Title: 0 Content: Zero (0) is a special number., Chapter 14, page 125 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 == 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).  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 == right|A comparison of the letter O and the number 0 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. thumbnail|right|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. == Related pages == * Absolute zero * Bakhshali manuscript * Division by zero * Number theory *Nothing