Zero (0) is a special number. If there are zero things, then there is nothing at all. For example, if a person has zero hats, that means they do not have any hats.

Symbol
Math with zero
History of zero
The place of zero as a number
0 as a number
0 as a numeral
The numerical digit zero
Telling zero and the letter O apart
Zeroes of a function
Related pages
Images for kids
See also

Symbol

The symbol for the number zero is "0". It is the additive identity of common numbers. This means that if a number is added to 0, then that number would remain unchanged.

Math with zero

Adding a number to zero results in that number. For example, adding zero to three gives three. In symbols:

3 + 0 = 3

Subtracting zero from a number always gives that number. For example, subtracting zero from three gives three. In symbols:

3 − 0 = 3

Subtracting a positive number from zero always makes that number negative (or, if a negative number is subtracted from zero, it makes the number positive). In symbols:

0 − 3 = −3

Multiplying a number by zero always gives zero. For example, multiplying forty-three by zero gives zero. In symbols:

43 × 0 = 0

Dividing zero by a number always gives zero. For example, dividing zero by forty-three gives zero. In symbols:

0 ÷ 43 = 0

Any number divided by zero has no answer. In symbols:

43 ÷ 0 has an undefined answer.

In particular, zero divided by zero has no answer. In symbols:

0 ÷ 0 has no answer. The following table includes all of the above examples along with other operations in a condensed, generalized form (where x represents any number).

Operation	Rule	Example
Addition	x + 0 = x	3 + 0 = 3
Subtraction	x - 0 = x	3 - 0 = 3
Multiplication	x × 0 = 0	5 × 0 = 0
Division	0 ÷ x = 0, when x ≠ 0	0 ÷ 5 = 0
Division	x ÷ 0 is undefined	5 ÷ 0 is undefined
Exponentiation	0 ^x = 0, when x ≠ 0	0⁵ = 0
Exponentiation	x ⁰ = 1, when x ≠ 0	5⁰ = 1
Root	√0 = 0
Logarithm	log_b(0) is undefined
Factorial	0! = 1
Sine	sin 0º = 0
Cosine	cos 0º = 1
Tangent	tan 0º = 0
Derivative	0' = 0
Integral	∫ 0 dx = 0 + C

History of zero

The idea of zero was first thought about in Babylon, India and in Central America at different times. Some places and countries did not know about zero, which may have made it harder for those people to do mathematics. For example, the year after 1 BC is AD 1 (there is no year zero). In India, zero was theorized in the seventh century by the mathematician Brahmagupta.

Over hundreds of years, the idea of zero was passed from country to country, from India and Babylon to other places, like Greece, Persia and the Arab world. The Europeans learned about zero from the Arabs, and stopped using Roman math. This is why numbers are called "Arabic numerals".

The place of zero as a number

Zero is almost never used as a place number (ordinal number). This means that it is not used like 1, 2, or 3 to indicate the order, or place, of something, like 1st, 2nd, or 3rd. An exception to this is seen in many programming languages.

Some other things about zero:

The number zero is a whole number (counting number).
The number zero is not a positive number.
The number zero is not a negative number, either.
The number zero is a neutral number.

Any number divided by itself equals one, except if that number is zero. In symbols:

0 ÷ 0 = "not a number."

In time, zero means "now". For example, when a person is counting down the time to the start of something, such as a foot race or when a rocket takes off, the count is: "three, two, one, zero (or go)". Zero is the exact time of the start of the race or when the rocket takes off into the sky.

0 as a number

0 is the integer that precedes the positive 1, and follows −1. In most (if not all) numerical systems, 0 was identified before the idea of 'negative integers' was accepted. It means "courageous one" in hieroglyphics.

Zero is a number which means an amount of null size; that is, if the number of brothers is zero, that means the same thing as having no brothers, and if something has a weight of zero, it has no weight. If the difference between the numbers of pieces in two piles is zero, it means the two piles have an equal number of pieces. Before counting starts, the result can be assumed to be zero; that is the number of items counted before one counts the first item, and counting the first item brings the result to one. And if there are no items to be counted, zero remains the final result.

While mathematicians all accept zero as a number, some non-mathematicians would say that zero is not a number, arguing that one cannot have zero of something. Others say that if one has a bank balance of zero, one has a specific quantity of money in that account, namely none. It is that latter view which is accepted by mathematicians and most others.

Normally speaking, there was no year zero between 1 BC and 1 AD. More specifically, almost all historians leave out the year zero from the proleptic Gregorian and Julian calendars (that is, from the normal calendar used in English-speaking countries), but astronomers include it in these same calendars. However, the phrase Year Zero may be used to describe any event considered so important, that someone might want to start counting years all over again from zero.

0 as a numeral

The modern numeral 0 is normally written as a circle or (rounded) rectangle. In old-style fonts with text figures, 0 is usually the same height as a lowercase x.

On the seven-segment displays of calculators, watches, etc., 0 is usually written with six line segments, though on some historical calculator models, it was written with four line segments. The four-segment 0 is not common.

The number zero (as in the "zero brothers" example above) is not the same as the numeral or digit zero, used in numeral systems using positional notation. Successive positions of digits have higher values, so the digit zero is used to skip a position and give appropriate value to the preceding and following digits. A zero digit is not always necessary in a different positional number system. Something called bijective numeration is a possible example of a system without zeroes.

The numerical digit zero

0 (zero) is also used as a numerical digit used to represent that number in numerals. It is used to hold the place of that digit, because correct placing of digits affects a numeral's value.

Examples:

In the numeral 10, which stands for one times ten and zero units (or ones).
In the numeral 100, which stands for one times a hundred plus zero tens plus zero units.

Telling zero and the letter O apart

The number 0 and the letter O are both round, though of different widths. The difference is important on a computer. For one thing, a computer will not do arithmetic with the letter O, because it does not know that it should have been a zero.

The oval-shaped zero and circular letter O came into use together on modern character displays. The zero with a dot in the centre seems to have begun as a choice on IBM 3270 controllers (this has the problem that it looks like the Greek letter theta). The slashed zero, looking like the letter O with a diagonal line drawn inside it, is used in old-style ASCII graphic sets that came from the default typewheel on the well-known ASR-33 Teletype. This format causes problems because it looks like the symbol $\emptyset$ , representing the empty set, as well as for certain Scandinavian languages which use Ø as a letter.

The rule which has the letter O with a slash and the zero without was used at IBM and a few other early mainframe makers; this is even more of a problem for Scandinavians, because it looks like two of their letters at the same time. Some Burroughs/Unisys computers display a zero with a backwards slash. And yet another convention common on early line printers left zero without any extra dots or slashes but added a tail or hook to the letter O so that it resembled an inverted Q or cursive capital letter O.

A German licence plate showing zeroes

The letters used on some European number plates for cars make the two symbols look different. This is done by making the zero rather egg-shaped and the O more circular, but most of all by cutting open the zero on the upper right side, so the circle is not closed any more (as in German plates). The style of letters chosen is called fälschungserschwerende Schrift (abbr.: FE Schrift), meaning "script which is harder to falsify". But those used in the United Kingdom do not make the letter o and the number 0 look different from each other, because there can never be any mistake if the letters are correctly spaced.

In paper writing you do not have to make the 0 (zero) and O (letter O) look different at all. Or you may add a slash across the zero in order to show the difference.

Zeroes of a function

Functions are explained in the Function (mathematics) article. If the function f(x) = 0, then x is called a zero (or root) of the function f. For example, if the function f(x) is x² − 1, then the zeroes of the function are +1 and −1, because f(+1) = (+1)² − 1 = 0, and f(−1) = (−1)² − 1 = 0.

Zeroes of a function are used because they are another way to talk about solving an equation, which is a main goal in algebra. If we want to solve an equation like x² = 1, then we can subtract the right-hand side of the equation from both sides, in this case 1. Whatever we get on the left-hand side, in this case x² − 1, can be called a function f(x). The right-hand side has to be zero, because we subtracted it from itself. So f(x) = 0. Finding the zeroes of this function is the same as solving this equation. In the paragraph before, the zeroes of this function are +1 and −1, so they are the solutions of this equation. We got this equation by subtracting the same thing from both sides, so we also have solutions to the equation we started with, in this case x² = 1. More generally, if we could find zeroes of functions, we could solve any equation.

Images for kids

$illustration of a fractured inscribed stone with pre-Columbian glyphs and icons$

The back of Epi-Olmec stela C from Tres Zapotes, the second oldest Long Count date discovered. The numerals 7.16.6.16.18 translate to September, 32 BC (Julian). The glyphs surrounding the date are thought to be one of the few surviving examples of Epi-Olmec script.
Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus
This is a depiction of zero expressed in Chinese counting rods, based on the example provided by A History of Mathematics. An empty space is used to represent zero.
The number 605 in Khmer numerals, from the Sambor inscription (Saka era 605 corresponds to AD 683). The earliest known material use of zero as a decimal figure.