We have now met our cast of characters. Let's have a look at them all lined up on the complex plane:
So you take the number at \(\pi\) and multiply it by \(i\). That's easy enough to visualise now. We look at both numbers in polar form. \(\pi\) is just \(\pi\angle 0\) and \(i\) is \(1\angle \tfrac{\pi}{2}\). We just need to multiply the absolute size (\(\pi\cdot 1\)) and we add the angles (\(0+\tfrac{\pi}{2}\)). That gets us to \(\pi\angle \tfrac{\pi}{2}\). That looks like this:
So we are now left with the job of trying to work out how to raise the number \(e\) to this new number \(i\cdot \pi\). To do this, we are going to have to work out how to multiply \(e\) to a complex number. To start with, lets go back and look at our definition of \(e\).
\[e=\lim_{n \to \infty} (1+\tfrac{1}{n})^n\]
Lets look only at the function bit:
\[(1+\tfrac{1}{n})^n\]
Let's now raise that bit to a power:
\[((1+\tfrac{1}{n})^n)^x\]
We now know that when raising something to a power and then to a power again, we can just multiply the powers. In other words the expression above really means take the bit in brackets and multiply \(n\) of them together. Now take that group of things being multiplied together and multiply \(x\) of those groups together. If you did that you would just end up with \(x\) groups of \(n\) things all multiplied together. So essentially you end up with \(n\cdot x\) numbers of the stuff in brackets being multiplied together. So we can just write this instead:
\[(1+\tfrac{1}{n})^{n\cdot x}\]
How does this help me work out \(e\) to a complex power? Well, what I know how to do is to multiply, divide and add complex numbers. I do not know how to raise a number to a complex exponent. So what I really want to do is to get that \(x\) away from the exponent. How do I do that?
Let's create a new variable \(m\). Let's define this to be \(n\cdot x\):
\[m=n\cdot x\]
Now I will multiply both sides of that equation by \(\tfrac{1}{n}\):
\[\begin{align*}
\tfrac{1}{n}\cdot m&=\tfrac{1}{n}\cdot n\cdot x\\
\tfrac{m}{n}&=\tfrac{n}{n}\cdot x\\
\tfrac{m}{n}&=1\cdot x
\end{align*}\]
We also need to multiply both sides by \(\tfrac{1}{m}\):
\[\begin{align*}
\tfrac{m}{n}\cdot \tfrac{1}{m}&=x\cdot \tfrac{1}{m}\\
\tfrac{m\cdot 1}{n\cdot m}&=\tfrac{x}{m}\\
\tfrac{m\cdot 1}{m\cdot n}&=\tfrac{x}{m}\\
\tfrac{m}{m}\cdot \tfrac{1}{n}&=\tfrac{x}{m}\\
1\cdot \tfrac{1}{n}&=\tfrac{x}{m}
\end{align*}\]
We now have two defintions to use:
\[\begin{align*}
n\cdot x&=m\\
\tfrac{1}{n}&=\tfrac{x}{m}\\
\end{align*}\]
And we just need to fit them into this equation:
\[(1+\tfrac{1}{n})^{n\cdot x}\]
We have both a \(\tfrac{1}{n}\) and a \(n\cdot x\), so let's have at it:
\[(1+\tfrac{x}{m})^{m}\]
There we go - mission accomplished. We have manged to get the \(x\) AWAY from the exponent. We can now say that:
\[e^x=\lim_{m \to \infty} (1+\tfrac{x}{m})^{m}\]
We can now see what \(e\) raised to a complex power is. First, though, let's just run through this with real powers to check it is working. If we square \(e\), then according to my calculator we should get \(7.389056099\). So let's try:
\[e^2=\lim_{m \to \infty} (1+\tfrac{2}{m})^{m}\]
If you set \(m\) to \(1000000\) you get an equation that looks like this:
\[\begin{align*}
\left (\frac{1000000}{1000000}+\frac{2}{1000000}\right )^{1000000}\\
\left (\frac{1000002}{1000000}\right )^{1000000}\\
\frac{1000002^{1000000}}{1000000^{1000000}}
\end{align*}\]
If you work out that horror you get \(7.389041321\), which agrees to four decimal places with the "real" answer. So we are on the right track! So what we can do now is to replace the \(2\) with \(i\pi\). That looks like this:
\[e^{i\pi}=\lim_{m \to \infty} (1+\tfrac{i\pi}{m})^{m}\]
Let's set \(m\) to a hundred to see how we get on. That makes our equation:
\[e^{i\pi}\approx (1+\tfrac{i\pi}{100})^{100}\]
Let's deal with the bit in brackets first. How do we do the division? It is unsurprisingly the opposite of multiplication. So instead of multiplying the absolute values together you divide the first by the second. Then you deduct the second angle from the first. Remember the number we are dividing looks like this:
In polar form (for division) that is \(\pi\angle \tfrac{\pi}{2}\). The polar form for the number doing the dividing (the one on the bottom) is \(100\angle 0\). So the maths looks like this:
\[\frac{\pi\angle \tfrac{\pi}{2}}{100\angle 0}\]
\[(\tfrac{\pi}{100})\angle (\tfrac{\pi}{2}-0)\]
\[(\tfrac{\pi}{100})\angle \tfrac{\pi}{2}\]
So it is going to be at the same angle but only a hundredth the distance away from the origin. We then want to add one onto that. That just moves the number one unit in the positive real direction. That looks like this:
We can now see the number on the complex plane. This is the thing that we are going to raise to the power one hundred. Raising to a power is all about multiplication, so we are going to want to put the number into polar form. So what is it in polar form? First the absolute value is (again using Pythagoras) the square root of one squared plus \(\left (\frac{\pi}{100}\right )^2\). So the absolute value is:
\[\sqrt{1^2+\left(\frac{\pi}{100}\right )^2}\]
\[\sqrt{1^2+\left(\frac{\pi^2}{100^2}\right )}\]
\[\sqrt{1+\frac{\pi^2}{10000}}\]
\[\sqrt{\frac{10000}{10000}+\frac{\pi^2}{10000}}\]
\[\sqrt{\frac{10000+\pi^2}{10000}}\]
Just looking at that, you can see that it is a number just a little big bigger than one. Why? Well ten thousand and a little bit divided by ten thousand is pretty close to one. And the square root of something pretty close to one, is even closer to one. You can compare that with the diagram:
That makes sense. What about the size of the angle? The sine function of the angle gives us the height of the point above the real axis, divided by the absolute value (to scale everything to the unit circle). If we work out that scaled down height, we can use the inverse of sine function to give us the distance around the unit circle to that point, and hence the size of the angle. The scaled height is \(\frac{\frac{\pi}{100}}{\sqrt{\frac{10000+\pi^2}{10000}}}\). Hmm. That looks like a nightmare, but it really isn't. We have already seen that the bit on the bottom (the absolute value) is pretty close to one. Any number divided by one is just itself. So really, we are interested in the bit on top. That tells us that the sine of the angle we are looking for is roughly \(\tfrac{\pi}{100}\). In fact, if you work it out it is \(0.0314004349\).
Now to actually DO the inverse sine function we could start drawing our unit circle, and then make very very precise measurements, or we could just use a calculator. Calculator it is then. The distance round the unit circle to a point \(0.0314004349\) above the real axis is \(0.0314055972\). So that's our angle in radians.
(If you think long enough about this you may well ask 'How the fuck does my calculator KNOW that this is the angle size that corresponds with that sine?' The calculator doesn't draw a circle and get out a ruler. Does it come with all the possible sine values for all possible angles to whatever number of decimal places? No. And before we have finished with this, you will find out what your calculator did to get this precise result).
If you look at the size of the angle, you may notice that it is pretty bloody close to the sine of the angle. In other words the distance around the unit circle to the point is almost exactly the same size as the height of the point above the real axis. Apparently this happens when your angle is very small. Why? Well, let's look at the diagram:
That's our number. Let's scale it to the unit circle (this does not change much because we know that the dashed line there is pretty close to one anyway), and use our old friends Imogen, Polly and Abby to see what is going on:
We can see the very small angle we are dealing with. What I am going to do next is to get rid of Raul because that is not relevant to this particular discussion. I am then going to move Imogen so that it is directly under Polly:
Now let's zoom in on the interesting bit:
Can you see that the orange line of Imogen is practically the same length as the section of circle round to Polly that it is pretty much obstructing? Let's move closer:
Can you see how the orange line and the blue line up to Polly are very close in length? That's exactly what I mean when I say that the sine of an angle is very close to the size of the angle in radians, when the angle is very small.
Anyway, enough distractions. We have our polar form number: \(\sqrt{\frac{10000+\pi^2}{10000}}\angle 0.0314055972\).
What are we going to do with it? The absolute value of the number is going to be multiplied by itself one hundred times. The angle is going to be added to itself one hundred times. Where does that take us? Well, the size is actually a number very close to one. If it was the square root of ten thousand divided by ten thousand, it would be one. The only thing that stops it being one is the \(pi^2\) on the top line. So it is roughly the square root of ten thousand and nine over ten thousand. That is very very close to one. So when we multiply it by itself one hundred times it should still be pretty close to one unit long. In fact it turns out to be a bit over \(1.05\), but not much. Now the angle. If you look closely you will see that the angle is, to four decimal places, one hundredth of \(\pi\). So what do you get if you multiply one hundredth of \(\pi\) by one hundred? \(\pi\)! In fact for our numbers you get to \(99.99\%\) of \(\pi\). So, our end result, is:
\[\left(\sqrt{\frac{10000+\pi^2}{10000}}\angle 0.0314055972\right)^{100}\approx 1.05\angle99.99\%\pi\]
Well, the angle \(\pi\) is of course half a circle, which makes the result, to within \(5\%\), negative one. So we can say, to within \(5\%\) that:
\[e^{i\pi}\approx (1+\tfrac{i\pi}{100})^{100}\approx -1\]
To generate the identity that started this whole thing, all we do is add one, an equals sign, and zero. So it looks like we are definitely on the right track. What about increasing the value for \(m\)? Well, I can't be bothered running though all the arithmetic again for a start. But lets try to imagine what would happen.
First the multiplication of \(\pi\) and \(i\) would proceed unchanged. But then we would divide that number by a much larger number, say a million. That would bring the point down to a millionth of \(\pi\) away from the real axis. We would still add one, which would take us out to a point very, very close to one. The angle would be much smaller and the absolute size would by much much closer to one.
Secondly, when we then raised the absolute size to the power of a million, it would stay much closer to one. And although the angle would be much smaller, we would then add a million of them together, getting us even closer to \(\pi\) as the total. So as the \(m\) number gets larger, the result gets closer and closer to negative one.
In fact, if you put set \(m\) to infinity, you will divide \(i\pi\) by infinity, getting an infinitely small number. One plus an infinitely small number is infinitely close to one. That number's imaginary part would be infinitely small. When you raised the absolute value of the number to infinity, it would stay at one. And the angle, which would be an infinith fraction of \(\pi\) multiplied by infinity would be exactly \(\pi\)! Which is exactly what we wanted to prove isn't it? Let off the fireworks and start the band we're done!
No. No we're not. This:
\[e^{i\pi}=\lim_{m \to \infty} (1+\tfrac{i\pi}{m})^{m}=-1\]
:gives us no real sense of what is going on here. In other words, sure, this formula tells us that raising \(e\) to \(i\pi\) gives you negative one, but it does not tell us WHY.
That's why this section is just called the intermission. This has been a test of all the concepts and tools that we have built up along so far. The good news is that they work. So we are doing the right thing. The problem at the moment is this whole 'limit' feature of our definition of \(e\). What we are going to do next, is to try to get rid of the whole limit approach to the puzzle altogether.
No comments:
Post a Comment