Complex numbers of the form r(cos φ + i sin φ) and re^(iφ)
We show z = a + bi as the point (a, b) in the plane. The distance from the origin to this point, r = √(a² + b²), is the modulus |z|, and the angle φ from the positive Ox direction to this vector is the argument; a = r cos φ, b = r sin φ. The argument is taken in 0 ≤ φ < 2π and found from the quadrant of the point: do not rely on tan φ = b/a alone, since it gives wrong angles in quadrants II and III. z = r(cos φ + i sin φ) is the trigonometric form and z = r · e^(iφ) is the exponential form (Euler’s formula e^(iφ) = cos φ + i sin φ). For example i = cos π/2 + i sin π/2 = e^(iπ/2) and −1 = e^(iπ). To go from trigonometric to algebraic form, compute cos and sin and multiply by r. The argument of zero is undefined.
On graph paper mark z = −√3 + i roughly (√3 ≈ 1.7), measure the angle of vector OA with a protractor and check that it is close to 150°, i.e. 5π/6 = 150°.