At first it might seem intuitive to think of fractions, negative numbers, maybe even decimals, but the more we stop to think about it, the more we might start to question their legitimacy and meaning. For instance, what does it mean to have -1 apples, or ¼ of a sheep? In fact, the number line we know today was very different 1000 years ago…
Before we used money as a currency, people bargained and traded using whatever they had, such as the amount of animals they own, like sheep. To count what they have, they used natural numbers, which included all positive integers. Over time, several problems came up: what if someone does not have enough sheep to pay me back with? What happens when the sheep does not belong to one person, but two? To answer these questions, the negative and rational (fractional) numbers were introduced to the number line. It was hard for people to interpret and visualize them at first, but people slowly accepted these numbers because they made calculations easier. Irrational numbers became part of the number line when Hippasus, a Pythagorean Philosopher, argued that some numbers, such as the square root of 2, cannot be represented by a fraction. From natural numbers to irrational numbers, these numbers together form the set of real numbers.
You might think that real numbers encompass all numbers in mathematics, and this is even what mathematicians thought until relatively recently, but there is actually another dimension of numbers that the real numbers do not include. Whenever we square a real number, we always end up with a result that is positive. It appears that we cannot square a number that gives a negative value, but yet we see these numbers come up in mathematics. So, someone decided to give this “mysterious” number a name: imaginary number i, which is defined to be √-1. But what does it mean?
Generally, we tend to think of numbers on a line, which is what we call the ‘numberline’. But, if we now consider a second dimension, the “imaginary dimension”, then a whole new world of numbers awaits. A number can be purely real or imaginary, or it can contain both a real and imaginary part which is what we call a complex number is. We can plot any complex number on an Argand Diagram, which is no different than any other coordinate plane you have seen, except the x-axis is now the “real” axis, and the y-axis is the “imaginary” axis. Here are some examples:
Even though the concept of imaginary numbers themselves might at first seem hard to comprehend, they can make other mathematical concepts much more intuitive to understand. One of the concepts is the Fundamental Theorem of Algebra, which states that:
“Every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots.“
Ok, but what does that all mean? Let’s take a quadratic equation, which is a polynomial with terms up to the power of 2, eg. x2 + 4x -1. The theorem is essentially saying that any quadratic equation in the form y = ax2 + bx + c (for some fixed numbers, a, b, c where in our example a = 1, b = 4 and c = -1), crosses the x-axis twice (or once if the root is repeated). This seems intuitive for an equation like y = x2 – 1, but what about y = x2 + 1?
The quadratic equation doesn’t touch the x-axis in the second graph (y = x2 + 1). But, if we set a third axis, the imaginary axis, in and out of the page (as shown below), we can see that the graph does touch the axis at the 2 points +i and -i (or + √-1 and – √-1).
So, what about ‘real-life’? We can see negative numbers in financial management, fractions in baking recipes, but where can we apply complex numbers? It turns out imaginary numbers are crucial in air traffic control. Who would have imagined…
Air traffic control uses RADAR (RAdio Detection And Ranging) to detect moving planes. When thinking of radar, I’m sure many of you will picture a spinning green needle that sweeps around a black circular disk, like the one shown below. The control centre has a rotating source that scans 360º every 2-3 seconds. The source sends a radio signal, with a frequency higher than those used for radio or TV broadcasts to avoid interference, and has a receiver that detects echoes from any object in its path. You can think of it as throwing a ball in a dark room and seeing whether or not it comes back to you. If it does, then something must be in front of you, and the longer it takes to come back, the further away the object must be.
Rather than using something small like a ball, the radio beam is shaped like a fan: narrow in the horizontal direction, and wide in the vertical direction to cover a large area. This method is known as primary radar. However, it is rarely used today because there are now too many planes in the sky. So instead, we use secondary radar, where a coded pulse sequence is sent to aircraft and a transponder on the plane generates a coded return.
But what has this got to do with complex numbers? Well, imagine the amount of data that the computer has to process if the control centre can sweep the entire circle in 2-3 seconds. Furthermore, the radar is scanning airplanes that can move up to 900 km/hr, making computations more difficult. Using imaginary numbers allows computers to calculate much quicker. The same calculations can be done with real numbers, but the plane would have moved somewhere else by the time the calculation is done!
The data that air traffic control centres receive often has a lot of data noise, and sometimes it can be hard for the radar to pick up the signal from the plane, similar to how it’s difficult to hear someone speaking next to you in a loud room. But there are ways that we can remove background noise to make the signal from the plane clearer. Background noise usually has a different frequency to the plane signal, and we can distinguish these frequencies by using something called a Fourier Transform. I won’t go into the maths here, because it gets quite complicated, but the theory of Fourier Transforms basically says that a random wiggly pattern can be made up by cosine and sine waves of different amplitudes and frequencies, and it tells you what these different waves are.
As you can see in the formula, using the Fourier Transform requires complex numbers. So, the question is what do cosine and sine functions have to do with complex numbers? This is where Euler’s formula comes in:
If this is all new to you, no doubt you’re thinking how does this all make sense? Where did this even come from? The answer lies in the Argand Diagram. If we look at the diagram of the unit circle below, we see that we can represent the real and imaginary part of a complex number with cos and sin. It’s just like how we can express x = cosθ and y = sinθ in our usual coordinate plane, except x and y are the real and imaginary dimensions respectively in the Argand Diagram. We call e iθ the polar form, because here we represent the complex number by its angle and radius (in this case the radius = 1). Similarly, we call cosθ + i sinθ, which takes the form a + bi, the rectangular form.
Puzzles 1 and 2
1. It’s probably difficult for you to believe that calculations with imaginary numbers are much simpler when interpreting radar, so why not give it a go yourself? To understand how radar signals are generated, we will look at how simple waves can be added together to create something more complicated. Let’s say the two waves both have a frequency of 5 Hz, and the runtimes are 5 and 6.05 respectively.
(a). Write an equation for both waves in the form s(t) = cos(k(t – φ)), where φ is the runtime in seconds (the time it takes for the signal to first appear) and k = 2π * frequency.
(b). Draw both waves on a graph. Also draw what happens when you add the 2 waves together (add the amplitudes of the waves together).
(c). We can also add the waves mathematically using the trigonometric identity:
But there is a simpler method using complex numbers. Knowing that cosθ is equal to the real part of e iθ, express the sum of the two waves in polar form. You should get e 10 π t (e -50 π i + e -60.5 π i).
(d). Ignoring the e 10 π t, convert your results from above to rectangular form (a + bi). Show that you get [cos(-157.08) + cos(-190.07)] + i [sin(-157.08) + sin(-190.07)]. Remember to use radians instead of degrees.
(e). Express the right-hand side of the expression above as one complex number (in the form a+bi). What angle corresponds to that complex number?
(f). Convert the expression back to polar form and multiply it back with the e10 π t we ignored in parts (d) and (e). You should get e10 π t – 0.25 π i. This format is very useful because it is much easier for computers to transfer and interpret information from one wave than two waves.
2. Let’s say the radar on the air traffic control tower is emitting a radio wave at a frequency of 100 gigahertz. It sends out a 1 nanosecond pulse with an amplitude of 1. In mathematical terms, the emission signal will be represented as cos(100 * 2πt). After a certain time, you receive the following signal:
(a). From other sources, you know this signal is the radar signal reflected off 2 planes. Noting the starting and ending time of this wave, how far do you think the planes are from the control centre? Remember that (distance) = (speed of light) x (time).
(b). Is the distance between the 2 planes realistic? What does this say about the height for the 2 planes relative to each other?
(c). What are the equations of the waves? They should be in the form cos(k(t – φ)), where t is in nanoseconds (10-9s) and φ is the runtime in nanoseconds.
(d). Add the 2 waves above together. The result should be in the form s(t) = ei k(t – φ), where is the runtime in nanoseconds and k = 2π * frequency. See question 1 for more detailed steps.
(e). The signals and waves we worked with above form what we call primary radar. Why do you think we rarely use primary radar nowadays (think about what the signal will look like in an area with a lot of planes)?
Imaginary numbers can also help us to better interpret waves. When thinking of waves, most people will imagine a periodic upward and downward motion across the page. However, we can also think of a wave as taking the x or y coordinate as you move around a circle. In fact, this is where the cos(x) and sin(x) functions come from. But, there’s a problem – a similar looking wave can actually represent two types of movement. We call these the positive and negative frequencies.
These can be explained using the concept of visualising waves as a circle as mentioned above. In the way we normally think of waves, we only take one-dimension (x or y) of the circle. What if we want to include both dimensions? This is what the imaginary dimension allows us to do. If we take x and y to be the real and imaginary dimension, we make a helix. A wave is basically a projection of a helix into the real dimension only. With a helix, we can see if the wave has a positive or negative frequency: the wave looks the same in the real dimension, but it has opposite directions in the additional imaginary dimension.
|Positive frequency||Negative frequency|
3. As mentioned above, we can think of waves as a projection of a helix onto the real domain. Let’s suppose the air traffic control centre received the following signal:
(a). Draw the projection of the helix on the real axis. What is the frequency of the signal?
(b). What would the conjugate of this helix look like? Remember for a complex number in the form a + bi, the conjugate will be a – bi.
(c). Draw the projection of both helices onto the imaginary axis. What happens when you add the 2 waves together? Remember, to add 2 waves together, you just add the height of the 2 waves together.
(d). Using what you discovered in part (c), how do we only obtain the real part of the signal?
[Hopefully you tried and worked through the question above, if not spoilers are up ahead!]
When an air traffic control tower receives these waves, it converts the imaginary parts of the waves back into the real dimension. The way we do this conversion is by adding the wave to its conjugate. If we have a complex value of a + bi, adding the conjugate a – bi gives us (a + bi) + (a – bi) = 2a, twice the real component. Given a complex number and its conjugate, we can also refer to these as the positive (a + bi) and negative frequencies (a – bi). If you look back at the graphs above, you can see that the waves in the imaginary dimensions are opposites of each other because of the +b and -b in the complex number and its conjugate form. In fact, a radio wave encodes these 2 pieces of information together, so that information can be preserved.
It’s crazy how imaginary numbers – which are hard to visualize – can have such important applications. “Imaginary” numbers are not imaginary at all, otherwise air travel might not have be possible!