Complex Numbers would definitely is one of my favourite topics in Pure Mathematics. Its a pity that H1 Mathematics will not to get to learn them.

Firstly, Complex Numbers is being used very widely in Engineering School, heads up. So its usefulness in real life is beyond just $z=x+iy$. The idea of represent two components into one is quite amazing. And because of these, many A-level questions are combining vectors with complex numbers and expect students to resolve them.

Secondly, the geometrical interpretations in Complex Numbers are really interesting. And yes, I do admit that there are times whereby examiners overkill the questions and expect students to see some unconceivable relationships in the Argand Diagram.

Thirdly, finding solutions for a mere $x^2 = -1$ is really helpful to exploring many channels in mathematics and open the floodgates to many possibilities in Mathematics. It allow us to solve equations, previously unsolvable. Of course, A-level students should be quick to relate the Conjugate Root Theorem here and expect questions combined with graphings, polynomials to come out to test their understanding on this theorems.

The last reason is quite nerdy. I find it really amazing how $1 = e^{i0}$. Yes many students will probably go meh. But here, we established a relationship between a real number, complex number and zero. Zero, itself is intriguing enough. And now we link these three components together into one equation. 🙂

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