= Interval arithmetic
{title2=$X=[a,b]$}
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<Interval arithmetic> calculates with <closed intervals> that contain the possible values of a quantity. If all we know is $x\in X=[a,b]$, an <interval enclosure> for a <function> $f$ must contain $f(x)$ for every allowed $x$. The result says what is guaranteed, including uncertainty in the input and, with <outward rounding>, <roundoff error> in the calculation. A wide answer can still be correct; a narrow answer is useful only if it contains every possible value.
The attraction is that a finite calculation can make a statement about infinitely many <real numbers> at once. An <interval enclosure> can certify that a <function> stays positive throughout a whole <closed interval>, or bound its <maximum> for <mathematical optimization>. For example, on $X=[0,1]$, the actual <image of a function> for $f(x)=x(1-x)$ is $[0,1/4]$, but direct <interval arithmetic> gives the wider $[0,1]$. <Interval subdivision> and the <interval square range> improve this answer in different ways.
Historically, Ramon E. Moore's 1962 Stanford dissertation studied <interval arithmetic> and automatic error analysis in digital computing; his book \i[Interval Analysis] followed in 1966. Eldon Hansen's account documents earlier reports as well: these dates describe important steps in developing and disseminating the subject, rather than a claim of sole invention in a particular year.https://interval.louisiana.edu/Moores_early_papers/bibliography.html[Eldon R. Hansen, Publications Related to Early Interval Work of R. E. Moore, 2001, references 5 and 8]{ref} Moore, R. Baker Kearfott and Michael J. Cloud later wrote \i[Introduction to Interval Analysis] (2009).https://interval.louisiana.edu/books.html[R. Baker Kearfott, Books written or edited, entry for Introduction to Interval Analysis]{ref}
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