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Homework 1 (MATH 1199 Fall 2019)
1. In the following problems, write the expression in the form a+bi for real numbers a and b.
(a) (2+i)(1−i)
(b) 21+i
(c) 1+i1−i+2i
2. Plot the number in the complex plane and calculate its modulus.
(a) 2+i
(b) 3−2i
(c) −2−7i
3. Draw a picture of the set in C described by the inequality.
(a) |z−1|=2
(b) |z−(1+i)|<1
(c) |z+1|≥2
4. Use the properties of complex conjugates to establish the following identities.
(a) ¯¯z−2i=z+2i
(b) ¯iz=−i¯z
5. Find the principal argument Arg of the following numbers.
(a) √22+√22i
(b) √32−12i
(c) 1−2−2i
6. Use Euler's formula, eiθ=cos(θ)+isin(θ) for θ∈R to compute e5π3i, and compute its modulus |e5π3i|.
7. Write the number in polar form z=reiθ by finding r=|z| and θ=Arg(z).
(a) 1+2i
(b) 1i
(c) (1−i)(1+i)
8. Write the number given in polar form in standard form z=a+bi by using Euler's formula appropriately.
(a) z=2eπ2i
(b) z=8e−5π6i
(c) z=22e14πi