I don't understand the underscore before 9 in "(7,_9)"? Did you mean (7,9)? or
(7,-9)? I'll assume the "_" was a typo and you meant (7,9).
Apparently you are using the ordered pair notation for complex numbers, which
is usually used at the beginning of a complex analysis course.
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The definition of addition is (a,b)+(c,d) = (a+c,b+d) which is equivalent to
(a+bi)+(c+di) = (a+c)+(b+d)i
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In ordered pair notation, the additive inverse of (7,9) is (-7,-9). That's
because (7,9) + (-7,-9) = (0,0), the additive identity.
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The definition of multiplication is (a,b)(c,d) = (ac-bd,ad+bc) which is
equivalent to
(a+bi)(c+di) = ac+adi+bci+bdi2 = ac+(ad+bc)i+bd(-1) = (ac-bd)+(ad+bc)i
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In order pair notation, we find the multiplicative inverse of (7,9) by
letting it be (u,v), then
(7,9)(u,v) = (7u-9v,7v+9u) = (1,0), the multiplicative identity. Then we have
the system of equations
which gives (u,v) = (7/130,-9/130)
Checking:
=
=
= 
= 
Edwin
