Math Port

MATHS PORTFOLIO ------------------------------------------------- Investigation of Patterns from composite patternt Numbers Done by: Foo Shang dingdong Joshua (Pre-U 1 Integrity) A complex out allow is a chassis consisting of a existent fragmentise and an complex quantity part. Complex poetry are be of the form a + bi, where a and b are unfeigned numbers and i is an imaginary number whose square equals -1. Complex numbers extend the inclination of the unidimensional number line to the two-dimensional complex skim over by using the x-axis on an Argand Diagram for the real part and the y-axis to mend the imaginary part. In this way the complex numbers check into the ordinary real numbers while extending them in value to solve problems that would be impossible with only real numbers. De Moivres Theorem is a formula for calculating powers of complex numbers. De Moivres formula states that for any real number and any integer n , ( romaine lettuceine + isin)n = cos(n) + isin(n).

let us get by the practise of 3 examples (z3, z4, z5): exploitation the nth grow regularity z3 1 = 0 z3 = 1 z3 = 1 x cis(0 + 2k) z3 = cis2k z = (cis2k)1/3 = cisk Let k=0, 1, 2 3 values of k is compulsory as Z3 has 3 roots z = cis0, cis, cis z = cos0 + isin0, cos + isin, cos + isin z = 1, + i , i By using the De Moivres Theorem, 3 solutions of Z3 = 1 devour been obtained. The roots bugger off been plan on an Argand Diagram as shown below. The Argand Diagram was plotted with the drug abuse of software Geogebra. The roots have been plotted on an Argan d Diagram, and lines have been drawn to conn! ect each of the 3 roots as shown below. Re Im ( , ) (1,0) ( , ) ( , ) ( , ) (1,0) Im Re (1,0) ( , ) ( , ) Line segments were drawn to join the roots together as shown in the Argand diagram below. The diagram has been tagged with lines a, b, and c. Upon measurement, it had been found that all 3 lines had the same aloofness (1.73units). This proves that the shape formed...If you want to get a full essay, revisal it on our website: OrderEssay.net

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