Pages

Sunday, 19 February 2012

What Are Imaginary Numbers?

There is a lot of confusion around imaginary numbers. What exactly are they? If they're not "real," what's the point of them? Surely you can't just invent numbers? The Aftermatter shall explain all this and more...


So what exactly is an imaginary number? It is most commonly represented by the letter "i," or in electronics as "j" (to avoid confusion with Current, which also has symbol "I" or "i"). "i" is normally known as the "imaginary unit," and is a constant in algebra. So, some examples of imaginary numbers would be:
i, 5i, -7i, 3.59i

In short, an imaginary number is any real number multiplied by the imaginary unit.

But what exactly is the imaginary unit?

It is defined as:
i2=-1 (or i=√-1)


The problem that mathematicians hit was that they could find no real number whose square was negative. If you square a positive number, the result is positive,  and if you square a negative number, the result is also positive -  as in 52=25 and -52=25,  and if you square zero then the answer is zero. Up until the 1500's mathematicians just considered equations such as x=√-1 to have no solutions. This meant that certain quadratic equations had no solutions, such as x2+1=0, but with the invention of imaginary numbers, it becomes ±i.

You may still consider this absurd, but if you think about it, they are no more "imaginary" than negative numbers! You can't actually have -3 cows in the real world, for example, but they are very useful for certain calculations, and so we use them. The name "imaginary" is quite misleading, it was actually coined by Descartes and used as a derogatory term about these numbers, trying to explain how nonsensical they were. Descartes, of course, was wrong, but unfortunately the name has stuck.

So, the square root of a negative number is simply the square root of the positive equivalent, multiplied by i. This is because we can express √-x as √x×√-1, which is equivalent to √x × ±i. Therefore, √-x = ±i√x. For example, √-4 = √4√-1 = i√4 = ±2i.

This is very hard to visualise; how big is i? Is it greater than 1, or less than 1? Where does it fit in the number line?

Well, we need to stop thinking of a single "number line" as such. The imaginary numbers have their own number line, which is at right angles to the "real" number line. This creates a number 'plane' rather than a number line; the crossing lines have brought the number lines into 2 dimensions. This plane is known as the 'complex plane,' and is shown in the diagram below.


As you can see from the diagram, no matter how much you add imaginary numbers, you will never get to a real number, and no matter how much you add real numbers, you will never get to an  imaginary number. In short, we can treat different and imaginary numbers as different dimensions. For example, increasing the height of a rectangle is completely independent of the width, and vice versa.

The powers of i have some rather interesting properties. They are shown below...
i0=1 (as is with any number to the power of 0)
i1=i (any number to the power of 1 is itself)
i2=-1 (this is the definition of i in the first place)
i= i2×i = -i (you can substitute the i2 with -1)
i= i2×i= -1×-1 = 1 (you can substitute both i2 for -1)
i5=i (1×i, we're back to where we started!)
i6=-1
etc.

So as you can see, the powers of i actually repeat through the pattern 1, i, -1, -i. But why does this happen? How can powers go round in a circle? Surely this violates all mathematical logic?

As you have probably noticed, imaginary numbers are quite different to real numbers. If we consider the complex plane, then addition of a real number is moving left or right on the graph, and multiplying by a real number is scaling by that number from the origin. This idea is quite familiar to us. If you think about it there is nothing unusual about this. But with imaginary numbers, addition is sliding up or down, and multiplication is actually rotation about the origin! When you multiply by i, you are actually rotating the number 90° counter-clockwise. This explains the powers of i; when it repeats, you have literally travelled in a circle!



There is where the mathematical joke comes from: What is 8×i? Answer? ∞!


(Rotate by 90°? Get it? No? Never mind...)


But what about numbers that fall on neither of these number lines? What are they?

These are known as complex numbers, numbers that have both a real and imaginary part, such as "3 + 6i." Complex numbers are of the form "a + bi", where a and b are both real numbers. The diagram below shows their position on the complex plane.

There are some important things to note about this. Firstly, technically speaking, all real and imaginary numbers are complex. For real numbers, b happens to be 0, and for imaginary numbers, a is 0. However, All complex numbers are not imaginary or real, 3+6i sits on neither line but in the middle of the plane.

Secondly, how do we measure "size" in the complex plane? How big is 3+6i? We call this the "magnitude," and the magnitude of a number z is expressed as |z|. The magnitude is the length of the line from the origin to the red dot on the diagram. This is the hypotenuse of a right angled triangle with sides 3 and 6, so by Pythagoras' Theorem we can find the magnitude.


We can also find the angle that the point makes with the Real number line and the origin, by finding the angle of the right angled triangle using trigonometry.


Now, adding complex numbers is very simple, simply add the two "a" terms, and add the two "b" terms, then you have your answer! For example, (3+6i)+(2+4i) = (3+2)+(6+4)i = 5 + 10i. All very intuitive, still only sliding, albeit in two dimensions.

Multiplication and division, however, are a little more complicated. 

Firstly, you have to find the magnitudes of both complex numbers. Secondly, you find the θ angle as we did above, the one the point formes with the Real line and the origin. The magnitude of the new number is the product of the two old magnitudes, and to find the new θ angle we add the two angles together. 

Let's take an example:
(3 + 6i)×(4+3i)
And then to add the angles...




We can check this by multiplying out the brackets fully...

The magnitude of this would be...

which agrees with our original method.

And θ would be the following...
(Note that the convention is for θ to be treated as the angle formed by the
 +∞ end of the real line, the origin, and the complex point in question. 
Anti-clockwise is considered positive. Thus, a negative imaginary number 
would have θ of 270° or -90°. This is why we have the "180-79.7" rather 
than simply 79.7).

So we can see that this multiplication method works. Division is merely the opposite: you divide by the magnitudes, and subtract the angles.

But why should we care? It's nice maths, but the numbers are imaginary, it doesn't affect the real world right? 

In fact, without the discovery of imaginary numbers our world would be very different. Imagine a world without radio waves. Obviously, radio wouldn't exist. Neither would our interview this week on Lauren Laverne's WebChat on Radio 6 (check it out at 43 minutes in). You'd turn on your TV and get nothing but black and white fuzz. There would be no WiFi. There would be no 3G. There would be no GPS. We couldn't have so many planes in the sky because air traffic control would be virtually impossible. We would be stuck in the dark ages of information.

The equations that allow devices to filter out the correct signal from other conflicting frequency (this conflict is known as impedance) is heavily reliant upon complex numbers. Without them, these devices would be unable to isolate a single one of the two waves below, and would only receive a garbled mismatch of the two waves. When you spread this across millions of possible frequencies, without being able to isolate one, there would be no discernible information at all.


In addition, in order to encode audio files such as MP3's from an analogue audio source, it is necessary to find the discrete Fourier Transform of the sound waves, to digitise the track so that it can be read by computers. This once again uses complex numbers.

Complex numbers are almost ubiquitous in computing. In order to calculate the movement of 3D graphics, we actually need two more dimensions of imaginary numbers, called j and k, to fully describe where the object is and what it is doing. This proves the point: we could have as many "dimensions" of imaginary numbers as we want to define: the original imaginary numbers were defined because one extra dimension was useful. Quaternions were defined because two more were found to be useful as well. At some point, use may be found for even more dimensions, and by the Point-Line-Plane postulate, we could theoretically create an infinite amount of number lines, and an infinite amount of numerical dimensions. Numbers without units really are quite abstract: As long as it is mathematically consistent, you can pretty much define anything that may be useful to you.

In conclusion, we hope you now have a better understanding of what imaginary numbers are, and also what they are used for. It can seem like an absurd idea at first, but just as the greatest minds struggled at first with 0 and the negative numbers when they were first proposed, you will eventually understand, and we hope we have helped you. The fact that you can define new sets of numbers does make us question how "real" our real numbers actually are; in fact, it can be argued that without units, real numbers are just as abstract as imaginary ones!


We hope you enjoyed this post. If you have anymore questions, you can follow us on twitter, @theaftermatter, email us at contactus@theaftermatter.com or search "The Aftermatter"on Facebook. We really like hearing your feedback or just talking about the posts or other physics and maths. We hope you enjoyed this post.

Theo Caplan.

Check out our last two posts:
What is Gravity? (2/2) - What actually makes up gravity? And how does it fit in with our other understanding of the universe?
What is Harmony? - Following on from our post What is Music?: How can physics and maths describe why some notes work with each other, and some don't?

What are we posting about next
What is the "Theory of Everything"? - What are the leading theories that unite quantum mechanics and Einstein's general relativity to describe the actions of everything in our universe?

2 comments:

  1. Great explanation of one of the toughest and the most confusing question which can not be easily explain or understand and in simple way we can say that imaginary numbers are those which can not be represent on number line.

    ReplyDelete
  2. I have recently become curious about the amazing powers of imaginary numbers. This is a stupendously interesting "popular" introduction. Very informative, thank you.

    ReplyDelete