Where does remainder collide?

Euclid's Elements (c. 300 BCE), in Book VII, set out the procedure now called the Euclidean algorithm, in which one whole number is divided by another and the part that cannot be divided evenly is carried forward as what is left over. Carl Friedrich Gauss's 1801 Disquisitiones arithmeticae made this leftover the central object of number theory, grouping whole numbers by the remainder they give on division, the idea he called congruence. Calculus later borrowed the same word for a different quantity, the remainder term that Joseph-Louis Lagrange described in his 1797 Théorie des fonctions analytiques, meaning the error left when a smooth function is approximated by a finite polynomial. In every mathematical use the referent is a leftover quantity produced by an exact operation, not a right in land waiting to take effect or a pile of unsold books.[1][2][3]

Unit

A number smaller than the divisor, left after integer division

Recovery means

Not applicable

Measured by

Computed as the amount left when the dividend is divided by the divisor, written r in a = qn + r

Sir Thomas Littleton's Tenures (c. 1481), the classic early treatise on English estates in land, described the remainder as an estate granted to take effect in a second party once an earlier estate, such as a life estate, comes to its natural end. William Blackstone's 1766 Commentaries on the Laws of England gave the settled account, distinguishing a vested remainder, ready to fall into possession the moment the prior estate ends, from a contingent remainder, which waits on some uncertain event. The referent is a present right to future possession of property, created at the same time as the estate before it, and it has nothing to do with a leftover number in arithmetic or unsold copies of a book.[6][4][5]

Unit

One future interest in land, held by a named party after the prior estate

Recovery means

Not applicable

Measured by

Identified from the granting instrument and classified as vested or contingent