Click here to purchase the entire book in PDF format. Who cares? Here's where the beauty of all this math actually becomes apparent. What we've essentially done is to make things look complicated in order to simplify working with the equations. We saw in the last two chapters how an arbitrary wave like a cosine with a peak amplitude of 0.93 and $\frac{\pi}{3}$ radians late could be expressed in a number of different ways. We can say $0.93 \cos (n + \frac{\pi}{3})$ or $0.4650 \cos(n) - 0.8054 \sin(n)$ or we can represent it with the complex number $0.4650 + j 0.8054$. I argued at the time that using these weird complex notations would make life simpler. For example, it's easier to add two complex numbers to get a third complex number than it is to try and add two waves with different amplitudes and delays. However, if you were the arguing type, you'd have pointed out that multiplying two complex numbers really isn't all that attractive a proposition. This is where Euler becomes out friend. Using Euler's identity, we can convert the complex representation of our waveform into a complex exponential notation as shown below $$0.93 \cos(\alpha + \frac{\pi}{3}) = 0.4650 \cos(\alpha) - 0.8054 \sin(\alpha)$$ (2.66) Which is represented as $$0.4650 + j 0.8054 = 0.93 e^{j \frac{\pi}{3}}$$ (2.67) There's a really important thing to remember here. The two values shown in Equation 1.67 are only representations of the values shown in Equation 1.66. They are not the same thing mathematically. In other words, if you calulated $0.93 \cos (\alpha + \frac{\pi}{3})$ and $0.4650 \cos(\alpha) - 0.8054 \sin(\alpha)$, you'd get the same answer. If you calculated $0.4650 + j 0.8054$ and $0.93 e^{j \frac{\pi}{3}}$, you'd get the same answer. However, the two answers that you just got would not be the same. We just use the notation to make life a little simpler. The nice thing about this, and the thing to remember is the way that the 0.93 and the $\frac{\pi}{3}$ on the left-hand side of Equation 1.66 correspond to the same numbers on the right-hand side of Equation 1.67. Of course, now the question is ``Why the hell do we go through all of this hassle?'' Well, the answer lies in the simplicity of dealing with complex numbers represented as exponents, but I will leave it to other books to explain this. A good place to start with this question is The Scientist and Engineer's Guide to Digital Signal Processing by Steven W. Smith and found at http://www.dspguide.comwww.dspguide.com. Figure 1.16: A quick guide to what things mean when you see a cosine wave expressed in exponential notation. Note that both angles ($\phi$ and $\theta$) can be expressed in either degrees or radians - just make sure that they're both the same.