Problem Set - The number systems $\mathbb{Z},\mathbb{Q},ℝ,$ and $ℂ$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

The number systems $\mathbb{Z},\mathbb{Q},ℝ,$ and $ℂ$

	Prove that ${\displaystyle \sum _{k=1}^{n}k}=\frac{1}{2}n(n+1)$.
	Prove that ${\displaystyle \sum _{k=1}^n(2k-1)}=n^2$.
	Prove that ${\displaystyle \sum _{k=1}^n(3k-2)}=\frac{1}{2}n(3n-1)$.
	Prove that ${\displaystyle \sum _{k=1}^n{k}^2}=\frac{1}{6}n(n+1)(2n+1)$.
	Prove that ${\displaystyle \sum _{k=1}^n{k}^3}=\frac{1}{4}{n}^2{(n+1)}^2$.
	Prove that ${\displaystyle \sum _{k=1}^n{k}^3}={\left({\displaystyle \sum _{k=1}^{n}k}\right)}^2$.
	Prove that ${\displaystyle \sum _{k=1}^n\frac{1}{k(k+1)}=\frac{n}{n+1}}$.
	Define $a_1=0$, $a_{2k}=\frac{1}{2}{a}_{2k-1}$ and $a_{2k+1}=\frac{1}{2}+a_{2k}$. Show that $a_{2k}=\frac{1}{2}-{\left(\frac{1}{2}\right)}^{\mathrm{k}}$.
	Prove that if $n\in {\mathbb{Z}}_{>0}$ then 3 is a factor of $n^3-n+3$.
	Prove that if $n\in {\mathbb{Z}}_{>0}$ then 9 is a factor of ${10}^{n+1}+3\cdot {10}^n+5$.
	Prove that if $n\in {\mathbb{Z}}_{>0}$ then 4 is a factor of $5^n-1$.
	Prove that if $n\in {\mathbb{Z}}_{>0}$ then $x-y$ is a factor of $x^n-y^n$.
	Give an example of $s\in \mathbb{Q}$ which has more than one representation as a fraction.
	Show that $\sqrt{2}\notin \mathbb{Q}$.
	Show that $\sqrt{3}\notin \mathbb{Q}$.
	Show that $\sqrt{15}\notin \mathbb{Q}$.
	Show that $2^{\mathrm{1/3}}\notin \mathbb{Q}$.
	Show that ${11}^{\mathrm{1/4}}\notin \mathbb{Q}$.
	Show that ${16}^{\mathrm{1/5}}\notin \mathbb{Q}$.
	Show that $\sqrt{2}+\sqrt{3}\notin \mathbb{Q}$.
	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 $z^{-1}=\frac{1}{|z{|}^2}(x-yi)$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)