Reciprocal Complex Numbers
Reciprocal
complex numbers are of the form a<+>ib,
where i equals the square root of negative one. This equals:
a <+> ib = aib = iab(a-ib)
a + ib
a2 + b2
= ab ( b + ia )
a2 + b2
= ( b <+> a ) ( b +
ia )
a b
Conversely,
a + ib = ab ( b <+> ia
)
a2 <+> b2
= ( b +
a ) ( b <+> ia
)
a
b
They follow
the usual rules of complex arithmetic, in reductions:
(a <+> ib) <+> (c <+>
id) =
(a<+>c) <+> i(b<+>d)
(a <+> ib) <-> (c <+>
id) =
(a<->c) <+> i(b<->d)
(a <+> ib) *
(c <+> id) = (ac<->bd)
<+> i(ad<+>bc)
1
/ (c <+> id) = (c/(c2<+>d2))
<-> i(d/(c2<+>d2))
The reciprocal
form of the cis function is sec(a)<+> i csc(a). Call this “sic(a)”;
these equations apply:
sic(a) = sec(a)<+> i csc(a)
= (1/cos(a))<+>
(1/(-isin(a))
= 1/ ( cos(a) -
isin(a) )
= (cos(a) +
isin(a))/(cos2(a)+ sin2(a))
= (
cos(a) + i sin(a) ) = cis(a)
So sic equals
cis, and has the same identities:
sic(a+b)
= sic(a) * sic(b)
sic(a-b)
= sic(a) / sic(b)
sic(Na) = sic(a)N
In
general a <+>
ib equals
ab (b + ia)
a2 + b2
=
ab sqrt(a2 + b2) (cos(q)
+ i sin(q))
a2 + b2
= ab (
cos(q) + i
sin(q) )
sqrt(a2+b2)
=
sqrt(a2<+>b2) * (
sec(q) <+> i csc(q)
)
-
where q is
the angle that a<+>ib makes with the positive
real axis.
From sic and
reciprocal-complex algebra we can derive these and other
reciprocal-trigonometric identities:
sec2(a)<+> csc2(a) = 1
sec(a+b) = sec(a)*sec(b) <-> csc(a)*csc(b)
csc(a+b) = sec(a)*csc(b) <+> csc(a)*sec(b)
sec(2a) = sec2(a) <-> csc2(a)
csc(2a) = ½ sec(a)*csc(a)
sec(3a) = sec3(a) <-> (1/3) sec(a)csc2(a)
csc(3a) = (1/3) sec2(a)*csc(a) <-> csc3(a)
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