# remainder

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## 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

## 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 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.

## remainder

[ri′mān·dər]
(mathematics)
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.

## 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.
References in periodicals archive ?
Remainder calculation function for easy division calculations to find a quotient and remainder
2055(e)(2)(A), an estate is allowed a deduction for a charity's remainder interest if the charity's interest is in any of certain designated charitable trusts, including a CRUT as defined under Sec.
In this paper we generalize the above result and prove the following: A non-locally compact paratopological group G has a remainder which is a p-space if and only if G is either a Lindelof p-space or a ([sigma]-compact space.
Looking to the Single Life Remainder Factors table and using Jack's mother's age of seventy, and the interest rate of 3.
The tax savings and increased cash flow offered by the use of a charitable remainder trust will often enable the grantor to use some or all of these savings to fund a wealth replacement trust for the benefit of his heirs, thereby providing for the tax effective replacement of property which had been transferred to the charitable remainder trust.
To be immediately deductible, the gift must be of real property or intangibles; a gift of a remainder interest in tangible personal property is deductible only when all intervening interests have expired or are held by parties unrelated to the donor.
3 : the number left over from the dividend after division that is less than the divisor <10 divided by 3 leaves 3 with a remainder of 1.
In determining the value of the charitable remainder interest for purposes of the deduction, straight-line depreciation is taken into account if the property is depreciable.
For this reason the NIMCRUT is perhaps the most popular type of charitable remainder trust, since it allows for the deferral of income to later years, much like a deferred-income retirement plan, and still allows the trustee to maximize the payout amount.
Charitable remainder trusts allow an individual to donate an asset to a charity and then generate a fixed or variable rate annuity on the value of that donation.
121-4(e)(2)(ii), that is, where the taxpayer applies the exclusion upon the sale of a remainder interest in the principal residence and later sells another interest in the residence, such as a life estate.
Herta has been in talks with PK Racing to drive a second car for the team for the remainder of the season.

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