Warm Up: Divide (Write your answer in standard form)
1 + 4i
6 ­ 2i
de Moivre, Abraham
The French­born mathematician Abraham de Moivre (May 26, 1667 ­ Nov. 27, 1754) was a pioneer in PROBABILITY theory and TRIGONOMETRY. He discovered the approximation of the BINOMIAL DISTRIBUTION known as the NORMAL DISTRIBUTION. He also investigated mortality statistics and the foundation of the theory of annuities and devised DE MOIVRE'S THEOREM, a trigonometric formula for obtaining powers and roots of complex numbers. A French Protestant, de Moivre emigrated (1685) to England following the revocation of the Edict of Nantes. In 1697 he was elected a fellow of the Royal Society. His book The Doctrine of Chances (1718) contained major advances in probability theory. Despite his scientific eminence, he subsisted mainly by tutoring mathematics and died in poverty. Bibliography: Smith, David E., History of Mathematics, vol. 1 (1923; repr. 1958). Product of Two Complex Numbers (in trig form)
z1 = r1 (cos θ1 + i sin θ1)
z2 = r2 (cos θ2 + i sin θ2)
z1z2 = r1r2[cos (θ1 + θ2) + i sin (θ1 + θ2)]
1
Example 1: Find the product of z1 = 2 cis 2π and z2 = 8 cis 11π
3 6
2
Quotient of Two Complex Numbers (in trig form)
z1 = r1 (cos θ1 + i sin θ1)
z2 = r2 (cos θ2 + i sin θ2)
z1 = r1 [cos (θ1 ­ θ2) + i sin (θ1 ­ θ2)] , z2 = 0
z2 r2
Example 2: Find the quotient of z1 = 24 cis 300 and z2 = 8 cis 75 3
de Moivre's Theorem
zn = rn (cos nθ + i sin nθ)
Ex 3: Find [3(cos 150 + i sin 150 )]4
Example 4: How would you find (1 + i)7 ? 4
Pg. 441 #51­66 (x3), 73­79 odd
5