What is a complex number?

A complex number is a number that consists of two parts, a real part, like any number you have been dealing with since you started learning maths, and an imaginary number, often tagged along with a i or a j, for example a+bi. The best way to imagine it is as a vector on a plane, the point not unlike one on a x,y axes but instead on a real, imaginary axes. 

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(FP1) Given k = q + 3i and z = w^2 - 8w* - 18q^2 i, and if w is purely imaginary, show that there is only one possible non-zero value of z


The curve C has parametric equations x=cos(t)+1/2*sin(2t) and y =-(1+sin(t)) for 0<=t<=2π. Find a Cartesian equation for C. Find the volume of the solid of revolution of C about the y-axis.


Prove by induction that for all positive integers n , f(n) = 2^(3n+1) + 3*5^(2n+1) , is divisible by 17.


Use De Moivre's Theorem to show that if z = cos(q)+isin(q), then (z^n)+(z^-n) = 2cos(nq) and (z^n)-(z^-n)=2isin(nq).