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Some Topics from MA347 in Remediation

\textit{$\C$  2010, Prof. George K. Francis, Mathematics Department,
University of Illinois}

\begin{document}
\maketitle

\section{Circular Numbers}
Long division of the natural numbers was probably the first algorithm you
every learned in grade school. (It is sheer and malicious nonsense to
holed that with the advent of calculators there is no longer any need for
teaching arithmetic in grade school.) To do it effectively, you needed to
know some multiplication tables. Actually, this is was unnecessary (though
very useful). If you had learned subtraction, you could solve
$42 \div 5 = 8 \ R \ 2$, or more attractively $42 = 5 (8) + 2$,
by subtracting $5$ from $42$ eight times, with a remainder of $2$.

\textbf{Review Exercise}

Apply this process to calculate the greatest common divisor of two
numbers $gcd(n,m)$, namely by subtracting the smaller from the larger,
until there is no smaller or larger. The last remainder is the gcd. Why?

\subsection{Clock Arithmetic}
Applying this concept to the integers, $\mathbb{Z}$, for a given positive
integer $m$, called the \textit{modulus} during this context, we identify
two integers \textit( modulo $m$ ), written $a \equiv b ( mod \ m)$,
if they have the same remainder when dividing by $m$. The equivalence
classes (review the concept of an \textit{equivalence class}) modulo $m$,
is a finite set, generally written by their chief representive, namely
the unique (why unique) integer $0 \le k < m$ in the class. This
set is written, classically, $\mathbb{Z}_m$, and this number system is
at least a ring (group under addition, plus distributive multiplication).
Sometimes (when?) this number system is a field (the nonzero elements
form a multiplicative group.)

\subsection{Circular Numbers}
When when try to extend this concept to the real numbers we have a problem
with division, since a real can be divided by any non-zero real without a
remainder. Never mind what this might mean. Reinstall long division by
continued subtraction. Again we have to be careful. For two positive reals,
$n > m$ continued subtraction of the smaller from the larger as long as
there is a larger", is well defined. The last non-negative remainder is the
\textit{residue} of $n$ \texit{modulo} $m$ .
It is always possible to assume $m >0$, since you can push the minus sign
into the numerator. The interesting case is when $n < 0$. Now you can
add $m$ until you get the residue $0 \le k < m$ of $n$ modulo $m$.

You have experience of this when you considered radians in trigonometry.
But be careful of the context. In physics, an angle of 720 degrees
is not the same as an angle of -360 degrees. (Why not? think of odometers.)
But in geometry we do consider these two angles to have the same measure.
In radians (aha, the word radians" has several meanings), we consider
$2.5 \pi$ the same angle measure as $0.5 \pi$.

Of course we can do arithmetic with circular numbers, in particular angle
measure. How would you add angles? At any event, you can always draw
a figure of a circle, reprenting a unit circle, and calculate angles there.