Interval (mathematics) Totally Explained

Everything about Interval Mathematics totally explained

In mathematics, an interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.For example, the set of all numbers

x

satisfying

0le xle 1

is an interval which contains

0

and

1

, as well as all numbers between them. The set of positive numbers is also an interval.

Notations for intervals

The interval of numbers between

a

and

b

including

a

and

b

is often denoted

[a,b]

. If one of the endpoints is to be omitted, then the corresponding square bracket is replaced by a parenthesis. Thus, in set builder notation, ť

(a,b)=.

The basic operations of interval arithmetic are, for two intervals [a, b] and [c,d] that are subsets of the real line (-∞,∞),

[a,b] + [c,d] = [a+ c, b + d]

[a,b] − [c,d] = [a− d, b −c]

[a,b] × [c,d] = [min(ac, ad, bc, bd), max (ac, ad, bc, bd)]

[a,b] / [c,d] = [min(a/c, a/d, b/c, b/d), max (a/c, a/d, b/c, b/d)] Division by an interval containing zero isn't defined under the basic interval arithmetic. The addition and multiplication operations are commutative, associative and sub-distributive: the set X (Y + Z ) is a subset of XY + XZ.
There are interval versions of standard algorithms, such as Newton's method. See also: comprehensive German Wikipedia article on

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