nLab interval arithmetic

Idea

Arithmetic for closed intervals of an ordered field.

Definition

Let $F$ be an ordered field. The closed intervals of $F$ can be represented by a subset of the Cartesian product set $F \times F$:

$$\mathcal{I}(F) \coloneqq \{(a, b) \in F \times F \vert (a \leq b)\}$$

The elements $[a, b] \in \mathcal{I}(F)$ of the set are the endpoints of the closed intervals, which we shall call intervals for short.

There is an embedding $i:F \to \mathcal{I}(F)$ defined by

$$i(a) \coloneqq [a, a]$$

Equality of intervals is defined as

$$([a, b] = [c, d]) \coloneqq (a = c) \times (b = d)$$

If $R$ has decidable equality, then $\mathcal{I}(R)$ has decidable equality as well.

Relation to the zero interval

A positive interval is defined by

$$([0, 0] \lt [a, b]) \coloneqq (0 \lt a)$$

A negative interval is defined by

$$([0, 0] \gt [a, b]) \coloneqq (0 \gt b)$$

A indefinite interval is defined by

$$([0, 0] \sim [a, b]) \coloneqq \neg(0 \lt a) \wedge \neg(0 \gt b)$$

Arithmetic of intervals

Zero is defined by

$$0 \coloneqq [0, 0]$$

Addition is defined by

$$[a, b] + [c, d] \coloneqq [a + c, b + d]$$

Negation is defined by

$$-[a, b] \coloneqq [-b, -a]$$

Subtraction is defined by

$$[a, b] - [c, d] \coloneqq [a, b] + (-[c, d]) = [a - d, b - c]$$

Intervals do not form an abelian group because $[a, b] - [a, b] = [a - b, b - a]$, and $[a - b, b - a] = 0$ if and only if $a = b$. However, they do form a commutative monoid under zero and addition.

One is defined by

$$1 \coloneqq [1, 1]$$

Multiplication is defined by

$$[a, b] \cdot [c, d] \coloneqq [\min(ac, ad, bc, bd), \max(ac, ad, bc, bd)]$$

Intervals form a commutative monoid under one and multiplication. Zero and multiplication also form an absorption monoid. Multiplication does not distribute over addition:

$$[1, 2] \cdot ([1, 3] + [-1, 3]) = [1, 2] \cdot [0, 6] = [\min(0, 6, 0, 12), \max(0, 6, 0, 12)] = [0, 12]$$

$$[1, 2] \cdot [1, 3] + [1, 2] \cdot [-1, 3] = [\min(1, 2, 3, 6), \max(1, 2, 3, 6)] + [\min(-1, -2, 3, 6), \max(-1, -2, 3, 6)] = [1, 6] + [-2, 6] = [-1, 12]$$

$$[0, 12] \neq [-1, 12]$$

The reciprocal is only defined for intervals that are positive or negative: $0 \lt [a, b]$ or $0 \gt [a, b]$, and is defined by

$$1/[a, b] \coloneqq [1/b, 1/a]$$

Division is only defined for intervals in the denominator that are positive or negative: $0 \lt [c, d]$ or $0 \gt [c, d]$, and is defined by

$$[a, b]/[c, d] \coloneqq [a, b] \cdot [1/d, 1/c] = [\min(a/c, a/d, b/c, b/d), \max(a/c, a/d, b/c, b/d)]$$

However, division of an interval by itself does not return the constant one: if $0 \lt [a, b]$ or $0 \gt [a, b]$, then

$$[a, b]/[a, b] = [\min(a/a, a/b, b/a, b/b), \max(a/a, a/b, b/a, b/b)] = [a/b, b/a]$$

and $[a/b, b/a] = 1$ if and only if $a = b$.

Ordering of the intervals

The linear order of the rational intervals is defined as

$$([a, b] \lt [c, d]) \coloneqq (b \lt c)$$

The opposite linear order of the rational intervals is defined as

$$([a, b] \gt [c, d]) \coloneqq (a \gt d)$$

The containment relation of the rational intervals is defined as

$$([a, b] \subseteq [c, d]) \coloneqq (a \geq c) \wedge (b \leq d)$$

The opposite containment relation of the rational intervals is defined as

$$([a, b] \supseteq [c, d]) \coloneqq (a \leq c) \wedge (b \geq d)$$

See also

• arithmetic

• interval

• ordered field

• interval cut

• Dedekind cut