We will now further develop our ideas of the complex plan for investigating what happens when we add and multiply complex numbers together. In order to do we will have to apply Euler's Formula together with some rules of indices.
ADDITION AND MULTIPLICATION IN THE COMPLEX PLANE
May want to check out:
THE COMPLEX PLANE
PROOF USING TRIGONOMETRY IDENTITIES
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VIDEO LECTURE
MAIN CONCEPTS
In the complex plane, adding two complex numbers together is the vector sum of the a and b. When multiplying two complex numbers together, we multiply the magnitudes and add the arguments. Get the Printer Friendly Version COMMENTS
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LESSON
In the previous section, we ended up with Euler’s Formula,
 which we describe as one of the more important formula, well if any formula can be said as unimportant, in mathematics. From Euler’s Formula, it allows us to re-represent, or write the complex number in another form.
to which we enforce again that Drawing from Euler’s Formula and the laws of indices we can find another way to rewrite the complex number
This becomes interesting because notice that the magnitude of the complex number Since we have developed ideas of the complex plane, we can now geometrically represent the addition of two complex numbers. Suppose we have the complex numbers
where
This shows that the product of
which should be fairly straight forward. In this chapter, we have learnt three different ways to represent a complex number them being the standard form, polar form and Euler’s form (though I don’t think that is what it is called) We have also found geometrical meanings of the addition and multiplication of complex numbers in the complex plane. All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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, the magnitude of 
and the argument is 
.
, which is
is only dependent on x as 
.
. The addition of the two complex numbers can be thought of as vector addition. The values of a are along the real axis and the values of b are along the imaginary axis. 
While geometrical representation of addition is fairly obvious, the process of multiplying two complex numbers, at least in standard form, does not easily give us a geometrical representation. In order find the meaning of multiplying two complex numbers in the complex plane, we first convert them using Euler’s formula as previously shown.
are the magnitudes and argument respectively. Then by multiplying both together applying the laws of indices, we have
is found by multiplying the magnitudes and adding the arguments, as illustrated below.
We conclude by using Euler’s Formula to rewrite the reciprocal of a complex number.
