3. Solving Equations with Complex Numbers

L3CALC

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All complex numbers can be expressed in the form of $z=x+iy$.

Equations can be made with these complex numbers, such as:

$x+3i=2-iy$

In complex numbers, real parts, equal to the real parts, and the imaginary numbers equal to the imaginary parts.

In the example above, the real part is the $x$ on LHS, and 2 on RHS. Since real equals real in complex numbers, $x=2$.

Same rule applies — imaginary equals imaginary.

$\therefore 3=y$

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Arithmetics of Complex Numbers

Adding & Subtracting:

Adding and subtracting complex numbers is very similar to basic algebra. $i$ acts as a variable, in very simple terms. For example:

$4i+6i=10i$

This is the same as when we use $x$ instead of $i$.

Another instance, if there is:

$(3+2i)+(-1+4i)$

… we can evaluate the answer by adding real by real, and imaginary by imaginary. $3+-1=2$, so the real part is $2$, $2i+4i=6i$, so the imaginary part is $6i$.

Therefore, the answer is $2+6i$.

Subtracting also works in the same way. You subtract real by real, and imaginary by imaginary.

These can also be plotted on the Argand diagram, since it is in the form $z=x+iy$.

Multiplying:

Multiplying also works in the same way as expanding brackets. For instance, when we have:

$(3+i)(2-3i)$

… we just expand the brackets like this:

$6-9i+2i-3i^2$

Since $i^2=-1$, we replace like so:

$6-7i+3$

$\therefore 9-7i$

Dividing

Dividing is a little bit different, as dividing by a non-existent number doesn’t work.

We do this by multiplying the conjugate (explaned in rationalization).