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1. Maths
a. the amount left over when one quantity cannot be exactly divided by another
b. another name for difference
2. Property law a future interest in property; an interest in a particular estate that will pass to one at some future date, as on the death of the current possessor



The remainder in an approximation formula is the difference between the exact and the approximate values of the expression represented by the formula. A remainder can take different forms depending on the nature of the approximation formula. The task of investigating a remainder usually consists in obtaining estimates for it. For example, corresponding to the approximate formula

we have the exact equality

where the expression R is the remainder for the approximation 1.41 for the numberRemainder and it is known that 0.004 < R < 0.005.

Remainders are constantly encountered in asymptotic formulas. For example, for the number π (x) of primes not exceeding x we have the asymptotic formula

where μ is any positive number less than 3/5. Here, the remainder, which is the difference between the functions π (x) an ∫x2 du/ln u for x2, is written in the form

O[xe-(In x)μ

where the letter O indicates that the remainder does not exceed the expression

Cxe-(In x)μ

in absolute value, C being some positive constant. Remainders are found in formulas that give approximate representations of functions. For example, in the Taylor formula

the remainder Rn (x) in Lagrange’s form is

where θ is a number such that 0 < θ < 1; θ generally depends on the values of x and h. The presence of 0 in the formula for Rn(x) introduces an element of indefiniteness; such indefinite-ness is inherent in many formulas for the remainder.

Remainders also occur in quadrature formulas and interpolation formulas.


The remaining integer when a division of an integer by another is performed; if l = m · p + r, where l, m, p, and r are integers and r is less than p, then r is the remainder when l is divided by p.
The remaining polynomial when division of a polynomial is performed; if l = m · p + r, where l, m, p, and r are polynomials, and the degree of r is less than that of p, then r is the remainder when l is divided by p.
The remaining part of a convergent infinite series after a computation, for some n, of the first n terms.


An interest in property that confers a right to possession in someone other than the grantor or his heirs upon the termination of a prior interest, such as following the death of a life tenant.