Remainder

remainder

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

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remainder [ri′mān·dər]

(mathematics)

The remaining integer when a division of an integer by another is performed; ifl=m·p+r,wherel, m, p,andrare integers andris less thanp,thenris the remainder whenlis divided byp.

The remaining polynomial when division of a polynomial is performed; ifl=m·p+r,wherel, m, p,andrare polynomials, and the degree ofris less than that ofp,thenris the remainder whenlis divided byp.

The remaining part of a convergent infinite series after a computation, for somen,of the firstnterms.

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remainder

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.

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Remainder 

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 number 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 x ≥ 2, 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 R_n (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 R_n(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.