We also get Euler’s formula, which I wrote about in a previous blog post: This is one of those really interesting formulas, since it takes two irrational numbers ( e and π) with the imaginary unit i, and the result is…a negative integer. With that, we can see that positive real numbers are just complex numbers where the phase φ = 0, and negative real numbers correspond to complex numbers with φ = π. One full circle is 2π radians (360°), a half-circle is π radians (180°), and a quarter circle is π/2 radians (90°). Now we can see that the phase φ represents an angle, but we need to use radians instead of degrees. Then the x- and y-coordinates of the end of the line are given by the cosine and sine of the angle as shown. The radius line (in blue) has a length of 1, and we’ll use that as the hypotenuse of a triangle. The legs of the triangle are the real and imaginary parts of the complex number. Think of it like this: the complex number is like one point on a right triangle, whose hippopotamus hypotenuse connects the number to the origin. We can write it as a sum of trigonometry functions: The exponential part of the equation is what we most care about, since r is just a scaling factor. The exponential number, like π, is a fundamental geometric quantity. Finally, e is the exponential number, roughly equal to 2.71828 (but going on to infinite number of digits). The relative weight to the real and imaginary portions are represented by φ (the Greek letter “phi”, pronounced either “fye” or “fee”, depending on how pedantic you want to be), which is known as the phase. Additionally, r is a positive number that represents the magnitude of the complex number: the distance from the origin of the complex plane to the point represented by z. All real (“ordinary”) numbers lie on the horizontal axis, and all imaginary numbers (which are just multiples of i) lie along the vertical axis.Īn equivalent but often more useful way of writing complex numbers uses the exponential function: In general, we write complex numbers as the sum of the real part and the imaginary part, and often find it useful to plot them on the complex plane, as shown at the left. Four points are plotted so you can see the correspondence between x and y coordinates and the real and imaginary parts of the complex numbers.Ĭomplex numbers are the combination of real and imaginary numbers. Complex numbers in boyshorts brief Illustration of the complex plane: the connection between complex numbers and points in two dimensions. People often ask the same question when they learn about imaginary numbers: if i is the square root of -1, what’s the square root of i? Is it a new kind of imaginary number, or can you write it in terms of “regular” imaginary numbers, or is it even a meaningful question? As you might expect, the question is not only meaningful, but leads to some interesting geometrical insights (even if most of us won’t find it as exciting as Zach Weiner’s character in his comic strip). Of course, imaginary numbers aren’t any more “imaginary” than many other useful mathematical constructs my feeling is that, if something in math can be used for something measurable, then it’s real in a meaningful sense. Most people are probably aware that the square root of -1 isn’t a regular real number: it’s the imaginary number, which we often write as i. The comic behind the link is very NSFW and very nerdy, so don’t click if you’re offended by either. When multiplying and dividing complex numbers we must take care to understand that the product and quotient rules for radicals require that both \(a\) and \(b\) are positive.A panel from “Saturday Morning Breakfast Cereal”.
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