Rationalizing the denominator when working with complex numbers means getting rid of complex numbers in the bottom part (denominator) of a fraction. This makes it easier to understand and work with the result. Let’s go through the steps together!

Why Should We Rationalize?

Having a complex number in the denominator, like ( a + bi ), can make things tricky. It’s usually better to have simpler denominators that are easier to deal with. By rationalizing, we change the expression so that the real and imaginary parts are clear and easy to see.

How to Rationalize

Let’s say we have a fraction with complex numbers, like this one:

$$\frac{3 + 4i}{1 - 2i}$$

In this case, the bottom part is ( 1 - 2i ). To rationalize it, we multiply both the top (numerator) and the bottom (denominator) by the conjugate of the bottom part.

The conjugate of a complex number ( a + bi ) is ( a - bi ). So, for ( 1 - 2i ), the conjugate is ( 1 + 2i ). Here’s how it looks when we multiply:

$$\frac{3 + 4i}{1 - 2i} \cdot \frac{1 + 2i}{1 + 2i} = \frac{(3 + 4i)(1 + 2i)}{(1 - 2i)(1 + 2i)}$$

Simplifying the Denominator

First, let’s make the bottom part easier:

$$(1 - 2i)(1 + 2i) = 1^2 - (2i)^2 = 1 - (-4) = 5$$

Now the bottom part is just the simple number 5, which is what we wanted!

Simplifying the Numerator

Next, let’s work on the top part:

$$(3 + 4i)(1 + 2i) = 3 \cdot 1 + 3 \cdot 2i + 4i \cdot 1 + 4i \cdot 2i = 3 + 6i + 4i - 8 = -5 + 10i$$

The Final Result

Putting everything together, we have:

$$\frac{-5 + 10i}{5}$$

Now, we can simplify this by dividing each part in the top by 5:

$$-1 + 2i$$

Conclusion

By rationalizing the denominator, we changed the original expression into ( -1 + 2i ), making it much cleaner and simpler.

To sum it up, rationalizing the denominator when dividing complex numbers helps us work with the math more easily and clearly. Always remember to use the conjugate of the complex number in the denominator, and your math will be a lot smoother!