Working with Complex Numbers

First, though, you'll probably be asked to demonstrate that you understand the definition of complex numbers.

• Solve 3 − 4i = x + yi for the values of x and y

---

To simplify complex-valued expressions, you combine "like" terms and apply the various other methods you learned for working with polynomials.

What are some examples of adding complex numbers?

• Simplify (2 + 3i) + (1 − 6i).

---

• Simplify (5 − 2i) − (−4 − i).

Many students find it helpful to insert the "understood 1" in front of the second set of parentheses (highlighted in red above). This can help the student better keep track of the "minus" being multiplied through the parentheses.

How do you multiply two complex numbers?

To multiply two complex numbers, use the techniques that you learned for multiplying binomials. (This may have been called "FOIL-ing" in your earlier studies.) Then remember that i ² = −1, and use this fact to simplify further.

What is an example of multiplying complex numbers?

• Simplify (2 − i)(3 + 4i).

---

As the example above demonstrates, FOILing works for this kind of multiplication, if you learned that method. You can also multiply vertically, if you prefer.

But whatever method you use, remember that multiplying and adding with complex numbers works just like multiplying and adding polynomials, except that, while x² is only ever just x², i² can be simplified to the value −1. You can use the exact same techniques for simplifying complex-number expressions as you do for polynomial expressions, but you can simplify even further with complexes because i² reduces to the number −1.

---

Adding and multiplying complexes isn't too bad. It's when you work with fractions (that is, with division) that things turn ugly.

Most of the reason for this ugliness is actually arbitrary. Remember back in elementary school, when you first learned fractions? Your teachers would gag if you used "improper" fractions. You couldn't say "³/₂"; you had to convert it to "1 ¹/₂". But now that you're in algebra, nobody cares, and you've probably noticed that "improper" fractions are often much more useful than are "mixed" numbers.

The corresponding issue with complex numbers has to do with denominators again. If you are told to simplify a fraction with the imaginary i in the denominator, you are expected to "rationalize" that denominator. Your instructors will become very cross with you if you leave any imaginaries lurking in denominators. So how do you handle this?

• Simplify

This was easy enough, but what if they give you something more complicated?

What is an example of rationalizing a complex denominator?

• Simplify

In the last step, note how the fraction was split into two pieces. This is because, technically speaking, a complex number is in two parts, the real part and the i part. They aren't supposed to "share" the denominator. To be sure your answer is completely correct, split the complex-valued fraction into its two separate terms.