Multiplying a non-zero complex number by \( \frac{1 - i}{1 + i} \) results in a rotation about the origin on an Argand diagram - HSC - SSCE Mathematics Extension 2 - Question 5 - 2016 - Paper 1

To find the rotation produced by multiplying by the complex number ( \frac{1 - i}{1 + i} ), we first need to calculate its argument. We can express this division in polar form by calculating the modulus and argument of the numerator and the denominator.

Modulus:

• For ( 1 - i ): $\sqrt{1^2 + (-1)^2} = \sqrt{2}$

• For ( 1 + i ): $\sqrt{1^2 + 1^2} = \sqrt{2}$

Argument:

• For ( 1 - i ): $\tan^{-1}\left(-1\right) = -\frac{\pi}{4}$ (in the fourth quadrant)

• For ( 1 + i ): $\tan^{-1}\left(1\right) = \frac{\pi}{4}$ (in the first quadrant)

Combining these:

The argument of ( \frac{1 - i}{1 + i} ) is: $\text{arg}(1 - i) - \text{arg}(1 + i) = -\frac{\pi}{4} - \frac{\pi}{4} = -\frac{\pi}{2}$

This indicates a rotation of ( -\frac{\pi}{2} ) radians, meaning a clockwise rotation.