Studying mathematics helps you to improve your problem solving skills, logical reasoning and the ability to think of objects in abstract ways. All forms of employment require knowledge of math in some form or another. When it comes to solving problems, wouldn’t you rather have advanced math skills than not. So, this is where multiplying complex numbers comes into play. To give you a head start, you should enroll in this Udemy course, Algebra I: Straight to the Point.
Complex Numbers and How to Multiply Them
Complex and imaginary numbers have been around since about the time of the Reformation and are one of the hardest topics to wrap your head around in algebra due to their abstract nature. The original issue that caused the creation of an imaginary number was the problem of solving the square root of a negative number. Traditionally the square root of a negative number always results in a positive outcome such as the square root of -2 is 4 as there was no logical way to come up with any other result.
How did Imaginary Numbers Evolve
Mathematicians as is their nature were not satisfied with this very simple solution to the problem of negative square roots and created and new imaginary number that could be used to simulate this mysterious problem. It was called “i” for imaginary and is the building block towards complex numbers. Before “i”, there was no way to represent the negative square root of a number because it could not be represented as “real” number.
Before moving on to complex numbers and their multiplication, it is important to understand the basics of imaginary numbers as a complex number is made up of an imaginary number and a real number. The basic premise of an imaginary number expressed mathematically is:
i = √-1 and following to i2 = (√-1)2 = -1
This is quite straightforward and shows how i represents the negative of a square root. However, you cannot take the expression further to;
i2 = (√-1)2 = √(-1)2 = √1 = 1
as “i” cannot equal both 1 and -1 and this shows by using an imaginary number you have broken the mould of the square root of a negative number always resulting in a positive result.
Basics of Complex Numbers
Complex numbers are created when you combine an imaginary number with a real number. This is often defined in its most basic form as a+bi with “a” representing the real number and “b” the imaginary amount. In order to fully understand the basics of a complex number you need to look at the first rule of complex numbers with solving a problem such as this:
6 – 3i = a + bi
The answer is very straightforward as it highlights the main concept of complex numbers that they can only be of equal value if the imaginary and real number are the same. Therefore the solution to this problem is that 6 = x and -3 = y.
A slightly simplified but useful way to think of operations using complex numbers is that they are treated in a similar way to binomials, so like terms are combined to get to the answer. This applies when adding, subtracting and multiplying complex numbers and just requires that you remember to combine the imaginary and real parts separately.
Multiplying Complex Numbers
Multiplying complex numbers is not as mind bending as it sounds and long as you follow a set of simple rules or steps. As with most things where you need to remember a sequence of steps there is a convenient acronym for multiplying complex numbers called FOIL. This stands for First, Outside, Inside, Last and gives you the basic steps for successfully multiplying two complex numbers. To understand this further, here are the definitions of the four terms in FOIL using the following example:
(x + yi) (a + bi)
- Firsts = x * a
- Outers = x * bi
- Inners = yi * a
- Lasts = yi * di
Put in mathematical terms you end up with this result:
(x + yi) (a + bi) = xa + xbi + yai + ydi2
Now if you put this into an actual equation you are able to solve it in a quite straightforward way. So you should be able to solve this equation easily using FOIL:
(4 + 3i) (1 + 5i)
Use the same order as the theoretical application above and you get:
= 4*1 + 4*5i + 3i*1 + 3i*5i
= 4 + 20i + 3i + 15i2
= 4 + 20i + 3i – 15
= -11 + 23i
The key to the solution of this problem is to remember that i2 is -1 so any number multiplied by i2 is going to be its negative counterpart, in this case 15i2 becomes minus 15.
Using a Faster Method than FOIL
This method can be harder to remember than FOIL as it does not have such an easy to remember acronym, but it is faster. The rule that is applied here is:
(x + yi) (a + bi) = (xa – yb) + (xb + ya)i
If you take the example used above and apply this rule then, you will end up with the same result. The previous example was:
(4 + 3i) (1 + 5i)
And the solution using the new rules is as follows:
(4*1 – 3 *5) + (4*5 + 3*5) + (4*5 + 3*1)i
= (4 – 15) + (20 + 3)i
= -11 + 23i
This works because you are using the FOIL method in the beginning and then applying the fact that i2 is equal to minus one and then gathering the like terms. It is just a faster way to get to the solution, but you can just follow FOIL and get to the same result if you have trouble remembering this formula.
Complex numbers are not as difficult as they immediately appear. It just takes practice and thought in order to comprehend their structures. If you need more help, you should check out these two Udemy courses, Advanced Algebra: Strategies for Success and Algebra I Made Easy. You can improve your math skills even further by checking out this blog, What is Volume in Math? Overview and Examples. With all of these resources, you’ll become a math whiz in no time!