LaTeX Exercise 1

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1 - Using LaTeX   % This is meant to be a section heading

LaTeX is a logical typesetting language % those last three words should be emphasised 
so headings such as the one just above should be put in with LaTeX commands. The same applies to lists like the one 
in section 2.   % Break the paragraph here
%Also give section 2 a name using \label and replace the "2" by a \ref
The LaTeX referencing system is very powerful and you should use it where possible. It saves a lot of   time in the long run. 





2 - Mathematics in LaTeX   % This is meant to be a section heading


Here are three important properties of the complex numbers C.  

(1) If z is a complex number then |z|=1 if and only if z lies on the unit circle in the complex plane. 

(2) If z=x+iy and w=u+iv then wz= (xu-yv) + i(xv+yu). %that last equation should be displayed on a line on its own

(3) Every polynomial with complex coefficients has a solution in the complex numbers.  This is called the "Fundamental theorem of Algebra".   % do those inverted commas look right? 

(4) The exponential function can be defined on complex numbers and

    exp(i theta) = cos theta + i sin theta.
    
    So, in particular, 
 
     i pi
    e       = -1. 


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