\subsection{A primer on Interval Arithmetic}
\label{prec_section}
In this section, we present briefly Interval Arithmetic and focus on its interaction with floating point approximation of reals. For more details on
Interval Arithmetic, the interested reader can consult a more extensive reference, such as~\cite{InterBook}.\medskip
\subsubsection{Bird's eye view on Interval Arithmetic.}
Interval arithmetic is a representation of reals by intervals that contain them. For instance, one can specify that a value $x$ is given with an error $\epsilon$ by considering the interval
$[x-\epsilon, x+\epsilon]$, or even manipulating the transcendent constant $\pi$ as the interval $[3.14, 3.15]$. Interval arithmetic is crucial the context of numerical
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The \emph{diameter} of the interval
$\inter{x}$ is defined as $\inter{x}^+ - \inter^{x}^-$, and its \emph{center} is the real $\frac{1}{2}(\inter{x}^+ + \inter^{x}^-)$.
We can now define abstractly an arithmetic on intervals:
\begin{definition} Let $\bowtie$ be a binary operation --- resp.\ f be a function ---, then the result $\inter{x} \bowtie \inter{y}$ of the operation between the intervals $\inter{x}$ and ${\inter{y}}$ --- resp $f(\inter{x})$, result of the application of $f$ --- is the smallest interval, in the sense of inclusion, containing \[ \{x\bowtie y | (x,y)\in \inter{x}\times\inter{y}\} \qquad \textrm{--- resp.~} \{f(x) | x\in \inter{x}\} \textrm{~---}.
\] \end{definition}
In the context of actual computations, requiring the equality in the above definition is in most case illusory, since reals cannot be represented exactly.
Yet, only the inclusion of the resulting interval in the last defined set
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\subsection{Representations of reals}
\subsubsection{Integral representation}
\label{sec:intgral_representation}
As evoked, real numbers can only be computationally handled at finite precision.
A naive yet convenient way to do so is to use integers to represent the bounds of intervals (specifically, integers made from the most-significant bits of the binary expansion of the desired reals). Formally we have:
\begin{definition}[Integral representation of reals]
\label{def:intgral_representation}
Let $x\in\RR$ be an arbitrary real number and $n>0$ a non-negative integer. Define its \emph{integral representation at accuracy\footnote{% We use here the denomination of "accuracy" instead of "precision" to avoid confusions with the floating point precision as defined in paragraph~\ref{sec:fp_representation}.}~% $n$} as the symetric interval centered on $\lfloor2^{n}x\rceil$ of diameter $1$: \[ \inter{x}_n = \left[\lfloor2^{n}x\rceil-\frac{1}{2}, \lfloor2^{n}x\rceil+\frac{1}{2} \right] \]
\end{definition}
\begin{remark} The interval $\inter{x}_n$ only needs to be represented as its center,
The real number system consists of five subsets of numbers (Blitzer, 2013). The first subset is natural numbers; natural numbers consist of positive counting numbers not including zero. The second subset is whole numbers; whole numbers consist of zero and natural numbers. The third subset is integers; integers are positive and negative whole numbers, as well as zero. The fourth subset is rational numbers; rational numbers are numbers that can be written in fraction form. The fifth and last subset is irrational numbers; irrational numbers are numbers that are not a perfect square, do not have a repeating or terminating decimal, and are not included in the whole numbers subset (Blitzer, 2013). Rational and irrational numbers are often the most difficult to understand out of these 5 subsets of real numbers. Simply put, rational numbers are any numbers that can be re-written as a simple fraction, and if a number cannot be defined as rational then it is defined as irrational. For example, the number 7 is a rational number because it can be re-written as , which is a simple fraction. The number 2.5 is also defined as rational because it can be re-written as , which, again, is a simple fraction. However, if the number π were defined, it would have to be irrational since it has neither a repeating decimal nor a terminating decimal, and cannot be written as a simple fraction.
Lastly, Niven showed how the integral of f(x) sin(x) cannot always be an integer for all n values. Since sin(x) is always bounded between 0 and 1 when x is bounded between 0 and π, it can be stated that f(x) sin(x) < f(x). Niven used what he has stated before: f(x) = xn(a –bx)n/n!, and showed how that (a – bx) < a when x is bounded between 0 and π. From this Niven stated that must be a smaller value than (a)n. Similarly, when x is bounded between 0 and π, the term xn must be between 0 and πn. Niven took all of these properties and functions and stated
We can use this information for simple division, multiplication and even when multiplying and dividing larger numbers. This information can be used when cooking, grocery shopping, building things etc..
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