Click here to purchase the entire book in PDF format. Absolute Value (also known as the Modulus) The absolute value of a complex number is a little weirder than what we usually think of as an absolute value. In order to understand this, we have to look at complex numbers a little differently: Remember that $j * j = -1$. Also, remember that, if we have a cosine wave and we delay it by $90^{\circ}$ and then delay it by another $90^{\circ}$, it's the same as inverting the polarity of the cosine, in other words, multiplying the cosine by -1. So, we can think of the imaginary component of a complex number as being a real number that's been rotated by $90^{\circ}$, we can picture it as is shown in Figure 1.12. Figure 1.12: The relationship bewteen the real and imaginary components for the number $(2 + 3j)$. Notice that the X and Y axes have been labeled the ``real'' and ``imaginary'' axes. Notice that Figure 1.12 actually winds up showing three things. It shows the real component along the x-axis, the imaginary component along the y-axis, and the absolute value or modulus of the complex number as the hypotenuse of the triangle. This is shown in mathematical notation in exactly the same way as in normal math - with vertical lines. For example, the modulus of $2+3i$ is written $\vert 2+3i \vert$ This should make the calculation for determining the modulus of the complex number almost obvious. Since it's the length of the hypotenuse of the right triangle formed by the real and imaginary components, and since we already know the Pythagorean theorem then the modulus of the complex number $(a + jb)$ is $$\vert a + jb \vert = \sqrt{a^{2} + b^{2}}$$ (2.53) Given the values of the real and imaginary components, we can also calculate the angle of the hypotenuse from horizontal using the equation $$\phi = \arctan \frac{imaginary}{real}$$ (2.54) $$\phi = \arctan \frac{b}{a}$$ (2.55) This will come in handy later.