Imaginary numbers are numbers that are not present on your normal coordinate plane. In quadratics, when the discriminant is less than zero and one has to take a negative square root, we say the solution is imaginary. This means that the answer can be written in terms of the imaginary "i".

$$ \text {Remember this} : i^2 = -1 $$

Multiplying, rationalizing, and calculating i to the nth power should be very easy as long as you memorize the statement above.

Example 1:

$$ \text {Calculate} \space (1+3\text{i})(5-6\text{i}). $$

Multiplying this one out should be easy:

$$ (1+3\text{i})(5-6\text{i}) = 5-6\text{i}+15\text{i}-18\text{i}^2 = 5+9\text{i}-18(-1) = 23+9\text{i} $$

Note above that the answer was written in the form a+bi, with the imaginary number coming after the constant. This is how you should always vocalize your answers.

Rationalizing imaginary expressions just requires multiplying by the conjugate. Here is an example:

$$ \text{Simplify} \space \frac{3-\text{i}}{4+5\text{i}} $$

The basic point is that there should be no "i"'s on the denominator. Thus multiply by the conjugate of 4+5i, which will just be 4-5i. Note that all that is being done to find the conjugate is making the +5i a -5i. The vice versa of that also holds true.

<aside> 💡 The conjugate of a+bi is always a-bi and the conjugate of a-bi is always a+bi.

</aside>

$$ \frac{3-\text{i}}{4+5\text{i}}=\frac{(3-\text{i})(4-5\text{i})}{(4+5\text{i})(4-5\text{i})}=\frac{12-15\text{i}-4\text{i}+5\text{i}^2}{16-25\text{i}^2}=\frac{12-19\text{i}+5(-1)}{16-25(-1)}=\frac{7-19\text{i}}{41} $$

It can be seen above that when the "i" was squared, it became -1 and turned the denominator into a constant.

Finally, here comes the easiest part: finding i to the nth power:

$$ \text{Find}\space -\text{i}^{25} $$

$$ -(\text{i})^{24}(\text{i})^1=-(\text{i}^2)^{12}(\text{i})=-(-1)^{12}(\text{i})=-\text{i} $$

This may look complex but you are essentially finding the closest even number under 25 (which is 24), and splitting that once more to get the desired i^2. This should take under 5 seconds to do as long as you practice.

Here is a handy pattern chart to memorize to enhance your speed: