Arithmetic with Complex Numbers

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Algebra Two $\longrightarrow$
Complex Numbers $\longrightarrow$
  • Add and subtract complex numbers by focusing on like terms
  • Multiply complex numbers using FOIL and simplify final answer to be one complex number
Lesson Description

We will combine what we know about arithmetic with imaginary numbers with what we know about binomial arithmetic to understand the proper way to add, subtract, and multiply complex numbers, including simplifying the final result.

Practice Problems

Practice problems and worksheet coming soon!


Same Arithmetic, New Numbers

Performing arithmetic with numbers of the form $a+bi$ is intuitive, and overall very in tune with the skills we have up to this point. While we won't spend a lot of time using this skill, we will need to be able to do it any time we need it.This short lesson will get you up to speed with addition, subtraction, and multiplication of complex numbers. Due to similarities with how we rationalize complex denominators » we will cover division in the next lesson.

Addition and Subtraction

Simply said, when you need to add or subtract complex numbers, combine like terms - referring to each the real and imaginary parts. For example,$$[4 - 3i] + [2 + 5i]$$$$= 6 + 2i$$The only common "attention to detail" mistake is that when you subtract a complex number, you need to treat it the same way you do when you subtract a binomial. For example,$$[2+i] - [-7 + 4i]$$$$=2 + i +7 -4i$$$$=9-3i$$similar to how$$(2+x) - (-7+4x) = 9-3x$$
Pro Tip
Adding and subtracting complex numbers should look a lot like adding variable expressions and binomials. In fact, when it comes to arithmetic with complex numbers in general, your best bet is to think of it as such.

Complex Multiplication

Multiplying complex numbers will still look familiar to skills we have from working with variables and binomials, but it will have an extra step thanks to the definition of $i$.Start off a complex number multiplication task with our ol' pal FOIL »:$$(4+9i)(3-7i)$$$$=12 -28i + 27i -63i^2$$Each term comes directly from the FOIL binomial multiplication process. However, we cannot simply combine the "$i$" terms and call it a day. We are required to do one more step, and recognize the fact that $i^2$ is really equal to $-1$. To see why this is true, recall that we typically define $i$ as$$i = \sqrt{-1}$$so that$$i^2 = -1$$Keeping that in mind, our problem becomes$$=12 -28i + 27i -63i^2$$$$\longrightarrow 12 -28i + 27i +63$$$$\longrightarrow 75 - i$$

Put It To Test

Perform the indicated arithmetic for each problem.
Example 1$$(3-2i) + (4+8i)$$
Show solution
$\blacktriangleright$ Add the real parts and imaginary parts each together.$$3 + 4 -2i + 8i$$$$=7 + 6i$$
Example 2$$(-7+4i)+(-6i-1)$$
Show solution
$\blacktriangleright$ Same deal, just be careful that the real and imaginary parts aren't always in the "standard" order.$$-7 -1 +4i -6i$$$$-8 -2i$$
Example 3$$(-3i-2)-(2i-2)$$
Show solution
$\blacktriangleright$ Take care with subtraction - remember to treat the situation as if you are subtracting a binomial.$$\longrightarrow -3i - 2 - 2i + 2$$$$0 -5i = -5i$$
Example 4$$(6)-(-4-3i)$$
Show solution
$$\blacktriangleright \,\, 6 + 4 + 3i$$$$10 + 3i$$
Example 5$$(5+3i) \times (-2i-3)$$
Show solution
$\blacktriangleright$ Even though the order is not "standard" $a+bi$, the best thing to do is just go for the FOIL right off the bat and clean up after. Remember that $i^2$ becomes $-1$.$$(5+3i)(-2i-3)$$$$-10i-15-6i^2-9i$$$$-10i-15+6-9i$$$$-9-19i$$
Example 6$$(2-2i) \times (5-8i)$$
Show solution
$\blacktriangleright$ Again, just FOIL and clean.$$10-16i-10i+16i^2$$$$10-16i-10i-16$$$$-6 -26i$$
Example 7$$(4i) \times (3+3i)$$
Show solution
$\blacktriangleright$ Since you can't really "FOIL" this one, just stick to my high level advice to treat $i$ like a variable, multiply, and clean up if you have any $i^2$ in the result.$$12i + 12i^2$$$$12i-12 \longrightarrow -12+12i$$
Lesson Takeaways:
  • Understand addition and subtraction of complex numbers as the combination of like terms - real numbers and imaginary numbers each
  • Multiply complex numbers using the FOIL method, again thinking of complex numbers as binomials
  • Simplify multiplication of complex numbers by turning any $i^2$ terms into real numbers

Lesson Metrics

At the top of the lesson, you can see the lesson title, as well as the DNA of Math path to see how it fits into the big picture. You can also see the description of what will be covered in the lesson, the specific objectives that the lesson will cover, and links to the section's practice problems (if available).

Key Lesson Sections

Headlines - Every lesson is subdivided into mini sections that help you go from "no clue" to "pro, dude". If you keep getting lost skimming the lesson, start from scratch, read through, and you'll be set straight super fast.

Examples - there is no better way to learn than by doing. Some examples are instructional, while others are elective (such examples have their solutions hidden).

Perils and Pitfalls - common mistakes to avoid.

Mister Math Makes it Mean - Here's where I show you how teachers like to pour salt in your exam wounds, when applicable. Certain concepts have common ways in which teachers seek to sink your ship, and I've seen it all!

Put It To The Test - Shows you examples of the most common ways in which the concept is tested. Just knowing the concept is a great first step, but understanding the variation in how a concept can be tested will help you maximize your grades!

Lesson Takeaways - A laundry list of things you should be able to do at lesson end. Make sure you have a lock on the whole list!

Special Notes

Definitions and Theorems: These are important rules that govern how a particular topic works. Some of the more important ones will be used again in future lessons, implicitly or explicitly.

Pro-Tip: Knowing these will make your life easier.

Remember! - Remember notes need to be in your head at the peril of losing points on tests.

You Should Know - Somewhat elective information that may give you a broader understanding.

Warning! - Something you should be careful about.

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