Complex numbers: One of the main mathematical concepts when it comes to describing physical systems, such as signals, and circuit analysis! Let's delve deep into what complex numbers are and more about its applications. What makes a number "complex"? And what is the "real" definition of a "real" number? (ha!) real vs. imaginary numbersReal numbers consist of integers, whole numbers, positive or negative, rational numbers and irrational numbers (or a combination of those!!). Imaginary Numbers on the other hand, consist of numbers with a property where it's square is negative. An imaginary number, i, represents the square root of -1, where i^2 = -1. NOTE: Sometimes imaginary numbers are denoted with a j. a complex numberA complex number is a number that's made up of a real part, and an imaginary part (that's what makes it 'complicated'). A complex number (usually denoted as z), is written in the form: a + bi, where a represents the real part of the complex number, and b represents the imaginary part of the complex number z. Note that a and b are real numbers, but since b is being multiplied by an imaginary number i, b is the imaginary part, or Im(z). A really interesting part about a complex number is that it can be represented graphically, just like how we plot coordinates in a Cartesian plane. (Like (2,4) on the Cartesian plane for example). The difference is we plot complex numbers in the Complex Plane, where the vertical axis is the imaginary axis (where imaginary numbers are plotted - i, 2i, 3i..), and the horizontal axis is the real axis (where real numbers are plotted - 1, 2, 3..). To plot a complex number a + bi, you would essentially move a steps horizontally (horizontal real axis) from the origin, and then b steps vertically or perpendicularly (vertical imaginary axis). The the magnitude of z, represents the distance from the origin [(0, 0)] to the complex number z's point. The distance is calculated by sqrt((difference in x axis)^2 + (difference in y axis)^2). So it's calculated by adding the squares of a and b and taking the square root of the sum. (Just like how you would calculate the distance in the cartesian plane or in a map, for example). For any real number, a one -dimensional number line would satisfy to plot the number. But for a complex number, we would require a 2 dimensional plane to plot it's imaginary part and real part. The really interesting part of this complex number representation is the angle, or the phase. To calculate the angle, we can use the distance between the origin and the point, r! We can say that a, the horizontal portion is equal to the cos(of the phase angle) * r. (Since cosine is adjacent over hypotonuse) and b the vertical portion is equal to sin(of the phase angle) * r. The tan of (the phase) is then equal to b/a! Since the r's cancel out when dividing. So you can then find the phase of the complex number (also called argument of z) by finding inverse tan of b/a. euler's formulaThe Euler's Law relates the these cosines and sines we see above with a complex exponential using a revolutionary formula! The Euler's formula states that: So essentially the component to the left of the equation is a complex exponential function, e raised to the power of (imaginary number * a phase/angle x). This is equal to the cos(x) + imaginary number i * sin(x). What makes this formula so intriguing is that there is a way to represent sines and cosines using complex exponentials, which can be very useful in a lot of applications. applicationsThere are numerous applications of complex numbers and complex exponentials! One of the applications is in Signals and Systems/Processing, where you can express periodic signals as a bunch of sines and cosines (sinusoidal functions in Fourier Analysis). These sinosoids can also be expressed as complex exponentials, since in LTI systems (Linear and Time Invariant systems), complex exponential inputs can make calculations much more efficient/easier! (These are also called eigenunctions of LTI systems!) Thank you!
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