0
Zero (0) is both a number and a numeral. The number zero is the size of the empty set: if you don't have any apples, then you have zero apples. The numeral or digit zero is used in positional number systems, where the position of a digit signifies its value, with successive positions having higher values, and the digit zero is used to skip a position. This use was introduced into mathematics relatively late, during the early 800s in Indian books. Prior number systems, such as the Babylonian or Greek, were not positional and did not use the digit zero at all, but still succeeded quite well for everyday purposes. The number zero was not accepted until later still.
In certain calendars it is common usage to omit the year zero when extending the calendar to years prior to its introduction: see proleptic Gregorian calendar and proleptic Julian calendar.
The following are some basic rules for dealing with the number zero. These rules apply for any complex number x, unless otherwise stated.
- Addition: x + 0 = x and 0 + x = x. (That is, 0 is an identity element with respect to addition.)
- Subtraction: x - 0 = x and 0 - x = -x.
- Multiplication: x · 0 = 0 and 0 · x = 0.
- Division: 0 / x = 0, for nonzero x. But x / 0 is undefined, because 0 has no multiplicative inverse.
- Exponentiation: x0 = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0x = 0.
Distinguishing Zero from O If your zero is centre-dotted and letter-O is not, or if letter-O looks almost rectangular but zero looks more like an American football stood on end (or the reverse), you're probably looking at a modern character display (though the dotted zero seems to have originated as an option on IBM 3270 controllers). If your zero is slashed but letter-O is not, you're probably looking at an old-style ASCII graphic set descended from the default typewheel on the venerable ASR-33 Teletype (Scandinavians, for whom slashed-O is a letter, curse this arrangement).
If letter-O has a slash across it and the zero does not, your display is tuned for a very old convention used at IBM and a few other early mainframe makers (Scandinavians curse *this* arrangement even more, because it means two of their letters collide). Some Burroughs/Unisys equipment displays a zero with a *reversed* slash. And yet another convention common on early line printers left zero unornamented but added a tail or hook to the letter-O so that it resembled an inverted Q or cursive capital letter-O.
A zero of a function f defined on a set S is an element x of S such that f(x) = 0. One of the most important unsolved problems in mathematics concerns the location of the zeros of the Riemann zeta function.
Zero can also mean to set to zero. Usually said of small pieces of data, such as bits or words (especially in the construction "zero out").
Zero means to erase; to discard all data from. This is often said of disks and directory, where "zeroing" need not involve actually writing zeroes throughout the area being zeroed. One may speak of something being "logically zeroed" rather than being "physically zeroed".
This article (or an earlier version of it) contains material from FOLDOC, used with permission. Update as needed.