Questions 1 - 9#

These questions examine various properties of complex numbers.

Q1#

If \(z_1 =2+i\) and \(z_2 =-1-3i/2\) calculate (a) \(z_1 + z_2\) (b) \(z_1 - z_2\)
(c) \(-iz_1z_2\)

(a) If \(i^2=-1\), what are \(i^3,\;i^4,\;i^5\), and \(i^6\)?

(b) What relationship links positive powers of i?

If \(z = a - ib\) what is \(i^3z\)?

If \(z=3+4i\) find \(z^2\) and the modulus and argument of \(z^2\).

Calculate \(z = (2 - 5i)(3 + i) + 3i\), and find the modulus and argument of the result.

Express the number \(\displaystyle \frac{5 - i}{2-3i}\) in the form \(z = a + bi\) and find its modulus and argument.

Simplify \(z = (2 - 5i)(3 + i)/(3 - i)\) and find the modulus and argument of the result.

Find the modulus and argument of

(a) \(\cos(\theta) - i \sin(\theta)\),

(b) \(1-i\tan(\theta)\), where \(0 \lt \theta \lt \pi/2\) in both cases.

If \(w=z^2\) and \(z=x+iy\) and \(w=u+it\), find \(u\) and \(t\).