Division by zero is an illegal operation, regardless of the value of the dividend. It has an undefined value, and therefore no meaningful value. No conclusions can be drawn, and usually the operation if otherwise unnoticed will produce a contradictory or illogical result. No numeric or abstract concept exists that explains or justifies a division by zero, and therefore, if such an expression is reached, we can conclude that a mathematical error or false premise is to blame. Most calculators return an error comment, sometimes a small capitalized E.

Example of the fallacy[edit]

Given the obviously true statement:

Each side may be multiplied by an arbitrary value:

The above is still true, given the fact that zero multiplied by any number is still zero. This is known as the zero property of multiplication.

If division by zero were permissible:

Simplifying by dropping out common factors, an obvious fallacy results. Any number can be made to equal any other number or expression via this error in algebra.

In math culture[edit]

Either as a joke or as a true misconception, some believe in the proof that 1=2 based on a trickery of algebraic manipulations

Define (as a premise):

Therefore:

(This statement is false, as proven below)

This, and proofs like it, all share the common algebraic error of a division by zero. As in the first example, above, zero equals zero, and from it any number can be made to equal any other number due to division by zero.

The transition from step four to step five involves the division by the term .

The trick in the math lies here, as a division by zero is obscured with a seemingly innocent algebraic expression. If the first line of the proof, , is valid then . And so, division by  IS a division by zero.

Reasoning for being undefined[edit]

Division can be thought of as being a multiplication of one input value by the multiplicative inverse of the other. That is, . If zero had a multiplicative inverse (a number by which we can multiply by our given number to obtain one), then division by zero would be possible. However, we know that  for all , so zero does not have a multiplicative inverse, and thus  is meaningless.

Wikipedia article is at the bottom

One of the commonest questions which the readers of this archive ask is: Who discovered zero? Why then have we not written an article on zero as one of the first in the archive? The reason is basically because of the difficulty of answering the question in a satisfactory form. If someone had come up with the concept of zero which everyone then saw as a brilliant innovation to enter mathematics from that time on, the question would have a satisfactory answer even if we did not know which genius invented it. The historical record, however, shows quite a different path towards the concept. Zero makes shadowy appearances only to vanish again almost as if mathematicians were searching for it yet did not recognise its fundamental significance even when they saw it.

The first thing to say about zero is that there are two uses of zero which are both extremely important but are somewhat different. One use is as an empty place indicator in our place-value number system. Hence in a number like 2106 the zero is used so that the positions of the 2 and 1 are correct. Clearly 216 means something quite different. The second use of zero is as a number itself in the form we use it as 0. There are also different aspects of zero within these two uses, namely the concept, the notation, and the name. (Our name "zero" derives ultimately from the Arabic sifr which also gives us the word "cipher".)

Neither of the above uses has an easily described history. It just did not happen that someone invented the ideas, and then everyone started to use them. Also it is fair to say that the number zero is far from an intuitive concept. Mathematical problems started as 'real' problems rather than abstract problems. Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five". If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer.

One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 BC survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 BC that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6.

The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 '' 6 we never find 216 ''. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases.

If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer "It's three fifty" means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended.