Interval arithmetic operations ($X+Y,X-Y,XY,X/Y$)

For closed intervals $X=[a,b]$ and $Y=[c,d]$, exact addition, subtraction and multiplication giveThese interval arithmetic operations describe all results $u\circ v$ for $(u,v)$ in the Cartesian product $X\times Y$: the two inputs are allowed to vary independently. Independence here means freedom to choose either input anywhere in its closed interval; it does not assert anything about probability. For example, $[1,2]+[3,5]=[4,7]$ and $[1,2]-[3,5]=[-4,-1]$.

All four endpoint products matter when signs change. For $X=[-2,3]$ and $Y=[-4,1]$, the endpoint products are $8,-2,-12,3$, so $XY=[-12,8]$. Taking just $ac$ and $bd$, even after sorting them, would give $[3,8]$ and lose valid negative products.

For division, if $0\notin Y$, the multiplicative inverse gives $1/Y=[1/d,1/c]$ and $X/Y=X(1/Y)$. Thus $[1,2]/[3,4]=[1/4,2/3]$. This ordinary finite-closed interval rule requires a denominator that excludes zero. When the denominator contains zero, division is undefined for some pairs and can become unbounded near zero; these formulas cannot supply a finite interval enclosure of all quotients. The operations above use exact endpoints; a computer implementation must also apply outward rounding.