“Out of intense complexities intense simplicities emerge.” — Winston Churchill
As I alluded to in the very first blog, the way in which we as mathematicians organize numbers is very special: We use what is called the Real Number System to categorize any number into a family of numbers with similar properties. While I will not go into the specific breakdown of the Real Number System here, I do want to mention the biggest family within the system. Consider a number, any number. The chances are high that your number is a member of what we call the real number family, the biggest family in the Real Number System. In fact, any number you see and place practical value to your daily life—such as 1, 0, sqrt(2), -3, π, ¼, or 0.5—is a member of the real numbers.
Now, at first, you may find it superfluous that mathematicians would create a family of numbers wherein every number known to mankind is accepted. You may also find it a tad redundant to call this family the “real” numbers; after all, how can numbers be imaginary? Well, it turns out that, against our better intuition, there are numbers that transcend the real family, and so classifying it is not a total waste as originally thought.
Consider the number sqrt(-1). As it is written currently, sqrt(-1) suggests that there is such a number that, when you multiply it by itself, you will get -1 as a result. Now, if such a number exists, let us assume it is positive. A positive number multiplied by a positive number is always positive. Therefore, our number cannot be positive because we want the result of the multiplication to be negative. The number cannot be zero either because zero times zero is zero, not -1. Therefore, we must now assume that the number is negative. A negative number multiplied by a negative number yields a positive number. Hence, our number cannot be negative either because we need the product with itself to be -1, which is not a positive number.
Thus, it appears that sqrt(-1) lies outside the boundaries of the real numbers, and so we call it an imaginary number. More particularly, we call sqrt(-1) the principle imaginary number, denoting it with the letter i. All other imaginary numbers can be expressed in terms of this principle imaginary number. For instance, consider sqrt(-4). We can rewrite sqrt(-4) as sqrt(4)*sqrt(-1) by properties of exponents. Notice sqrt(4) simplifies to 2. Also notice that sqrt(-1) becomes i. Therefore, sqrt(-4) is equivalent to 2i.
If we now take these imaginary numbers and add to them a real number, we get what is called a complex number. In general, complex numbers take the form a + bi, where a is a real number and bi is an imaginary number. -3 + 5i, for example, is a complex number with a real component -3 and an imaginary component 5i.
Complex numbers can be graphed in what is called the Argand plane. You can think of this plane as a modified version of the familiar rectangular plane, where the x-axis becomes the real axis and the y-axis becomes the imaginary axis. If a complex number is expressed in the form a + bi, then the graph of the complex number in the Argand plane is given by moving a units along the real axis and b units along the imaginary axis.
The diagram above has a lot of variables thrown in it; however, the most important thing to see for now is that the complex number x + yi was graphed by moving x units on the real axis and y units on the imaginary axis.
I will now explain the origin of the other variables you see in the diagram. An important note that you should be made aware of before I do so is that there are many different ways to graph a complex number a + bi in the Argand plane. One such way is to do exactly what we have been doing: Move a units along the real axis and then move b units along the imaginary axis. Another way to define a complex number in the Argand plane is to give a direction of travel, denoted by an angle θ from the real axis, and then specify the traveling distance required to reach the complex number on the plane, denoted by a radius r stemming from the origin to the complex number’s location.
Notice in the diagram that x, y, and r form a right triangle. Therefore, we can use trigonometric principles that we acquired last time to establish relationships between x, y, r, and θ. From trigonometry, we can determine that x/r = cos(θ) and y/r = sin(θ). If we multiply both sides of each equation by r, we get x = r*cos(θ) and y = r*sin(θ), which now coincides with the extra variables in the diagram.
Hopefully, we can now draw a very important conclusion about complex numbers. Namely, for any complex number of the form a + bi, there is a corresponding form r*cos(θ) + r*i*sin(θ), or alternatively r*(cos(θ) + i*sin(θ)). It is the expression cos(θ) + i*sin(θ) that we will really begin investigating in the coming weeks.
Given a complex number a + bi, find the corresponding r and θ values in terms of a and b. (Hint: Look at the right triangle formed by a, b, and r.)