By Evans, Les.
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Extra resources for Complex Numbers and Vectors (MathWorks for Teachers)
1 + i 2 - 3i 2 + 3 + 2i - 3i 2 + 3i # 2 - 3 i = 4+9 5-i = 13 i 5 = 13 - 13 In general terms: z1 x1 + iy1 z2 = x2 + iy2 x + iy x - iy = x 1 + iy1 # x2 - iy2 2 2 2 2 = x1 x2 + y1 y2 + i^x2 y1 - x1 y2h x22 + y22 This process is analogous to the rationalisation of a real surd expression. 5 allows students to explore not only the operation of complex number divisions, but also some of its geometric properties. 5 Consider the complex numbers z1 = 1 + i , z2 = 1 + 3 i and z3 = 3 + i . a b z z z Evaluate z1 , z1 and z2 .
5: An Argand diagram plotting the complex number z = 3 + 2i (z = x + yi where x, y d R and i 2 = - 1) It was through the work of mathematicians who used their imaginations, and did not allow the boundaries of their own or of the creations of others to limit their thinking, that we can come to the realisation that it is possible to visualise a ‘length’ that is less than zero. It may not occur in the realm of ‘reality’, we need to move into the world of imagination. If we concentrate and refuse to limit our view of what may be possible, we can suddenly bring into our view of the world a ‘length’ that can be measured as a negative.
We can obtain results using technology. Exact answers can be obtained by using equation-solving functions of computer algebra systems, and programs that will provide roots for polynomial equations are freely available for graphics calculators. This family of graphics calculator programs will give roots and factors and perform division of polynomials. 3-i 7 3+i 7 and are a conjugate pair. It is worth noting that 2 2 We would expect a conjugate pair when factorising a quadratic, given the nature of the quadratic formula.