# 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 ∫^{x}2 *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.

## remainder

[ri′mān·dər]*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*.

*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*.

*n,*of the first

*n*terms.