Problem Set - The number systems Z, Q, R and C

## The number systems $ℤ,ℚ,ℝ,$ and $ℂ$
 Prove that $\sum _{k=1}^{n}k=\frac{1}{2}n\left(n+1\right)$. Prove that $\sum _{k=1}^{n}\left(2k-1\right)={n}^{2}$. Prove that $\sum _{k=1}^{n}\left(3k-2\right)=\frac{1}{2}n\left(3n-1\right)$. Prove that $\sum _{k=1}^{n}{k}^{2}=\frac{1}{6}n\left(n+1\right)\left(2n+1\right)$. Prove that $\sum _{k=1}^{n}{k}^{3}=\frac{1}{4}{n}^{2}{\left(n+1\right)}^{2}$. Prove that $\sum _{k=1}^{n}{k}^{3}={\left(\sum _{k=1}^{n}k\right)}^{2}$. Prove that $\sum _{k=1}^{n}\frac{1}{k\left(k+1\right)}=\frac{n}{n+1}$. Define ${a}_{1}=0$, ${a}_{2k}=\frac{1}{2}{a}_{2k-1}$ and ${a}_{2k+1}=\frac{1}{2}+{a}_{2k}$. Show that ${a}_{2k}=\frac{1}{2}-{\left(\frac{1}{2}\right)}^{k}$. Prove that if $n\in {ℤ}_{>0}$ then 3 is a factor of ${n}^{3}-n+3$. Prove that if $n\in {ℤ}_{>0}$ then 9 is a factor of ${10}^{n+1}+3\cdot {10}^{n}+5$. Prove that if $n\in {ℤ}_{>0}$ then 4 is a factor of ${5}^{n}-1$. Prove that if $n\in {ℤ}_{>0}$ then $x-y$ is a factor of ${x}^{n}-{y}^{n}$. Give an example of $s\in ℚ$ which has more than one representation as a fraction. Show that $\sqrt{2}\notin ℚ$. Show that $\sqrt{3}\notin ℚ$. Show that $\sqrt{15}\notin ℚ$. Show that ${2}^{1/3}\notin ℚ$. Show that ${11}^{1/4}\notin ℚ$. Show that ${16}^{1/5}\notin ℚ$. Show that $\sqrt{2}+\sqrt{3}\notin ℚ$. Give an example of $s\in ℝ$ which has more than one decimal expansion. Compute the decimal expansion of $\pi$ to 30 digits. Compute the decimal expansion of $2\pi$ to 30 digits. Compute the decimal expansion of ${\pi }^{2}$ to 30 digits. Compute the decimal expansion of $-\pi$ to 30 digits. Compute the decimal expansion of ${\pi }^{-1}$ to 30 digits. Show that .9999... = 1.00000... . Compute the decimal expansion of $\sqrt{2}$ to 30 digits. Let $z=x+yi$ with $x,y\in ℝ$. Show that ${z}^{-1}=\frac{1}{|z{|}^{2}}\left(x-yi\right)$.