First Principles AI

Questions

Question 1
Find the multiplicative inverse of each complex number.
(i)
(-4, 7)
(ii)
(√2, -√5)
(iii)
(1, 0)
Solution Theory
Question 2
Separate the real and imaginary parts (write as a + bi).
(i)
(2 - 7i) / (4 + 5i)
(ii)
((-2 + 3i)^2) / (1 + i)
(iii)
i / (1 + i)
(iv)
((4 + 3i)^2) / (4 - 3i)
Solution Theory
Question 3
Prove that a complex number equals its conjugate if and only if it is real.
Solution Theory
Question 4
For z in C, show the following identities.
(i)
(z + z̄)/2 = Re(z)
(ii)
(z - z̄)/(2i) = Im(z)
(iii)
|z|^2 = z·z̄
Solution Theory
Question 5
Given z1, z2, z3, express a required expression in the form a + bi.
()
If z1 = 2 + i, z2 = 3 - 2i, z3 = 1 + 3i, express (z1 z3)/z2 in the form a + bi.
Solution Theory
Question 6
If z1 and z2 are given, evaluate the following.
()
Let z1 = 2 + 7i and z2 = -5 + 3i. Evaluate:
(i)
|2z1 - 4z2|
(ii)
|3z1 + 2z2|
(iii)
|-7z2 + 2z2̄|
(iv)
(z1 + z2)^3
Solution Theory
Question 7
Show that a given expression involving powers of i equals 0 (for all n in N).
Solution Theory
Question 8
Find the least positive value of n satisfying a complex-number condition.
Solution Theory
Question 9
Show that a statement about i^r holds for n in N with a remainder condition.
Solution Theory