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Math 254 Review information on
arithmetic for complex numbers
Complex numbers are deﬁned in terms of an “imaginary” number
i
which
is a square root of

1. Of course no real number could have a negative
square, so
i
is a new kind of number. It turns out that using numbers of the
form
a
+
bi
(complex numbers) where
a
and
b
are real numbers, you can ﬁnd
roots for any polynomial, and in fact, any polynomial can be factored as a
product of a constant and terms of the form
x

r
, where the
r
’s are complex
numbers. Complex numbers are very useful in diﬀerential equations, so it is
worth reviewing some basic facts about them.
To add or subtract two complex numbers, one simply adds (or subtracts)
the corresponding real and imaginary parts.
a
+
bi
+
c
+
di
= (
a
+
c
) + (
b
+
d
)
i,
(
a
+
bi
)

(
c
+
di
) = (
a

c
) + (
b

d
)
i.
Multiplication is also straightforward
(
a
+
bi
)(
c
+
di
) =
ac
+
adi
+
bci
+
bdi
2
=
ac
+
adi
+
bci

bd
= (
ac

bd
)+(
ad
+
bc
)
i.
Division is a bit trickier.
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