Remainder, when referring to books, is a book that was unsold at the list price, and has been marked so that it will not be sold by or returned to a bookstore for full list price. The mark is almost always made with a black permanent marker, across the top or bottom of the pages when the book is closed. The mark is called a remainder mark.

In mathematics, the remainder is the amount "left over" after division, and technically subtraction also. For example, the remainder of 10 / 4 is 2, because 4 goes into ten twice, with two left over. This is often how children are taught to handle division results that would otherwise be non-integral. You'll see this written as 10 / 4 = 2 r 2, the 'r' standing for remainder.

Of course, once we get into the realm of rational numbers, we don't have any need for remainders... do we? Well, not arithmetically speaking, no. However, they do have some algebraic use - they're the key part of Euclid's proof that there is no largest prime number, for one!

Remainder can also apply not just to numbers but to functions. That is to say, we can divide one function by another and examine the remainder. This also has many uses in function theory.

We can also find the remainder by working modulo a certain number ie, 13 mod 6 is 1, and therefore 13 / 6 has a remainder of 1. In C and related computer languages, mod is represented by the % symbol; we can find the remainder without explicity finding the quotient - that is, the real result of the division.

Remainders are generally thought of as a childish concept from arithmetic but are actually remarkably useful in proofs. I hope this was illuminating.

Re*main"der (r?-m?n"d?r), n. [OF. remaindre, inf. See Remain.]

1.

Anything that remains, or is left, after the separation and removal of a part; residue; remnant.

"The last remainders of unhappy Troy."

Dryden.

If these decoctions be repeated till the water comes off clear, the remainder yields no salt.
Arbuthnot.

2. Math.

The quantity or sum that is left after subtraction, or after any deduction.

3. Law

An estate in expectancy, generally in land, which becomes an estate in possession upon the determination of a particular prior estate, created at the same time, and by the same instrument; for example, if land be conveyed to A for life, and on his death to B, A's life interest is a particuar estate, and B's interest is a remainder, or estate in remainder.

Syn. -- Balance; rest; residue; remnant; leavings.

Re*main"der, a.

Remaining; left; left over; refuse.

Which is as dry as the remainder biscuit
After a voyage. Shak.