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Short notes on Complex Numbers Arithmetic.
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ArithmeticBefore proceeding in this section let me first say that I’m assuming that you’ve se
complex numbers at some point before and most of what is in this section is goin
you. I am also going to be introducing subtraction and division in a way that you p
seen prior to this point, but the results will be the same and aren’t important for th
sections of this document.
In the previous section we defined addition and multiplication of complex numbe
i2 = −1 is a consequence of how we defined multiplication. However, in practice
multiply complex numbers using the definition. In practice we tend to just multiply
numbers much like they were polynomials and then make use of the fact that we
i2 = −1.
Just so we can say that we’ve worked an example let’s do a quick addition and m
complex numbers.
Example 1 Compute each of the following.
(a) (58 − i) + (2 − 17i)
(b) (6 + 3i) (10 + 8i)
(c) (4 + 2i) (4 − 2i)
Hide Solution #
As noted above, I’m assuming that this is a review for you and so won’t be goin
here.
(a) (58 − i) + (2 − 17i) = 58 − i + 2 − 17i = 60 − 18i
(b) (6 + 3i) (10 + 8i) = 60 + 48i + 30i + 24i2 = 60 + 78i + 24 (−1) =
(c) (4 + 2i) (4 − 2i) = 16 − 8i + 8i − 4i2 = 16 + 4 = 20
It is important to recall that sometimes when adding or multiplying two complex nu
might be a real number as shown in the third part of the previous example!
The third part of the previous example also gives a nice property about complex n
(a + bi) (a − bi) = a2 + b2
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