Complex Numbers
Problems
- Find the roots of $x^4-1=0$.
- What is the value of $(1+i)(1+i^2)(1+i^3)(1+i^4)$?
- What is the value of $1+ i^2 + i^4 + i^6 + \dots i^{2n}$?
- What is the smallest positive integer $n$ such that $\left( \frac{2 i}{1+i}\right )^n$is a positive integer?
- Explain why in general the expression $a + i b > c + i d$ is not meaningful for complex numbers. Are there any special values of $a,b,c,d$ where such inequalities become meaningful?
- Suppose $\theta_1, \theta_2, \theta_3$ are the three angles (in radian) of a triangle. What is the value of $e^{i \theta_1} e^{i \theta_2} e^{i \theta_3}$?
- Let $\omega$ be one of the complex cubic roots of unity (i.e. $\omega$ is a root of $x^3-1=0$). Then simplify the following expression
$$
(1-\omega)(1-\omega^2)(1+\omega^4)(1+\omega^8)
$$
- If $\omega$ is the same as in the previous problem. Then simplify
$$
2 (3 + 5 \omega + 3\omega^2) + (3+ 3 \omega + 5 \omega^2)
$$
- Find the square root of $5+12 i$ and $-15-8i$ and express them in the algebraic form (i.e. $a+ i b$ form)
Find even more problems in the attached worksheet
worksheet.pdf
Vectors and Matrices
For the following vectors $\ket{u}$ and $\ket{v}$, find the inner products $\braket{u|v}$ and $\braket{v|u}$. Also, find the norms $||u||$ and $||v||$. (Problems 1 -3).
- $\ket{u} = 5i \ket{b_1} + e^{i \theta} \ket{b_2}$ , $\ket{v} = 3 \ket{b_1} + 2e^{i \theta} \ket{b_2}$ , where $\theta$ is a real constant and $\ket{b_i}$ form an orthonormal basis.