page_title: Interval arithmetic
section_title: Interval operators
hierarchical_section_title: Interval arithmetic / Interval operators
context_page_description: Interval arithmetic is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Numerical methods using interval arithmetic can guarantee reliable, mathematically correct results. Instead of representing a value as a single number, interval arithmetic represents each value as a range of possibilities. For example, instead of estimating the height of someone as exactly 2.0 metres, using interval arithmetic one might be certain that that person is somewhere between 1.97 and 2.03 metres.
  Mathematically, instead of working with an uncertain real, one works with the ends of an interval that contains. In interval arithmetic, any variable lies in the closed interval between and. A function, when applied to, yields an uncertain result; produces an interval which includes all the possible values for for all.
  Interval arithmetic is suitable for a variety of purposes. The most common use is in software, to keep track of rounding errors in calculations and of uncertainties in the knowledge of the exact values of physical and technical parameters.
context_section_description: An binary operation  on two intervals, such as addition or multiplication, is defined by
  In other words, it is the set of all possible values of , where  and  are in their corresponding intervals. If  is monotone in each operand on the intervals, which is the case for the four basic arithmetic operations (except division when the denominator contains ), the extreme values occur at the endpoints of the operand intervals. Writing out all combinations, one way of stating this is
  provided that  is defined for all  and .
  For practical applications this can be simplified further:
  Addition: 
  Subtraction: 
  Multiplication: 
  Division:
  where
  The last case loses useful information about the exclusion of . Thus, it is common to work with  and  as separate intervals. More generally, when working with discontinuous functions, it is sometimes useful to do the calculation with so-called multi-intervals of the form  The corresponding multi-interval arithmetic maintains a set of (usually disjoint) intervals and also provides for overlapping intervals to unite.
  Interval multiplication often only requires two multiplications. If ,  are nonnegative,
  The multiplication can be interpreted as the area of a rectangle with varying edges. The result interval covers all possible areas, from smallest to the largest.
  With the help of these definitions, it is already possible to calculate the range of simple functions, such as  For example, if ,  and :
  .
caption: Multiplication of positive intervals
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