# Article

# Article

- Mathematics
- Algebra and number theory
- Division

- Mathematics
- Arithmetic
- Division

# Division

Article By:

**Blumenthal, Leonard M. **Formerly, Department of Mathematics, University of Missouri, Columbia, Missouri.

Last updated:2014

DOI:https://doi.org/10.1036/1097-8542.202200

**In arithmetic and algebra, the process of finding one of two factors of a number (or polynomial) when their product and one of the factors are given.** The symbol ÷ now used mostly in elementary English and American arithmetics to denote division first appeared in print in an algebra by J. H. Rahn published in Zurich in 1659. Division is more often symbolized by the double dot :, the bar —, or the solidus /; thus *x*:*y*, $\frac{x}{\text{y}}$, or $\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$y$}\right.$ indicates division of a number *x* by a number *y*. Considered as an operation inverse to multiplication, $\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$y$}\right.$ is a symbol denoting a number whose product with *y* is *x*. Another way to base division upon multiplication is provided by the concept of the reciprocal of a number. If *y* is any number (real or complex) other than 0, there is a number, denoted by $\raisebox{1ex}{$\text{1}$}\!\left/ \!\raisebox{-1ex}{$y$}\right.$ and called the reciprocal of *y*, whose product with *y* is 1. Then $\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$y$}\right.$ is the symbol for the product of *x* and $\raisebox{1ex}{$\text{1}$}\!\left/ \!\raisebox{-1ex}{$y$}\right.$. This view of division furnishes a means of extending the concept to objects other than real or complex numbers. A whole number is divisible by 2 if its last digit is so divisible, and by 4 if the number formed by the last two digits is so divisible. It is divisible by 3 or 9 according to whether the sum of its digits is thus divisible, respectively, and is divisible by 11 if the difference between the sum of the digits in the odd and the even places can be so divided.* See also: ***Addition**; **Algebra**; **Multiplication**; **Numbering systems**; **Number theory**; **Subtraction**

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