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ideas

 * proofs of De Moivre and Euler
 * explanation of the Argand diagram (Im Z, Re Z)

Manipulation
The basic mathematical operations are done the same way as with reals keeping in mind that $$i^2 = -1$$.

Adding complex numbers is done by separately summing the real parts and the imaginary parts

$$\,(a_1+b_1i)+(a_2+b_2i)=(a_1+a_2)+(b_1+b_2)i$$

$$\,(a_1+b_1i)-(a_2+b_2i)=(a_1-a_2)+(b_1-b_2)i$$

and multiplication is done the same way as with reals

$$\,(a_1+b_1i)\times(a_2+b_2i)$$ $$\,=a_1(a_2+b_2i)+b_1i(a_2+b_2i)=a_1a_2+a_1b_2i+a_2b_1i+b_1b_2i^2$$

but $$i^2=-1$$, so $$\,b_1b_2i^2=-b_1b_2$$ and $$(a_1+b_1i)\times(a_2+b_2i)=(a_1a_2-b_1b_2)+(a_1b_2+a_2b_1)i$$.

To do division on complex numbers we must first explain the concept of the complex conjugate.

Complex conjugate
The complex conjugate of a complex number Z=x+iy is the number x-iy denoted Z*, the sum and the product of a complex number and its conjugate are always real, because $$(x+iy)(x-iy)=(x^2-y^2)+i(xy-xy)=x^2-y^2$$ and $$(x+iy)+(x-iy)=2x$$. This property of the complex conjugate is useful when doing division on complex numbers, to convert the denominator of a complex fraction to a real number, all we have to do is to multiply the numerator and the denominator by the complex conjugate of the denominator.

$$\,\frac{x+iy}{a+ib}=\frac{(x+iy)(a-ib)}{(a+ib)(a-ib)}=\frac{(xa+yb)+i(ay-xy)}{a^2-b^2}$$

$$\,=\frac{xa+yb}{a^2-b^2}+i\frac{ay-xy}{a^2-b^2}$$

Polar representation of complex numbers
In the Argand diagram, any complex number can be determined by its distance from the origin (the point of origin is the point 0+0i) and the angle it makes with the positive side of the real axis. The number's distance from the origin is called its modulus and the angle it makes with the positive side of the real axis is called its argument. The modulus (|Z|) of a number Z can be calculated using the pythagorean theorem and is equal to $$\sqrt {x^2+y^2}$$, the argument (arg Z) equals $$tan^{-1} \frac y x$$.