Problem Set - Number Systems and functions

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: 16 February 2010

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 ${\displaystyle \frac{-15+i}{4+2i}}$ and graph the result.
	Compute ${\left(3-2i\right)}^3$ and graph the result.
	Compute $\sqrt{2i}$ and graph the result.
	Compute ${\displaystyle \frac{1}{a+bi}}$ 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 ${\displaystyle \frac{6-i}{4+2i}}$ 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 ${\displaystyle 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{32}+\frac{1}{64}+\cdots }$.
	Compute $1·2,1·2·3,1·2·3·4,1·2·3·4·5$ and $1·2·3·4·5·6$.
	Compute ${\displaystyle \frac{1}{1·2}+\frac{1}{1·2·3}+\frac{1}{1·2·3·4}+\frac{1}{1·2·3·4·5}+\frac{1}{1·2·3·4·5·6}+\cdots }$.

Functions

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

Function identities

	Explain why ${\displaystyle \frac{1}{1-\mathrm{x}}=1+x+x^2+x^3+\cdots }$.
	Explain why ${\displaystyle \frac{x^n-1}{x-1}=1+x+x^2+\cdots +x^{n-1}}$.
	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 ${\displaystyle e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}+\cdots }$.
	Explain why $\ln x$ is the inverse function to $e^x$.
	Verify the identity $e^{x+y}=e^x{e}^y$.
	Verify the identity ${\displaystyle e^{-x}=\frac{1}{e^x}}$.
	Verify the identity ${\left(e^x\right)}^n=e^{nx}$.
	Verify the identity $e^0=1$.
	Verify the identity $\ln \left(xy\right)=\ln x+\ln y$.
	Verify the identity $-\ln x=\ln \left(1/\mathrm{x}\right)$.
	Verify the identity $\ln x^n=n\ln x$.
	Verify the identity $\ln 1=0$.
	Explain why ${\displaystyle \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots }$.
	Explain why ${\displaystyle \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots }$.
	Verify the identity $e^{ix}=\cos x+i\sin x$.
	Verify the identity ${\cos }^{2}x+{\sin }^{2}x=1$.
	Verify the identity $\sin \left(-x\right)=-\sin x$.
	Verify the identity $\cos \left(-x\right)=\cos x$.
	Verify the identity $\sin \left(x+y\right)=\sin x\cos y+\cos x\sin y$.
	Verify the identity $\cos \left(x+y\right)=\cos x\cos y-\sin x\sin y$.
	Verify the identity ${\displaystyle \cos x=\frac{e^{ix}+e^{-ix}}{2}}$.
	Verify the identity ${\displaystyle \sin x=\frac{e^{ix}-e^{-ix}}{2i}}$.
	Explain why ${\displaystyle \cosh x=1+\frac{x^2}{2!}+\frac{x^4}{4!}+\frac{x^6}{6!}+\cdots }$.
	Explain why ${\displaystyle \sinh x=x+\frac{x^3}{3!}+\frac{x^5}{5!}+\frac{x^7}{7!}+\cdots }$.
	Verify the identity $e^x=\cosh x+\sinh x$.
	Verify the identity ${\cosh }^{2}x-{\sinh }^{2}x=1$.
	Verify the identity $\sinh \left(-x\right)=-\sinh x$.
	Verify the identity $\cosh \left(-x\right)=\cosh x$.
	Verify the identity $\sinh \left(x+y\right)=\sinh x\cosh y+\cosh x\sinh y$.
	Verify the identity $\cosh \left(x+y\right)=\cosh x\cosh y+\sinh x\sinh y$.
	Verify the identity ${\displaystyle \cosh x=\frac{e^x+e^{-x}}{2}}$.
	Verify the identity ${\displaystyle \sinh x=\frac{e^x+e^{-x}}{2}}$.

trigonometric function identities

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

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)

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