Problem Set - Number Systems and Functions

## Numbers

 What are the positive integers and why do we care? What are the nonnegative integers and why do we care? What are the rational numbers and why do we care? What are the real numbers and why do we care? What are the complex numbers and why do we care? What do $2+3,2+\frac{5}{2},\frac{4}{9}+\frac{5}{7},2+1.4$ and $2+\sqrt{2}$ mean? What do ${x}^{2},\frac{1}{x}$ and $\sqrt{x}$ mean? What do $a+b$, $a+\frac{b}{c}$ and $\frac{a}{b}+\frac{c}{d}$ mean? What do ${2}^{3},{2}^{\frac{5}{7}},{\left(\frac{2}{3}\right)}^{\frac{5}{7}},{2}^{x}$ and ${2}^{\sqrt{2}}$ mean? What do ${a}^{b},{a}^{\frac{b}{c}},{\left(\frac{a}{b}\right)}^{\frac{c}{d}},{2}^{x}$ and ${x}^{2}$ mean? What do ${x}^{x}$ and ${x}^{\sqrt{x}}$ mean?

## Computing with complex numbers

 Find a complex number $z$ such that $z+w=w$ for all complex numbers $w$. Find a complex number $z$ such that $zw=w$ for all complex numbers $w$. Compute $\left(3-7i\right)+\left(2+5i\right)$ and graph the result. Compute $\left(-12+3i\right)-\left(7-5i\right)$ and graph the result. Compute $\left(4+8i\right)\left(2-3i\right)$ and graph the result. Compute $\frac{-15+i}{4+2i}$ and graph the result. Compute ${\left(3-2i\right)}^{3}$ and graph the result. Compute $\sqrt{2i}$ and graph the result. Compute $\frac{1}{a+bi}$ and graph the result, where $a,b\in ℝ$. Compute $\left(3-5i\right)+\left(7+2i\right)$ and graph the result. Compute $\left(5-2i\right)-\left(3-6i\right)$ and graph the result. Compute $\left(2-4i\right)\left(3+2i\right)$ and graph the result. Compute $\frac{6-i}{4+2i}$ and graph the result. Compute ${1}^{\frac{1}{4}}$ and graph the result. Compute ${16}^{\frac{1}{4}}$ and graph the result. Compute ${\left({27}^{\frac{1}{3}}\right)}^{4}$ and ${27}^{\left(4+\frac{1}{3}\right)}$ and graph the result. Compute $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\cdots$. Compute $1·2,1·2·3,1·2·3·4,1·2·3·4·5$ and $1·2·3·4·5·6$. Compute $\frac{1}{1·2}+\frac{1}{1·2·3}+\frac{1}{1·2·3·4}+\frac{1}{1·2·3·4·5}+\frac{1}{1·2·3·4·5·6}+\cdots$.

## Functions

 What is ${x}^{2}$? What is ${e}^{x}$? What is $\mathrm{sin}x$? What is $\mathrm{cos}x$? What is $\mathrm{tan}x$? What is $\mathrm{cot}x$? What is $\mathrm{sec}x$? What is $\mathrm{csc}x$? What is $\mathrm{sinh}x$? What is $\mathrm{cosh}x$? What is $\mathrm{tanh}x$? What is $\mathrm{coth}x$? What is $\mathrm{sech}x$? What is $\mathrm{csch}x$? What is $\sqrt{x}$? What is $\mathrm{ln}x$? What is ${\mathrm{sin}}^{-1}$? What is ${\mathrm{cos}}^{-1}$? What is ${\mathrm{tan}}^{-1}$? What is ${\mathrm{cot}}^{-1}$? What is ${\mathrm{sec}}^{-1}$? What is ${\mathrm{csc}}^{-1}$? What is ${\mathrm{sinh}}^{-1}$? What is ${\mathrm{cosh}}^{-1}$? What is ${\mathrm{tanh}}^{-1}$? What is ${\mathrm{coth}}^{-1}$? What is ${\mathrm{sech}}^{-1}$? What is ${\mathrm{csch}}^{-1}$?

## Function identities

 Explain why $\frac{1}{1-x}=1+x+{x}^{2}+{x}^{3}+\cdots$. Explain why $\frac{{x}^{n}-1}{x-1}=1+x+{x}^{2}+\cdots +{x}^{n-1}$. Find all possibilities for ${c}_{0},{c}_{1},{c}_{2},\dots$ so that $f\left(x\right)={c}_{0}+{c}_{1}x+{c}_{2}{x}^{2}+\cdots$ satisfies $f\left(x+y\right)=f\left(x\right)f\left(y\right)$. Explain why ${e}^{x}=1+x+\frac{{x}^{2}}{2!}+\frac{{x}^{3}}{3!}+\frac{{x}^{4}}{4!}+\frac{{x}^{5}}{5!}+\frac{{x}^{6}}{6!}+\cdots$. Explain why $\mathrm{ln}x$ is the inverse function to ${e}^{x}$. Verify the identity ${e}^{x+y}={e}^{x}{e}^{y}$. Verify the identity ${e}^{-x}=\frac{1}{{e}^{x}}$. Verify the identity ${\left({e}^{x}\right)}^{n}={e}^{nx}$. Verify the identity ${e}^{0}=1$. Verify the identity $\mathrm{ln}\left(xy\right)=\mathrm{ln}x+\mathrm{ln}y$. Verify the identity $-\mathrm{ln}x=\mathrm{ln}\left(1/x\right)$. Verify the identity $\mathrm{ln}{x}^{n}=n\mathrm{ln}x$. Verify the identity $\mathrm{ln}1=0$. Explain why $\mathrm{cos}x=1-\frac{{x}^{2}}{2!}+\frac{{x}^{4}}{4!}-\frac{{x}^{6}}{6!}+\cdots$. Explain why $\mathrm{sin}x=x-\frac{{x}^{3}}{3!}+\frac{{x}^{5}}{5!}-\frac{{x}^{7}}{7!}+\cdots$. Verify the identity ${e}^{ix}=\mathrm{cos}x+i\mathrm{sin}x$. Verify the identity ${\mathrm{cos}}^{2}x+{\mathrm{sin}}^{2}x=1$. Verify the identity $\mathrm{sin}\left(-x\right)=-\mathrm{sin}x$. Verify the identity $\mathrm{cos}\left(-x\right)=\mathrm{cos}x$. Verify the identity $\mathrm{sin}\left(x+y\right)=\mathrm{sin}x\mathrm{cos}y+\mathrm{cos}x\mathrm{sin}y$. Verify the identity $\mathrm{cos}\left(x+y\right)=\mathrm{cos}x\mathrm{cos}y-\mathrm{sin}x\mathrm{sin}y$. Verify the identity $\mathrm{cos}x=\frac{{e}^{ix}+{e}^{-ix}}{2}$. Verify the identity $\mathrm{sin}x=\frac{{e}^{ix}-{e}^{-ix}}{2i}$. Explain why $\mathrm{cosh}x=1+\frac{{x}^{2}}{2!}+\frac{{x}^{4}}{4!}+\frac{{x}^{6}}{6!}+\cdots$. Explain why $\mathrm{sinh}x=x+\frac{{x}^{3}}{3!}+\frac{{x}^{5}}{5!}+\frac{{x}^{7}}{7!}+\cdots$. Verify the identity ${e}^{x}=\mathrm{cosh}x+\mathrm{sinh}x$. Verify the identity ${\mathrm{cosh}}^{2}x-{\mathrm{sinh}}^{2}x=1$. Verify the identity $\mathrm{sinh}\left(-x\right)=-\mathrm{sinh}x$. Verify the identity $\mathrm{cosh}\left(-x\right)=\mathrm{cosh}x$. Verify the identity $\mathrm{sinh}\left(x+y\right)=\mathrm{sinh}x\mathrm{cosh}y+\mathrm{cosh}x\mathrm{sinh}y$. Verify the identity $\mathrm{cosh}\left(x+y\right)=\mathrm{cosh}x\mathrm{cosh}y+\mathrm{sinh}x\mathrm{sinh}y$. Verify the identity $\mathrm{cosh}x=\frac{{e}^{x}+{e}^{-x}}{2}$. Verify the identity $\mathrm{sinh}x=\frac{{e}^{x}+{e}^{-x}}{2}$.

## trigonometric function identities

 Verify the identity $\mathrm{tan}\left(x+y\right)=\frac{\mathrm{tan}x+tany}{1-\mathrm{tan}x\mathrm{tan}y}$. Verify the identity $\mathrm{sin}\left(x/2\right)=±\sqrt{\frac{1-\mathrm{cos}x}{2}}$. Verify the identity $\mathrm{cos}3x={\mathrm{cos}}^{3}x-3\mathrm{cos}x{\mathrm{sin}}^{2}x$. Verify the identity $\mathrm{sin}3x=3{\mathrm{cos}}^{2}x\mathrm{sin}x-{\mathrm{sin}}^{3}x$. Verify the identity ${\mathrm{sin}}^{2}A{\mathrm{cot}}^{2}A=\left(1-\mathrm{sin}A\right)\left(1+\mathrm{sin}A\right)$. Verify the identity $\mathrm{tan}B=\frac{\mathrm{cos}B}{\mathrm{sin}B{\mathrm{cot}}^{2}B}$. Verify the identity $\frac{\mathrm{tan}V\mathrm{cos}V}{\mathrm{sin}V}=1$. Verify the identity $\mathrm{sin}E\mathrm{cot}E+\mathrm{cos}E\mathrm{tan}E=\mathrm{sin}E+\mathrm{cos}E$. Verify the identity $\frac{1}{{\mathrm{sec}}^{2}x}+\frac{1}{{\mathrm{csc}}^{2}x}-1=0$.