Chapter 2, Introduction to Complex Numbers Video Solutions, Linear Algebra

Problem 1

Express the following complex numbers in the form $x+y i$ for $x, y \in \mathbb{R}$ :
(a) $(2+3 i)+(4+i)$
(b) $(2+3 i)^{2}(4+i)$
(c) $\frac{2+3 i}{4+i}$
(d) $\frac{1}{i}+\frac{3}{1+i}$
(e) $(-i)^{-1}$
(f) $(-1+i \sqrt{3})^{3}$

Nick Johnson

Numerade Educator

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Problem 1

Let $n \in \mathbb{Z}_{+}$ be a positive integer, let $w_{0}, w_{1}, \ldots, w_{n} \in \mathbb{C}$ be distinct complex numbers, and let $z_{0}, z_{1}, \ldots, z_{n} \in \mathbb{C}$ be any complex numbers. Then one can prove that there is a unique polynomial $p(z)$ of degree at most $n$ such that, for each $k \in\{0,1, \ldots, n\}, p\left(w_{k}\right)=z_{k}$
(a) Find the unique polynomial of degree at most 2 that satisfies $p(0)=0, p(1)=1$, and $p(2)=2$.
(b) Can your result in Part (a) be easily generalized to find the unique polynomial of degree at most $n$ satisfying $p(0)=0, p(1)=1, \ldots, p(n)=n ?$

Eduard Sanchez

Numerade Educator

01:56

Problem 2

Compute the real and imaginary parts of the following expressions, where $z$ is the complex number $x+y i$ and $x, y \in \mathbb{R}$ :
(a) $\frac{1}{z^{2}}$
(b) $\frac{1}{3 z+2}$
(c) $\frac{z+1}{2 z-5}$
(d) $z^{3}$

M Hassan Anwar

Numerade Educator

01:46

Problem 3

Find $r>0$ and $\theta \in[0,2 \pi)$ such that $(1-i) / 2=r e^{i \theta}$.
4. Solve the following equations for $z$ a complex number:
(a) $z^{5}-2=0$
(b) $z^{4}+i=0$
(c) $z^{6}+8=0$
(d) $z^{3}-4 i=0$

Linh Vu

Numerade Educator

04:16

Problem 4

Solve the following equations for $z$ a complex number:
(a) $z^{5}-2=0$
(b) $z^{4}+i=0$
(c) $z^{6}+8=0$
(d) $z^{3}-4 i=0$

Micah Hurewitz

Numerade Educator

02:32

Problem 5

Calculate the
(a) complex conjugate of the fraction $(3+8 i)^{4} /(1+i)^{1} 0$.
(b) complex conjugate of the fraction $(8-2 i)^{1} 0 /(4+6 i)^{5}$.
(c) complex modulus of the fraction $i(2+3 i)(5-2 i) /(-2-i)$.
(d) complex modulus of the fraction $(2-3 i)^{2} /(8+6 i)^{2}$.