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\title{Math 251--252: arithmetic expected without calculators}

\author{N.~Christopher Phillips}

\date{29 March 2026}

\begin{document}

\maketitle

\indent
Calculators, of any kind, are prohibited on all quizzes, exams,
etc.\  in Math 251.
There are apparently calculators available with the capability of
connecting to the internet, and ones with computer algebra systems
capable of some symbolic differentiation and integration.
Obviously such calculators can't be allowed,
but I have no way to tell whether any particular calculator
has these capabilities.
Therefore the only thing I can do is a blanket prohibition.

This means that you are expected to be able to do some arithmetic
without a calculator.
For example, you can't rely on a calculator to evaluate things
like the following:
\[
17 + 33,
\qquad
2 \cdot 9{.}007,
\qquad
\sin (0),
\qquad
\sqrt{49},
\andeqn
0{.}02^2.
\]

This file is about the sorts of arithmetic that it is expected
you can do without a calculator,
and some things you will not be expected to do
(except possibly in extra credit problems).


\section{Arithmetic expected without calculators}\label{Sec_5321_Calc}

\indent
Here is a (not necessarily complete) list of arithmetic you might
be expected to do.
%
\begin{enumerate}
% [label=$\mathrm{(\arabic*)}$]
%
\item\label{I_ArNCalc_5325_Add}
Any two digit addition, such as $34 + 78$ or $34 + (-78)$.
Also arithmetic which is similarly easy because of zeroes:
$34{,}000 + 78$, $34{,}000 + 78{,}000$,
$0{.}0034 + 0{.}0078$, $17{.}0034 + 0{.}0078$, etc.
%
\item\label{I_ArNCalc_5325_Mult}
Any single digit multiplication (up to $9 \cdot 9$,
multiplying anything by $2$, $3$, or any power of~$10$,
or combinations (such as $20 \cdot 311 = 6220$),
and anything which is similarly easy because of zeroes,
such as $0{.}03 \cdot 0{.}007 = 0{.}00021$
or $0{.}02^2 = 0{.}0004$.
%
\item\label{I_ArNCalc_5325_Dv}
Division by $2$, $3$, $4$, and~$5$ when it comes out evenly,
and division by $10$, $100$, etc.\  even if it doesn't come out evenly.
(Do not replace $\frac{33}{10}$ with $3{.}3$ when exact answers are
specified, which is most of the time.
If the problem statement doesn't contain a decimal point,
then almost certainly no decimal points should appear anywhere
in the solution or final answer.)
%
\item\label{I_ArNCalc_5325_Sq}
Square roots of perfect squares:
\[
\sqrt{0} = 0,
\qquad
\sqrt{1} = 1,
\qquad
\sqrt{4} = 2,
\qquad
\sqrt{9} = 9,
\qquad \ldots, \qquad
\sqrt{100} = 10,
\]
as well as cases which are similarly easy because of zeroes,
such as
\[
\sqrt{900} = 30,
\qquad
\sqrt{10{,}000} = 100,
\qquad
\sqrt{250{,}000} = 500,
\qquad
\sqrt{0{.}04} = 0{.}2,
\]
etc.
%
\item\label{I_ArNCalc_5325_Cube}
A few small cube roots and higher,
including $\sqrt[3]{8} = 2$, $\sqrt[3]{27} = 3$,
$\sqrt[4]{16} = 2$, and anything similarly easy because of zeroes.
%
\item\label{I_ArNCalc_5325_Rec}
Expressions like
\[
\frac{1}{0{.}1} = 10,
\qquad
\frac{1}{0{.}0001} = 10{,}000,
\qquad
\frac{1}{0{.}02} = 50,
\]
etc.
%
\item\label{I_ArNCalc_5325_Fcn}
Values of elementary functions at special points,
such as
\[
\sin (0) = 0,
\qquad
\cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2},
\qquad
\sin \left( \frac{\pi}{2} \right) = 1,
\qquad
\cos (3 \pi) = - 1,
\]
\[
e^0 = 1,
\qquad
\ln (1) = 0,
\qquad
\ln (e) = 1,
\qquad
\arctan (1) = \frac{\pi}{4},
\]
and many similar things.
%
\item\label{I_ArNCalc_5325_ord}
I don't expect decimal approximations to $\sqrt{2}$ or $\pi$
(see below), but you are expected to know
\[
1 < \sqrt{2} < \sqrt{3} < 2 < \sqrt{5} < \cdots
\andeqn
2 < e < 3 < \pi < 4.
\]
%
\end{enumerate}
%

\section{Making easy arithmetic hard}\label{Sec_5321_NC}

\indent
Here are examples of calculations that might appear which are easy
if done the right way,
but become unsuitable without a calculator if done the wrong way.

\begin{exa}\label{E_5326_Mosq}
Suppose a mosquito takes off from a log and buzzes
above a pool of stagnant water.
Its height $h (t)$ above the surface of the water,
in millimeters, at time~$t$ seconds,
is modelled by $h (t) = 279 + 10 t^2 - t^3$.
You are asked for the average upwards velocity
between $1$ and $3$ seconds.

This is
\begin{align*}
\frac{h (3) - h (1)}{3 - 1}
& = \frac{279 + 10 \cdot 3^2 - 3^3 - (279 + 10 \cdot 1^2 - 1^3)}{2}
\\
& = \frac{279 + 90 - 27 - (279 + 10 - 1)}{2}
\\
& = \frac{\cancel{279} + 90 - 27 - \cancel{279} - 10 + 1}{2}
  = 27.
\end{align*}
You make the arithmetic much harder if you evaluate
\[
h (3) = 279 + 10 \cdot 3^2 - 3^3 = 342
\]
rather than cancelling $279$ as above.
\end{exa}

\begin{exa}\label{E_5326_Factor}
You are given (or have calculated)
\[
g' (x) = 3 x^2 - 24 x = 3 x (x - 8),
\]
and want to know the sign of $g' (9)$.
It is easy to see that
\[
g' (9) = 3 \cdot 9 \cdot (9 - 8) = 27 > 0.
\]
{\textbf{Don't}} use
\[
g' (x) = 3 \cdot 9^2 - 24 \cdot 9.
\]
In fact, even to find the sign of $g' (35)$, you can write
\[
g' (9) = 3 \cdot 35 \cdot (35 - 8) = 3 \cdot 35 \cdot 27,
\]
which is obviously positive {\textbf{without}} actually
multiplying out $3 \cdot 35 \cdot 27$.
\end{exa}

\begin{exa}\label{E_5326_Unneeded}
You are given
\[
g (x) = x^3 - 12 x^2 + 16,
\]
and you are asked to find the values of~$x$ at which there is a
local minimum.
You find $x = 8$.
The problem as given {\textbf{does not ask for}} the value of the
function at $x = 8$.
That is, you {\textbf{don't}} need to calculate $g (8)$.
Read the problem to see what is actually asked for!
\end{exa}

\section{Arithmetic not expected}\label{Sec_5321_NExp}

\indent
I will not expect on exams numerical calculations of the following types.
(However, as above, I might well expect calculations which might look
like these if done without paying attention.
Also, such things might appear in extra credit problems.)
%
\begin{enumerate}
% [label=$\mathrm{(\arabic*)}$]
%
\item\label{I_NArNCalc_5325_Poly}
If $f (x) = x^3 - 6 x^2 + 11 x - 9$,
I won't expect calculations of $f (7)$ or $f (- 9)$.
(But I might expect a calculation of $f (10)$.
Also, some calculations I do expect might turn out like this
if done the wrong way.
See Section~\ref{Sec_5321_NC}.)
%
\item\label{I_NArNCalc_5325_Dec}
I don't expect you to know decimal approximations to
$\frac{1}{3}$, $\frac{1}{7}$, $\sqrt{2}$, $\sqrt{3}$, $\pi$, or~$e$.
In fact, most questions will ask for exact answers,
and such decimal approximations, not being exact,
will be considered wrong.
If the answer is $2 \sqrt[3]{70}$, just leave it like that.
(But see (\ref{I_ArNCalc_5325_Add}), (\ref{I_ArNCalc_5325_ord}),
(\ref{I_ArNCalc_5325_Sq}), and (\ref{I_ArNCalc_5325_Cube})
in Section~\ref{Sec_5321_Calc}.)
%
% \item\label{I_NArNCalc_5325_}
%
% \item\label{I_NArNCalc_5325_}
%
\end{enumerate}
%



\end{document}
