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 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.

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 ?
We investigate local metrizability of remainders of a paratopological group.
The second gift is the remainder interest (the trust corpus) to his son, Tom.
For example, for a charitable remainder trust, the CRAT, CRUT and PIF requirements reduce the incentive and opportunity for the trustee to invest the trust assets in risky securities to obtain higher returns for the noncharitable income beneficiaries, resulting in a smaller remainder interest for the charity if the securities decrease in value.
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In a comprehensive remake of the law of remainders, section 2-707 provides that, unless the trust instrument provides otherwise, all future interests in trust are contingent on the beneficiary's surviving the distribution date.
Our support of the Rock Bottom Remainders 2010 Wordstock tour is aligned with our mission to focus on education and literacy, critical social issues for many communities across the country," said Kathleen Ryan Mufson, vice president, Pitney Bowes Foundation.
The Service has indicated its intent to deny exclusions for gifts to holders of Crummey withdrawal powers when (1) the powerholders have no beneficial interest in the trust other than their withdrawal powers; (2) the powerholders hold only contingent remainder interests or discretionary income interests; or (3) the facts and circumstances indicate a prearranged understanding that the powerholder will not exercise his or her withdrawal power.
Billed as the "literary world's answer to the Rolling Stones," the Rock Bottom Remainders also features the musical talents of other top writers including Mitch Albom, Kathi Goldmark, Greg Iles, Stephen King and Ridley Pearson.
Example 2: N placed property in trust, retaining a life estate in trust income with the remainder to his daughter M or to her estate.
The gift tax value of the remainder interest equals the value of the property transferred to the trust reduced by the actuarial value of the retained annuity interest,
Grantor trusts were qualified S shareholders even though trustees have a contingent remainder interest.