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\begin{document}



The word {\em positive} is usually explained to mean that the \htmladdnormallink{number}{http://planetmath.org/encyclopedia/Number.html} under consideration is greater than zero. \,Without the \htmladdnormallink{relation}{http://planetmath.org/encyclopedia/Surjective2.html} ``$>$'', the positivity of (real) numbers may be defined specifying which numbers of a given number kind are positive, e.g. as follows.

\begin{itemize}
\item In the set $\mathbb{Z}$ of the \htmladdnormallink{integers}{http://planetmath.org/encyclopedia/RationalInteger.html}, all numbers obtained from 1 via \htmladdnormallink{addition}{http://planetmath.org/encyclopedia/Addition.html} are positive.
\item In the set $\mathbb{Q}$ of the \htmladdnormallink{rationals}{http://planetmath.org/encyclopedia/MathbbQ.html}, all numbers obtained from 1 via addition and \htmladdnormallink{division}{http://planetmath.org/encyclopedia/Reduction2.html} are positive.
\item In the set $\mathbb{R}$ of the \htmladdnormallink{real numbers}{http://planetmath.org/encyclopedia/MathbbR.html}, the numbers defined by the \htmladdnormallink{equivalence classes}{http://planetmath.org/encyclopedia/EquivalenceClass2.html} of non-zero decimal \htmladdnormallink{sequences}{http://planetmath.org/encyclopedia/Sequence.html} are positive; these sequences (\htmladdnormallink{decimal expansions}{http://planetmath.org/encyclopedia/Dyadic.html}) consist of \htmladdnormallink{natural numbers}{http://planetmath.org/encyclopedia/NaturalNumber.html} from 0 to 9 as \htmladdnormallink{digits}{http://planetmath.org/encyclopedia/PlaceSystems.html} and a single \htmladdnormallink{decimal point}{http://planetmath.org/encyclopedia/DecimalPoint.html} (where two decimal sequences are \htmladdnormallink{equivalent}{http://planetmath.org/encyclopedia/EquivalenceClass2.html} if they are identical, or if one has an \htmladdnormallink{infinite}{http://planetmath.org/encyclopedia/InfiniteSubset.html} tail of 9's, the other has an infinite tail of 0's, and the leading portion of the first sequence is one lower than the leading portion of the second).
\end{itemize}

For example, $1+1+1$ is a positive integer, $\frac{1+1}{1+1+1+1+1}$ is a positive rational and \,$5.15115111511115...$ is a positive real number.

If $a$ is positive and \,$a+b = 0$, then the \htmladdnormallink{opposite number}{http://planetmath.org/encyclopedia/NegativeAsANoun.html} $b$ is {\em negative}.

The sets of positive integers, positive rationals, positive (real) \htmladdnormallink{algebraic numbers}{http://planetmath.org/encyclopedia/AlgebraicNumber.html} and positive reals are \htmladdnormallink{closed under}{http://planetmath.org/encyclopedia/ClosureProperty.html} addition and \htmladdnormallink{multiplication}{http://planetmath.org/encyclopedia/Multiplication.html}, so also the set of positive \htmladdnormallink{even numbers}{http://planetmath.org/encyclopedia/OddInteger.html}.



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