Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers z_{1} = r_{1}(cos φ_{1} + i sin φ_{1}) and z_{2} = r_{2}(cos φ_{2} + i sin φ_{2}), because of the well-known trigonometric identities
- $ \cos(a)\cos(b) - \sin(a)\sin(b) = \cos(a + b) $
- $ \cos(a)\sin(b) + \sin(a)\cos(b) = \sin(a + b) $
we may derive
- $ z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).\, $
In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by i corresponds to a quarter-turn counter-clockwise, which gives back i^{2} = −1. The picture at the right illustrates the multiplication of
- $ (2+i)(3+i)=5+5i. \, $
Since the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or π/4 (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula
- $ \frac{\pi}{4} = \arctan\frac{1}{2} + \arctan\frac{1}{3} $
holds. As the arctan function can be approximated highly efficiently, formulas like this—known as Machin-like formulas—are used for high-precision approximations of π.
Similarly, division is given by
- $ \frac{z_1}{ z_2} = \frac{r_1}{ r_2} \left(\cos(\varphi_1 - \varphi_2) + i \sin(\varphi_1 - \varphi_2)\right). $
References
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