Chapter 1

In Exercises 1 through 9 compute the given arithmetic expression and give the answer in the form \(a+b t\) for \(a, b \in \mathbb{R}\). $$ i^{3} $$

Compute the given arithmetic expression and give the answer in the form \(a+b t\) for \(a, b \in \mathbb{R}\). $$ i^{4} $$

\(i^{4}\)

Compute the given arithmetic expression and give the answer in the form \(a+b t\) for \(a, b \in \mathbb{R}\). $$ (-i)^{35} $$

4\. \((-i)^{35}\) 7. \((2-3 i)(4+i)+(6-5 i)\)

Compute the given arithmetic expression and give the answer in the form \(a+b t\) for \(a, b \in \mathbb{R}\). $$ (4-i)(5+3 i) $$

5\. \((4-i)(5+3 i)\)

Compute the given arithmetic expression and give the answer in the form \(a+b t\) for \(a, b \in \mathbb{R}\). $$ (8+2 i)(3-i) $$

6\. \((8+2 i)(3-i)\)

Compute the given arithmetic expression and give the answer in the form \(a+b t\) for \(a, b \in \mathbb{R}\). $$ (2-3 i)(4+i)+(6-5 i) $$

7\. \((2-3 i)(4+i)\) 10. Find \(|3-4 i|\)

Compute the given arithmetic expression and give the answer in the form \(a+b t\) for \(a, b \in \mathbb{R}\). $$ (1+i)^{3} $$

8\. \((1+i)^{3}\)

Compute the given arithmetic expression and give the answer in the form \(a+b t\) for \(a, b \in \mathbb{R}\). $$ (1-i)^{5} \text { (Use the binomial theorem.) } $$

9\. \((1-i)^{5}\) (Use the binomial theorem)

10\. Find \(|3-4|\).

11\. Find \(\mid 6+4 i]\).

In Exercises 12 through 15 write the given complex number \(z\) in the polar form \(|z|(p+q i)\) where \(|p+q i|=1\). 12\. \(3-4 i\)

Write the given complex number \(z\) in the polar form \(|z|(p+q i)\) where \(|p+q i|=1\). $$ -1+1 $$

13\. \(-1+1\)

Write the given complex number \(z\) in the polar form \(|z|(p+q i)\) where \(|p+q i|=1\). $$ -3+5 i $$

In Exercises 16 through 21 , find all solutions in \(\mathrm{C}\) of the given equution. $$ z^{4}=1 $$

Find all solutions in \(\mathrm{C}\) of the given equution. $$ z^{4}=-1 $$

17\. \(z^{4}=-1\)

Find all solutions in \(\mathrm{C}\) of the given equution. $$ z^{3}=-8 $$

18\. \(z^{3}=-8\)

Find all solutions in \(\mathrm{C}\) of the given equution. $$ z^{3}=-27 i $$

19\. \(z^{3}=-27 i\)

Find all solutions in \(\mathrm{C}\) of the given equution. $$ z^{6}=1 $$

Find all solutions in \(\mathrm{C}\) of the given equution. $$ z^{6}=-64 $$

21\. \(z^{6}=-64\)

In Exercises 22 through 27 , compute the given expression using the indicated modular addition. $$ 10+_{17}16 $$

23\. \(8+106\)

In Exercises 22 through 27 , compute the given expression using the indicated modular addition. $$ 20.5+_{25}19.3 $$

25\. \(\frac{1}{2}+1 \frac{1}{8}\)

26\. \(\frac{3 \pi}{4}+3 \frac{37}{2}\)

27\. \(2 \sqrt{2}+\sqrt{31} 3 \sqrt{2}\)

In Exercises 29 through 34 , find all solutions \(x\) of the given equation. $$ x+157=3 \text { in } \mathbb{Z}_{15} $$

Example \(1.15\) asserts that there is an isomorphism of \(U_{8}\) with \(\mathrm{Z}_{8}\) in which \(\zeta=e^{i(x / 4]} \leftrightarrow 5\) and \(\zeta^{2} \leftrightarrow 2\). Find the element of \(\mathbb{Z}_{8}\) that corresponds to each of the remaining six elements \(\zeta^{\mathrm{in}}\) in \(U_{\mathrm{g}}\) for \(m=0,3,4,5,6\), and \(7 .\)

There is an isomorphism of \(U_{7}\) with \(Z_{7}\) in which \(\zeta=e^{(2 \pi / 7)} \leftrightarrow 4\). Find the element in \(Z_{7}\) to which \(\zeta^{m}\) must correspond for \(m=0.2 .3,4,5\), and 6 .

Why can there be no isomorphism of \(U_{6}\) with \(Z_{6}\) in which \(\zeta=e^{i(a / 3)}\) corresponds to \(4 ?\)

Derive the formulas $$ \sin (a+b)=\sin a \cos b+\cos a \sin b $$ and $$ \cos (a+b)=\cos a \cos b-\sin a \sin b $$ by using Euler's formula and computing \(e^{i a} e^{i b}\).

Derive the fornulas $$ \sin (a+b)=\sin a \cos b+\cos a \sin b $$ and $$ \cos (a+b)=\cos a \cos b-\sin a \sin b $$ by using Euler's formula and computing \(e^{j u} e^{i b}\).

a. Dertive a formula for \(\cos 3 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\) using Euler's formula. b. Derive the formula \(\cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta\) from part (a) and the identity \(\sin ^{2} \theta+\cos ^{2} \theta=1\). (We will have use for this identity in Section 32.)

a. Dertve a formula for \(\cos 3 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\) using Euler's formula. b. Derive the formula \(\cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta\) from part (a) and the identity \(\sin ^{2} \theta+\cos ^{2} \theta=1\). (We will have use for this identity in Section 32.)