What is the sum of 3 apples and 4 oranges? This classic statement highlights the difficulties in comparing the incomparable. At least, incomparable in one regard. Because, of course, it’s commonplace and perfectly correct to reply that the answer is “7 fruit.” In this context of different fruits, adding together apples and oranges makes perfect sense: All that’s needed is a different perspective. And just like the idea of 3 + 2i apples seems like nonsense when discussing how much fruit to buy, when talking in the framework of the complex plane these “imaginary operations” takes on a far more intuitive meaning.
Basic Structure
The complex number system subsumes the entire real number line, adding an imaginary term to any real number that corresponds to its height in the complex plane. We can write this as \( z=a+bi \), with “a” and “b” being any 2 real numbers that scale the real and imaginary components and “z” being the substitute for “x” to represent a complex variable. In the complex plane, we package a and b in the point \( (a,b) \) to represent the horizontal and vertical distances from z = 0, or the origin. Finally, since our coordinate plane uses horizontal and vertical coordinates, we know that by the distance formula \( d= \sqrt{a^2+b^2} \). When dealing with complex numbers, this distance from the origin is denoted as |z| and known as the “modulus.”
Adding Numbers in the Plane
Ok, with these ground rules laid out, let’s experiment a bit. Given 2 complex numbers \( z_{1}=a+bi \) and \( z_{2}=c+di \), what is \( z_{1}+z_{2} \)? Remember, we can add the real and imaginary parts of both numbers, so the sum is just \( (a+c)+i(b+d) \). That makes sense algebraically, but what does this look like in the plane?
Well, if we graph 2 random numbers \( 1+2i \) (Blue Line) and \( 3+i \) (Red Line) in the plane along with their sum (Purple Line) we see the following:
The dashed red and blue lines are duplicates of their solid counterparts, but the dashed line is placed at the tip of the solid complex number of the opposite color. Notice that this tail-to-tip addition produces the same result as what the algebra leads us to: \( (1+3)+(2+1)i=4+3i \). This is what complex addition looks like in the plane (what would subtraction look like?). No longer just adding apples and oranges, right?
Basics of Complex Multiplication
Let’s move on to complex multiplication. Multiplying z by a real number k scales the width and height of our complex number by a fixed ratio, so we know this just makes it k times longer. What about \( z*ki \)? Remember, multiplication by \( i \) simply rotates our number 90 degrees in the plane, so the resulting number has the same magnitude as zk, just rotated 90 degrees.
Things get a little muddier when we try and examine the effect of multiplying 2 complex numbers with both real and imaginary parts, but is there any interpretation of this expression that utilizes the concepts we’ve already covered?
\( z_{1}*z_{2}=(a+bi)(c+di) \)
We know that multiplication by a real constant scales the original number and multiplication by an imaginary constant scales and rotates it by 90 degrees. Fortunately, the distributive property allows us to break our expression apart into those exact components. If we multiply a+bi by the real part of c+di followed by the imaginary part we can expand as follows:
\( (a+bi)(c+di) \)
\( = c(a+bi)+di(a+bi) \)
\( = (ac-bd)+i(bc+ad) \).
Algebraically, this is no different than other methods of expansion, but geometrically this turns our problem into a sum of 2 scaled complex numbers, with one at a 90-degree angle from the other. This means that our complex multiplication is nothing more than an application of… well, still complex, but much simpler addition and single-term multiplication. To illustrate this, we can fall back on our earlier model of addition and try out a couple of different complex products shown below. Be sure to try and visualize where the product will carry our numbers based on the lengths of the factors.
We can try this for a few numbers. Try and guess where the products will end up by noting the positions of the complex factors instead of multiplying them out:
The Complex Conjugate
The last example gave us a strange result: The multiplication cancels the imaginary components and maps the product onto the real number line! Taking a look at the factors in the plane, we can see that they appear to be a reflection of one another across the real axis. Could this be a general property of all such factors of this formula? In the coordinate plane, reflection in the horizontal axis is numerically equivalent to negating the vertical coordinate, so if we apply it to the complex plane and multiply \( (a+bi)(a-bi) \) we find:
\( a(a+bi)-bi(a+bi) \)
\( a^2+abi-abi-(bi)^2 \)
\( a^2+b^2 \)
This should look familiar: It’s the distance of \( a+bi \), as well as \( a-bi \), from the origin. In other words, any complex number multiplied by its reflection in the real axis outputs a real number, specifically, one equal to the square of their modulus. We call this special counterpart to \( a+bi \) the “complex conjugate” \( \overline z \), and it will be a frequent partner in our journey.
Searching For Symmetry
If complex multiplication is still difficult to visualize, don’t worry. Our current model of adding scaled and rotated versions of the factors is inelegant, and it’s not immediately clear how it translates to division, unlike with addition and subtraction where the direction of the negative complex is just reversed. Indeed, nothing about division makes much sense at the moment. The expression \( \frac {a+bi}{c+di} \) doesn’t have an easy separation into a real and imaginary component. How can we get rid of our imaginary denominator? The answer is. of course, the complex conjugate! We’ve seen that a number multiplied by its conjugate yields a real number, so all we have to do is multiply the top and bottom by \( c+di \) to cancel it out!
\( \frac {(a+bi)(c-di)}{(c+di)(c-di)} \)
\( =\frac {(ac+bd)+i(bc-ad)}{c^2+d^2} \)
Looking back to our simplified expression for complex multiplication we can see some vague symmetries, but no striking similarities in form. This expression seems to have a notable quality though: It features a complex number divided by the magnitude of the divisor \( c+di \). Let’s try graphing this side-by-side with multiplication, starting with 2 of the same number to solely analyze the relationship between operations:
Placing these inverse operations side-by-side in the plane, can you spot any patterns? One that may jump out is in their angles: The green product \( -2+4i \) is perpendicular. Specifically, while the angle of the original factors was 45 degrees, the product is at an angle of 90 degrees and the quotient is at 0 degrees. Are these just markers of using the same complex numbers, or is there some more general symmetry at play here? Graphing two different complex numbers along with the angles they make with one another we find the following:
Multiplication as Angle Addition
The odd jumble of angles looks confusing, but if we break this image down we find that the green and purple numbers are once again perpendicular, and in particular, that many angles match up again: The angle between the red and blue factors looks congruent to that of the quotient and real axis, and the angle of the green product with the blue factor \( 1+3i \) appears congruent to that of the red factor \( 1+i \) with the real axis. The angle between the factors is the difference between their angles with the real axis, so this means the purple quotient’s angle appears to be the difference between the angle of the divisor and dividend!
Likewise, the sum of the factors’ angles with the real axis seems to match precisely with the angle of the product, with \( 1+i \)‘s angle with the real axis matching that between the green and blue complex numbers and the rest being composed of the angle of the blue factor.
Can a similar pattern be seen in the magnitudes? Well, if we calculate the magnitudes of the complex numbers \( |z_{1}|=\sqrt {10}, |z_{2}|=\sqrt {2} \), we find that their product is equal to the magnitude of the green complex product, \( \sqrt {20} \). Similarly (as we saw earlier in the form of the quotient), \( \frac {|z_{1}|}{|z_{2}} = \sqrt{5} \), which equals the plotted quotient’s magnitude.
A Better Way?
If we slog through the algebra these speculations can be confidently proven or disproven, but a little time exploring the geometry will be more rewarding and more interesting than direct, unintuitive algebraic manipulation. Our current notation would make such algebra cumbersome- after all, our current form of complex numbers doesn’t provide direct access to either the modulus or the angle, which are the two critical components of our musings. Is there any way we can revise our notation so that it applies these core concepts?
Holding x or y constant lets us trace a vertical and horizontal line respectively in the plane. But what if we wanted to hold the modulus, or radial distance from the origin, constant? We know that the modulus is defined as \( |z| = \sqrt {a^2+b^2} \), so let’s set this distance to 1 and figure out if there’s any way we can change a and b so that this equation is always true:
\( a^2+b^2 =1 \)
If you’re familiar with trigonometric identities then the solution may pop out immediately: The Pythagorean Identity says that for all angles \( x=\theta, \sin^2(x)+\cos^2(x) = 1 \). This means that, for all complex numbers a distance 1 away from z = 0, \( a=\cos(x) \) and \( b=\sin(x) \). We can generalize this with a scale factor: If we set the distance equal to r we can generalize our equation as follows:
\( r=r\sqrt{\sin^2(x)+\cos^2(x)} \)
\( =\sqrt{r^2(\cos^2(x)+\sin^2(x))} \)
\( =\sqrt{(r\cos(x))^2+(r\sin(x))^2}=r\cos(x), b=r\sin(x) \).
Complex Multiplication Visualized
This gives us the exact sort of notation we desired for our conjectures: \( z=r(\cos(x)+i\sin(x)) \), defining every point in the complex plane in terms of the angle it makes with the real axis and its modulus r. This notation immediately allows us to verify our 2 conjectures in a single, elegant stroke, for if we try multiplying 2 complex numbers \( z_{1} = r_{1}(\cos(x_{1})+i\sin(x_{1})) \) and \( z_{2} = r_{2}(\cos(x_{2})+i\sin(x_{2})) \) we find:
\( (r_{1}(\cos(x_{1})+i\sin(x_{1})))(r_{2}(\cos(x_{2})+i\sin(x_{2}))) \)
\(= r_1r_2(\cos(x_1)\cos(x_1)+i\cos(x_{1})\sin(x_2)+i\sin(x_1)\cos(x_2)-\sin(x_1)\sin(x_2) \)
\(=r_1r_2(\cos(x_1)\cos(x_1)-\sin(x_1)\sin(x_2)+i(\cos(x_{1})\sin(x_2)-\cos(x_1)\sin(x_2)) \)
Once again, anyone who is familiar with trig identities should find that this configuration of sine and cosine rings a bell: these are the angle sum and angle difference formulas for cosine and sine! This means that our complex multiplication essentially amounts to our final expression below:
\( r_1r_2(\cos(x_1+x_2)+i\sin(x_1+x_2)) \)
…which, as we can see, multiplies the distances and adds the angles! This is easy enough to verify for division, and the symmetry between the two operations is far more explicit in this form anyway.
Phew. That was a lot of formulas and relationships to unpack. The hardest part of learning about complex numbers for me was visualizing them, so, to supplement the algebraic tricks involved in their various identities and formulas I developed a penchant for playing around with these numbers in their natural habitat of the plane rather than in a list of equations. Test out the geometry of these operations yourself below, with 2 complex numbers \( a+bi \) and \( c+di \) at your disposal to manipulate with the operations we covered.
This effectively provides us with the basic syntax we will need in the future to work with the complex plane. We’ve explored how to perform the 4 basic operations on complex numbers, and in the process found an important link between multiplication and rotation that is at the heart of why this number system is so invaluable to mathematics. One interesting parallel that you may pick up on is the similarity of complex multiplication and properties of exponents: Just like multiplying two powers adds their exponents and multiplies their coefficients, multiplying our complex numbers in trigonometric form adds their angle and multiplies their lengths. Next post I’m hoping to make that correlation clear by exploring one of the imaginary numbers’ most famous identities, and maybe putting our notation to the test with the paradoxical problem that this expression poses.
Thanks for your blog, nice to read. Do not stop.
Glad you liked it, thanks so much for the support!