A Treatise on the Theory of Determinants by Thomas Muir, revised by W.H. Metzler

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By Thomas Muir, revised by W.H. Metzler

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3 Consider the complex numbers z1 = 1 + i , z2 = 1 + 3 i and z3 = 3 + i . a Express z1, z2 and z3 in polar form. b Evaluate z1 z2 , z1 z3 and z2 z3 using the polar form. c Plot z1, z2 and z1 z2 on an Argand diagram. d Plot z1, z3 and z1 z3 on an Argand diagram. e Plot z2 , z3 and z2 z3 on an Argand diagram. DIVISION When we divide one complex number by another we are attempting to simplify the expression z1 x1 + iy1 z2 = x2 + iy2 z 1+i For example if z1 = 1 + i and z2 = 2 + 3i then z1 = 2 + 3i .

Comment on your findings. 53 MATHSWORKS FOR TEACHERS Complex Numbers and Vectors It is possible to solve equations of the form zn - an where n ! N and a ! R using the polar form of complex numbers. This can be demonstrated by starting with the complex number z = rcis]qg. We will start by raising z to a series of powers beginning with z2. z2 = r # rcis]q + qg = r2 cis]2qg z 3 = r # r2 cis]2q + qg = r 3 cis]3qg z 4 = r # r 3 cis]3q + qg = r 4 cis]4qg This pattern continues to zn = rn cis]nqg. q q This would also suggest that n z = z = r cis b n l = n r cis b n l .

1 1 2 Express the following cubic expressions in depressed form. a x3 - 15x2 + 81x - 175 b x3 + 8x2 + 25x + 26 c 2x3 - 21x2 + 68x - 29 d 2x3 - 25x2 + 102x - 130 Using the depressed form, find the roots of these cubic expressions. It is also possible to use computer algebra to solve cubic equations for the general case and for specific examples. It would be opportune to compare results using the methods available to Fontano with those that can be achieved using modern technology. We can use computer algebra systems to find the depressed form of the general cubic equation.

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