# [texhax] dagger

PR Stanley prstanley at ntlworld.com
Tue Apr 27 19:03:17 CEST 2010

Hi folks,
Could you please tell me what the dagger macro is doing in the document below?

By the way, in case you were wondering about the weird LaTeX, it was
produced using an OCR engine ( inftyreader ) designed for converting
maths to LaTeX.
Thanks, Paul
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\documentclass[a4paper,12pt]{article}
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\usepackage{amsmath}
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\usepackage{fancybox}
\usepackage{bm}

\begin{document}
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$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}'\ v\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\rangle\ovalbox{\tt\small REJECT} \mathrm{f}\ovalbox{\tt\small REJECT} z\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{k}\ovalbox{\tt\small REJECT} X\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}$
\end{center}
NUMBERS: BIG, BIGGER, AND BIGG ER AGAI N [!]
{\bf QUIZ 1}: {\bf Every number ever seen by any person is small}.
Is this statement true[?]
{\bf Warm-up exereises}:
1. Which number is bigger: $333,33\times 444,444$ or $222,222\times 666,667$ [?] What is the
difference[?]
2. Which number is bigger, (50-17) (50[+]17) or $50^{2}$[?]
3. If we write all of the digits of the two numbers $2^{2008}$ and
$5^{2008}$ in a row, how
many digits will there be[?]
{\bf How big are faetorials}[?]
Have you ever seen a symbol like 5 [!] [?] We use it to denote the
product of all consecutive
integers from 1 through 5: $5!=1\cdot 2\cdot 3\cdot 4\cdot 5$. So
5[!] is not very big, right[?]
What about 10[!][?] Well, it's not too big, either: $10!=3,628,800$. But $20!=$
2,432,902,008,176,640,000. Looks kind of big, does it[?] Do you want
to calculate 100[!] by
hand[?] Will a calculator help[?] A computer[?]
Not much. So it will be better to try and answer all the following
questions without
calculating the actual numbers.
4. Start with 100[!]. Find the sum of all its digits; then find the
sum of all the digits of
this new number. Continue this process until you obtain a one-digit
number. What
is it[?]
5. Which of the two numbers is bigger:
(a) 1000[!] or (500[!])(500[!])
\begin{center}

\end{center}

\begin{center}

$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}'\ v\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\rangle\ovalbox{\tt\small REJECT} \mathrm{f}\ovalbox{\tt\small REJECT} z\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{k}\ovalbox{\tt\small REJECT} X\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}$
\end{center}
{\bf Whieh are bigger, faetorials or powers}[?]
As with factorials above, powers don't look too big in the beginning:
$2^{2}=4,2^{10}=1024$.
But $2^{50}$ is already so big that if you use a number line with the
scale of 1 millimeter, the
distance from 0 to $2^{50}$ will be about 1,125,899,906
$\mathrm{km}$, which is about 7.5 times farther
than the distance from the Earth to the Sun[!]
{\bf Whieh is bigger}:
10. (a) 100[!] or $100^{100}$
(b) 100[!] or $2^{100}$
(c) 100 [!] or 3100
(d) 100[!] or $10^{100}$
(e) Can you find an integer $\mathrm{n}$ such that
$n^{100}<100!<(n+1)^{100}$ [?]
11. 99[!] or $50^{99}$ [?] (Hint: look back at the warm-up problem
involving 50)
12. 100[!] or $50^{100}$
13. $2^{100\dagger}$ or $(2^{100})!$
14. Consider the number 32[!] Which prime numbers divide it[?] It is
obviously even,
but what is the largest power of 2 that divides 32[!] [?] What is the
largest power of 3
that divides it[?] the largest power of 5[?] of 7[?] If you continue
questions about consecutive primes, you will finally write 32[!] as a
product of
prime numbers (similarly to writing, say, $180=2^{2}\cdot 3^{2}\cdot 5$).
15. Which is bigger: $2^{5^{1}}$ or $(2^{5})!$ [?] (Hint: use the
previous problem.)
16. Generalize Problem 12 and compare $n!$ with $(n\mathit{1}2)^{n}$
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