how to use this table, just as
we calculated logs 2 before, let us now use Table 22-3 to fnd log+g¿.
--- Trang 402 ---
Table 22-3
Successive Squares of 10/1024 — 1 - 0.0022486¿
z/1024 1 1.00000 + 0.00225¿*
2/512 2 1.00000 + 0.00450¿
¿/256 4 0.99996 + 0.00900;
z/128 8 0.99984 + 0.01800;
¿/64 16 0.999386 + 0.03599;
7/32 32 0.99742 + 0.07193¿
z/16 64 0.98967 + 0.14349/
7/8 128 0.95885 + 0.28402;
¡/4 256 0.83872 + 0.54467:
¡z/2 512 0.40679 + 0.91365:
z/1 1024 | —0.66928 + 0.74332¡
* Should be 0.0022486;
Which of the numbers in Table 22-3 do we have to multiply together to
get a pure imaginary result? After a little trial and error, we discover that to
reduce z the most, it is best to multiply “512” by “128” 'This gives 0.13056 +
0.99159/. "Then we discover that we should multiply this by a number whose
imaginary part is about equal to the size of the real part we are trying to remove.
Thus we choose “64” whose 2-value is 0.14349, since that is closest to 0.13056.
This then gives —0.01308 + 1.00008/. Now we have overshot, and must đ¿uide
by 0.99996 + 0.009007. How do we do that? By changing the sign of ? and
multiplying by 0.99996 — 0.00900/ (which works if z2 + 2 = 1). Continuing in
this way, we fnd that the entire power to which 10 must be raised to glve ¿ 1s
(512 + 128 + 64 — 4— 2+ 0.20)/1024, or 698.20//1024. Tf we raise 10 to that
power, we can get ¿. Therefore logo ¿ = 0.68184:.
22-6 Imaginary exponents
To further investigate the subject of taking complex imaginary powers, let
us look at the powers of 10 taking swccess¿ue pouers, not doubling the power
each time, im order to follow Table 22-3 further and to see what happens to those
mỉnus signs. This is shown in Table 22-4, in which we take 10”, and just keep
--- Trang 403 ---
Table 22-4
Successive Powers of 107⁄8