By P. Howard

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**Additional info for Solving Ordinary Differential Equations with Matlab**

**Example text**

On the other hand, if s::/:nq, we can write s=nq+r, where O

Write down all the sixth roots of unity in the form a+ib (a. " real). 2. I'. 3. )1/8. 4. Show that if \"'\=1 and gt",= -i. then .... ""'1. s. Prove that. i~3.. (i) aI+11'=(a+b-)(aw+bwl)(awl +bw), (ii)(a+b+c)8+(a+bw+cuff+(a+bw'+cw)8=3(aI+11'+£,+6abc). 6. Show that the complex numbers 0, ". " correspond to the vertices of an equilateral triangle if and only if "1+"1=",,. 7. Deduce from Ex. 6 that the numbers "'I' "'I' .... correspond to the vertices of an equilateral triangle if and only if -1+·1+-1=·...

A solution of this equation is called an nIh root of unity. Evidently, 02'=1 is a solution, whatever the value of n, and this is clearly the only real 33 ROOTS OF UNITY positive solution. When n is even, Z= - 1 also satisfies the equation, but no real number other than ±1 can possibly occur amongst the roots of unity for any n. However, the situation is quite different, when we admit complex numbers as solutions. 1). 1), we find that z1I=lzl"=1. 1) and hence Izi =1. Therefore, a root of unity is necessarily a number of unit modulus, and we may put z=exp i8.