![]() ![]() ![]() Are you familiar with representing complex numbers in polar form? e i Ø = cosØ + isinØ, so if z = re i ø, this is a number with magnitude r and forming an angle ø with the real axis. The fact that it's (-b,a) and not (b,-a) (which is also perpendicular) makes it a rotation by 90 degrees counterclockwise. Multiply (a+bi)*i = ai-b right? This corresponds to a vector (-b,a) which is perpendicular to (a,b) (take the dot product). This could also represent a vector (a,b). This is more of an explanation of why we define it in this way.Ī simpler explanation: Suppose you have a+bi in the complex plane. In a construction I think it's more usual to define multiplication by i as a rotation in pi/2. This isn't a formal construction, but it provides a first idea of why it is. i has a modulus of 1 and an argument of pi/2, because of this, multiplication by i represents a turn in pi/2. This is how we multiply in the complex plane, we multiply the moduli together and add the arguments. Then we can also show that R e iT * r e it = Rr e i(T+t) Every complex number can be represented uniquely by Rcos(t)+iRsin(t) where R>=0 and 0<=t<2pi (because every point on a plane can be represented as such). The modulus represents the distance on the complex plane between the point and the origin, and the argument represents the angle. Because of this, we can define a modulus and an argument to any complex number. It can be proven that defining e it = cos(t) + i sin(t) has similar properties to the real function e x (hence the notation) - in fact, their power series are the same. The way most people are introduced to them is (I think) by just introducing a number i so that i 2 =-1 and relying on the fact that we can manipulate it as if it were any other number. Well there's a hundred ways of looking at it, depending on how you want to construct the set of complex numbers. ![]()
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