Complex Arithmetic

Now that we know complex numbers exist, it is natural to ask: what can we do with them? Since they are numbers, we expect to be able to perform operations on them: to add, subtract, multiply, and divide, at the very least. But how does that work? How can we understand adding two complex numbers or multiplying them?

As we know from the last chapter, when Cardano and Bombelli solved equations using complex numbers, they tried as much as possible to treat complex numbers the same way they would real numbers. The one exception to this rule was the fact that i2=, but otherwise they assumed that the same rules of arithmetic that they applied to real numbers would apply to imaginary and complex numbers.

Adding and Subtracting

We can add and subtract complex numbers just like any algebraic expression, by adding or subtracting their real and imaginary parts separately. You use “i” just like any other mathematical constants or variables, like π or x. With that in mind, let’s try adding and subtracting some complex numbers:

You may have noticed that complex numbers are quite similar to 2-dimensional vectors. In fact, we can represent each complex number as a vector in the complex plane, with the horizontal component corresponding to the real part imaginary part and the vertical component corresponding to the imaginary part real part. We can then add or subtract complex numbers in much the same way we would add or subtract vectors. We can think of adding or subtracting complex numbers as translating rotating reflecting, or shifting, points in the complex plane.

In seeing how we can add and subtract complex numbers, we can already see one of the reasons why complex numbers are useful: like vectors, they can be used to represent 2-dimensional quantities like movement, velocity, and force. But, it remains to ask, then: why not just use vectors if both can be used in similar ways? What makes complex numbers uniquely useful for mathematicians, scientists, and engineers, as they seek to understand, analyze, design and create?

The answer lies in the third basic operation: multiplication. It’s not possible to multiply two vectors – we’ve seen operations like the dot and cross products, but these weren’t really “multiplication”. However, we said that complex numbers should behave just like the real numbers we already know, which means we must be able to multiply them.

Multiplying Complex Numbers

To see how we might multiply complex numbers algebraically, consider a simple example: (2+i)(1-i). We can multiply complex numbers in much the same way we multiply real numbers. We can use the distributive associative commutative property and then we can combine all of the real parts and all of the imaginary parts separately, keeping in mind that i^2=. It might look something like this.

algebra flow (1+i)(1-i) 1(1+i)-i(1+i) Distribute each term in the first binomial to the second binomial. 1+i-i-i^2 Distribute again. 1+0i-i^2 Combine the imaginary parts. 1-(-1) Use the fact that i^2=-1 to simplify. 2

This makes sense algebraically, but what does it mean geometrically? Let’s start by imagining a simple object formed in the complex plane. Take a square, for example, with vertices at the following four points: 1+i, 2+i, 1+2i, and 2+2i. What would happen if we multiplied all of these points by another complex number, like 1-i? We have already multiplied (1+2i)(1-i). Let’s now take the other points on the square and multiply them by 1-i:

Let’s think about what happened to the square, visually, as each point was multiplied by 1-i. We can see that it was rotated reflected translated dilated by an angle of clockwise and dilated rotated reflected translated by a factor of . You can get a further sense of how complex multiplication works visually by playing with the diagram above. Pick a shape of your choice, and then drag the vector shown. It will show you what happens when you multiply all of the points that make up your shape by the complex number represented by the vector.

As you play with the diagram, record what you notice and what you wonder below.

You may have noticed that as you rotate one of the factors around the plane, the entire shape rotates with it. In other words, as you change the angle of rotation of each factor, that makes the product also rotate an equal amount. Similarly, if you stretch or shrink one of the factors, the product also stretches or shrinks proportionally. To see this most clearly, we can look at what happens if we only change the angle or only change the magnitude of each factor. Play around with the diagram below to get a sense of how this works.

We have seen that complex multiplication causes rotations and dilations, but you may be wondering: Why did multiplying each number by 1- i rotate the square by exactly 45or expand it by a factor of 2? To understand exactly why those particular transformations occurred, we need to look more closely at the number 1-i itself, and to understand it a bit differently.

So far we have been representing complex numbers as points or vectors in the complex plane. We have noted that the real part of each complex number corresponds to the horizontal component of the vector that represents it, and the imaginary component corresponds to the vertical component. Though this is one way to describe complex numbers, it is not the only way.

We can also describe them in terms of the angle that they make with the positive x-axis, and their distance from the origin. We often label this angle , and refer to it as the argument of a complex number. We call a complex number’s distance from the origin r, and refer to it as the number’s modulus. Sometimes the modulus is also written as z, – it’s the 2-dimensional version of absolute value, the distance of numbers from the origin. To see this visually, consider the diagram below, which shows how any point in the complex plane can be represented in terms of and r.

If we want to describe any complex number, a+bi, in terms of its modulus r, and its argument, , we could use our knowledge of trigonometry to solve for a and b in terms of r and . We could write that a=r cos () r sin () r tan (), and b= r sin () r cos () r tan (). Putting it all together, we can write the whole complex number as:

diagram

This is known as the polar form of a complex number. If this looks familiar to you from learning about polar coordinates, you are right to notice many similarities between the way we represent coordinates in polar form and how we represent complex numbers in polar form.

To write1-i in the polar form, let’s plot it in the complex plane shown above, and then attempt to find the values of r and that correspond to that point. We can find r using Pythagoras’ Theorem, and this gives us r=. In this case, we might be able find just by inspection, but we could also use the inverse tangent function and the fact that the point is in Quadrant IV Quadrant I Quadrant II Quadrant III . This gives us = . Putting everything together, we can write 1-i in polar form as .

This is a surprising discovery! In some sense, the fact that multiplying a complex number by 1-i causes that number to rotate 45 clockwise and expand by sqrt(2) is built into the number 1-i itself. If we had multiplied by another complex number, we would have generated a completely different combination of rotations and dilations. These transformations can be predicted in advance, and they can be seen much more clearly when we write complex numbers in polar form.

Add short practice wherein students convert between polar and Cartesian form

Even with an understanding of polar form, you may still be wondering exactly how multiplying complex numbers works. We can understand and visualize complex multiplication in many different ways, three of which are shown in the tabs below. As you play with each one, think about what each visual reveals about complex numbers that may be missing (or less clear) from the others.

Each of these visuals highlights different aspects of complex multiplication. From the first, we see that we can find the argument of the product by simply adding subtracting multiplying dividing the arguments of the two factors. Similarly, we can find the modulus (r) of the product by multiplying adding subtracting dividing the moduli of those two factors.

From the second and especially the third visuals, we can see that multiplying by a complex number does not just transform the other number that is being multiplied; it transforms the entire plane. Each complex number describes a unique set of rotations and dilations that happen to any point in the plane when it is multiplied by that particular complex number.

Dividing Complex Numbers

Now that we have a fuller understanding of complex multiplication, let’s finally consider what it might mean to divide two complex numbers. Take, for example, the following quotient:

1-2i3+4i=?

At first glance, this seems much more complicated than multiplication. In multiplication, we just use the distributive property as we would with variables or whole numbers. How can we use what we know about division to find a way to perform it on complex numbers?

We know that division is the “inverse” of multiplication, so it seems reasonable that in order to divide two complex numbers we their arguments, rather than adding them, and their modulii, rather than multiplying them. This turns out to be true, but the process of converting between Cartesian and Polar forms can be cumbersome. Especially if the complex number is already in Cartesian form, there is a much simpler method.

You may have noticed in some of the examples above that sometimes multiplying two complex numbers gives a purely real number. As it turns out, there is a way to turn any complex number into a purely real number using multiplication. It relies on a concept called the complex conjugate. We use the notation z* to represent the complex conjugate of a number z. For any complex number z=a+bi, its complex conjugate, z*=a - bi. For example the complex conjugate of 3+4i is , and the complex conjugate of -2-2i is .

To understand why the complex conjugate is useful here, let’s first think about it geometrically. In the diagram below, drag the two complex numbers, so that they are conjugates of each other. Try to find the conjugates of numbers in each different quadrant, and think about what you notice about each pair of conjugates and their product.

diagram

Geometrically, complex conjugates reflect a complex number about the x-axis y - axis line y=x . They have the same opposite modulus as the original complex number, but the opposite same argument. Because the arguments are opposites, when the two conjugates are multiplied together the resulting complex number has an argument of and rests along the real number line. This proves that multiplying a complex number by its conjugate always results in a real number.

You may be wondering how this relates to division, and in particular to our original problem: 1-2i3+4i. It is useful here to think of division as the “inverse” of multiplication. Take a basic division fact, like 15/5=3. Why is this fact true? We can come up with many reasons, but they all essentially boil down to the fact that 5 groups of 3 and 3 groups of 5 are both 15. In other words, division is defined by its relationship to multiplication, and all division problems can be thought of as questions about multiplication. For example, 15/5 can be thought of as asking a question: What number, when multiplied by 5, gives 15? Similarly, 1-2i3+4i can be thought of as asking: what number, when multiplied by 3+4i, yields 1-2i?

Because dividing by real numbers is much simpler to understand than dividing by complex numbers, one strategy we can use is to take the expression1-2i3+4i, and turn it into an equivalent expression where the denominator is a purely real number. This is where the complex conjugate comes in: if we multiply the denominator by its complex conjugate, we should get a purely real number. In order to keep the expressions equivalent, we must multiply both the numerator and denominator by the conjugate of the denominator. For our problem it would look like this:

algebra flow Math Commentary 1-2i3+4i3-4i3-4i This method is equivalent to multiplying our original fraction by . (1-2i)(3-4i)(3+4i)(3-4i) We multiply the numerators and the denominators separately. We can expand and simplify both the numerator and the denominator. Notice that the denominator is now a purely real number. -15-25i We sImplify completely. This is the quotient we were looking for.

This is a good amount to take in, so let’s start by making sure we did our division correctly. If 1-2i3+4i=-15-25i, then it should also be true that (3+4i)(-15-25)=. To convince yourself that this works, take a moment to verify that this statement is true.

You may be left wondering why this method works: it has to do with the properties of complex conjugates. To make a long story short, because any number has the same modulus, and the opposite argument as its conjugate, conjugates are uniquely suited for the division operation, which is in many ways just an “undoing” of multiplication. Let’s get some more practice with dividing complex numbers by solving the problems below:

example problems Problem Solution 2i1+i 1+i -4i 4i 4-2i-3+2i -16-2i13 2+2i2-i2 i

So there you have it! Just as with real numbers, we can add, subtract, multiply and divide any pair of complex numbers. We can do this, for the most part, by following the same rules we use to operate on real numbers: the distributive, associative, and commutative properties. What’s more; we found a way to understand complex number operations geometrically; by thinking of them as vectors in the complex plane. In some senses, we found that complex numbers are even better than vectors. In particular, when we write complex numbers in polar form, it gives us a simple and elegant way to understand 2-D rotation and dilation.

This property of complex numbers; that they can naturally describe expansion, contraction, and especially rotation is at the heart of their usefulness and their beauty. As we saw in the course on trigonometry, there is a deep connection between circular, rotational motion and wave behavior. Oftentimes when scientists and engineers deal with waves - whether radio waves received by your cell phone, quantum mechanical waves that describe the behavior of subatomic particles, or physical waves like the shock waves after an earthquake - complex numbers are the most simple and insightful way to describe these waves. In the coming chapters, we will further explore how complex numbers can be used to describe rotation and wave behavior, and how they enable us to find solutions to many important scientific and mathematical questions, both ancient and modern.