Physics 214 1/23/07 Notes on Complex Numbers Let a, b, x, y, r, φ , θ and ρ be real numbers. Defne i as i 2 =-1 so, For b n = 0, ib is a purely imaginary number and a + ib is a complex number. Any complex number z can be written in 2 ways (cartesian and polar) : z = x + iy = re iθ (1) An important identity is the Euler Identity : e iθ = cos θ + i sin θ (2) Combining the above 2 equations, we obtain z = x + iy = re iθ = r cos θ + ir sin θ (3) so we have x = r cos θ and y = r sin θ Consider the 2-dimensional xy-plane : every complex number z corresponds to a point in the xy-plane. We call this xy-plane the complex plane. Here, x = Re ( z ) is the real part oF z , y = Im ( z ) is the imaginary part oF z , θ is the phase oF z , and r = r x 2 + y 2 is the magnitude oF z . Let us consider the addition and the multiplication oF 2 complex numbers : z and w = a + ib = ρe iφ . Using the complex plane, it is easy to see that the addition oF z and w is given by z
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