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A Brief Introduction to Complex Numbers

Age 14 to 18 Challenge Level:
You are probably very familiar with $\mathbb{N}$, the set of natural numbers $1,2,3,4...$
The next set of numbers you met when you were younger might have been the integers, $\mathbb{Z}$, the positive and negative whole numbers.
You will also have met the rationals, $\mathbb{Q}$, numbers that can be written in the form $\frac{a}{b}$ where $a$ and $b$ are whole numbers which are coprime.
Finally, you will have come across irrational numbers such as $\sqrt2$ and $\pi$; these, together with the rationals, form the set of real numbers $\mathbb{R}$.

This problem introduces the set of complex numbers, $\mathbb{C}$

When you add together $2+2i$ and $3-i$, where does the real part of the answer come from? Where does the imaginary part of the answer come from?

What about when you multiply?