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{\bf  Math 210, Spring 2019\hfill Jerry L. Kazdan}

\Large

\begin{center}
{\LARGE\bf Big Numbers}
\smallskip

\end{center}

\begin{itemize}
\item List these numbers from smallest to largest: 
$2^{121}$, \ $9^{55}$, \ $7^{88}$,

N := number of seconds since the birth of our universe.

\item Age of the universe (14 billion years) in seconds:
$$
\begin{aligned}
N =& 14*10^9\text{ years }*\frac{365\text{ days}}{\text{year}}*
\frac{24\text{ hours}}{\text{day}}
*\frac{3600\text{ seconds}}{\text{hour}}\\
=& 441,504,000*10^9  \approx 4.4*10^{17}\text{ seconds}.
\end{aligned}
 $$

\item 1 billion = $1,000,000,000 = 10^9$ \medskip

\item $2^{121} =(2^{11})^{11} =2048^{11} \approx 2.7\ 10^{36}$\medskip

\item $9^{55}=(9^5)^{11} = 32805^{11} \approx 3.0\ 10^{52}$\medskip

\item $7^{88}=(7^8)^{11} = 5764801^{11} \approx 2.3\ 10^{74}$\medskip

\item $e^{100} \approx 2.68\ 10^{43}$\medskip

\item $15! \approx 1.3 \ 10^{12}$\hskip 1cm 
$25! \approx 1.5\ 10^{25}$

\end{itemize}
\bigskip

{\large \bf Example 1. } Let $S_N := 1+\frac{1}{2}+\cdots+\frac{1}{N}$.  Find
an estimate for $N$ so that $S_N>100$. 
\smallskip

\comments{
{\sc Solution: } Observe that 
$$
1 +\frac{1}{2}+\cdots \frac{1}{N} > \int_1^{N+1} \frac{1}{t}\,dt =  \ln(N+1)
$$
and
$$
\frac{1}{2}+\cdots \frac{1}{N} < \int_1^N \frac{1}{t}\,dt = \ln N.
$$
That is
}
{\sc Answer}\footnote{
R. P. Boas, Jr. and J. W. Wrench, Jr. found the exact number: 
$$15092688622113788323693563264538101449859497$$
{\it Amer.~Math.~Montly\,}, Vol.~78, No.~8, Oct., 1971, pp 864--870,
DOI: 10.2307/2316476, Stable URL: https://www.jstor.org/stable/2316476} 
\ By the idea behind the {\it integral test}
$$
\ln(N+1) < S_N < 1 +\ln N 
$$ 
Thus, to insure that $S_N>100$ pick $\ln(N+1)>100$, that is,
$N+1>e^{100}\approx 2.7*1 0^{43}$.\smallskip

On the other hand, if $1+\ln N <100$, then $S_N<100$.  Here we can pick
any $N<e^{99} \approx 10^{43}$. 
\bigbreak

\bigskip

{\large \bf Example 2. } Consider a stock that is sold on the New York Stock
Exchange.  If someone is Paris places a buy order and 1/100 of a second
later someone in New York buys the same stock, which order is executed
first?
\medskip

Some Data:

Speed of light in a vacuum \vtop{ 
$\approx 300,000$  km/sec \\
$\approx 186,000$ miles/sec\\ 
$\approx 1$ foot/ns $\approx$ $1$ m$/3.3$\,ns \ (ns\,=\,nanoseconds)\\
$\approx 4,000$ miles/$.0215$ sec \ (3,625 miles from Paris to New York)
}
\bigskip


\vfill
[Last revised: \today]
\end{document}
