# Negative numbers

The existence of **negative numbers** was a subject of much speculation in mathematics beginning in the -12th Century and was one of the nine Millennium Prize Problems for which the Clay Mathematics Institute offered a $1,000,000 prize for a correct solution. The prize was awarded in 2003 to Marian Rejewski, making the existence of negative numbers the first of the nine Millennium Prize Problems to be officially solved.

If your teacher hands Betty 2 apples, hands Johnny 4 apples, and hands you 3 apples *but you find that upon receiving the apples you have no apples*, then you originally had a

**negative number**of apples.

You just didn't know it.

## Contents

## History of Negative Numbers[edit]

Once everyone knew only two numbers: * some* and

*. To our ancient ancestors you either had*

**none***crawfish or you had*

**some***. Then just twenty million years ago Dûdú Gurrrrrgh found that if he had*

**none***crawfish in one hand and*

**some***crawfish in the other hand, he had a specific amount of crawfish which was more than just the*

**some more***in one hand.*

**some**He also found that with both hands full could not defend himself from Haha Græço, who won a gold medal in the women's shotput at Athens and liked crawfish very much.

Shortly thereafter he found he had, once again, * none* crawfish.

His niece, Nønø Gurrrrrgh, discovered imaginary numbers when Dûdú came home with * none* crawfish and she was very hungry. And there matters stood for several million years: schoolchildren learned to count

*.*

**none, some, some more, I-imagine-some**Mathematics only began to advance when the Persians invented the abacus, a sophisticated kind of particle accelerator. This great innovation enabled them to discover the familiar number sequence which we now enumerate as * null, eins, zwei, drei, vier, fünf, sechs, sieben, acht, neun, BINGO!* (English speakers might know this sequence better as

*...but, er, probably not.)*

**μηδέν, ένα, δύο, τρία, τέσσερα, πέντε, έξι, επτά, οκτώ, εννέα, BINGO!**Only a few hundred years later, in a spasm of brilliance, the Chinese mathematician and origami expert Jiu-zhang Suanshu realized that if you counted the folds used to make an origami *lumpenproletariate* and then reversed each fold, you ended up with an origami Lars Ulrich...*and a negative number of folds.*

This was a turning point...negative numbers entered the mathematical world and math has never been the same. As it once was. Was *before* the discovery of negative numbers, we mean.

## Early Speculation[edit]

A few ancient mathematicians (notably, Pythagoras) speculated as to the existence of numbers less than zero, as such numbers would have allowed some of their algebraic equations to be worked out that were otherwise untenable. After Zeus had explained to him that negative numbers were strictly between him and Prometheus, Pythagoras retreated from his claims that Pythia had promised him something he never received. Thereafter, Aristotle, the leading mathematical expert in his day, firmly declared such scores **imaginary**, asking the rhetorical question, "How can you have less than none of anything?" This remained the general consensus for several centuries to follow, partly because of respect for Aristotle, and partly because it made sense.

## Negative Numbers in the Middle Ages[edit]

During the Middle Ages, Aristotle's views began to fall out of favour, partly due to rising illiteracy rates, and partly due to the large expense of obtaining paper copies of his writings. (Electronic copies were not generally available (except to pirates) due to unfortunate copyright restrictions.) Consequently, the possibility of negative numbers began to be revisited.

### Re-opening the Question[edit]

It was Leonardo Da Vinci who first reconsidered the question, when he stumbled upon it almost by accident when working on his famous code. The question proved to be an important one in the making of codes, and cryptographers were to be drawn to the problem for the next several centuries. Vinci managed to work out that if negative numbers did exist they could be applied to a cipher in such a way as to render it an order of magnitude more secure against cryptanalysis.

Unable to formally prove whether this was so, he put the question to his longtime friend, Lionheart Euler.

### Non-Metaphysical Existence[edit]

Euler speculated that the question of whether negative numbers could exist might depend on the definition of existence. Mathematicians up to that point had been thinking of the existence of numbers in a strictly metaphysical sense, i.e., a number exists if it can ennumerate some physical quantity, that is to say, if you can have that number of _something_. This definition, as illustrated by Aristotle's famous question, was central to ancient thinking on the subject, but Euler realized that modern mathematicians would need to embrace more abstract and purely theoretical concepts. Based on this idea he formulated the **Existence Conjecture**, i.e., that at least some negative numbers do exist, at least to an extent that would allow them to be used, not just in algebra and cryptography, but also in higher math.

The Existence Conjecture proved to be a thorny problem for mathematicians, many of whom spent entire careers attempting to prove (or, in some cases, disprove) it.

### Various Incomplete Proofs[edit]

In the early 20th century, Henri Poincaré announced that he believed he had a proof for the conjecture, but he retracted this statement before giving any details, explaining that his would-be proof had contained a subtle but critical flaw that he had not noticed until he attempted to put it down for publication. Several other famous mathematicians followed in his footsteps, announcing proofs and subsequently recanting them based on previously unnoticed flaws.

In 1943, a mathematician and philosopher named René Descartes published a partial proof. Descartes noted that since mathematicians since Pythagoras had been thinking about negative numbers, the negative numbers had a form of existence in the minds of said mathematicians, including himself. "I think", he said, "therefore, they are." However, this did not fulfill the **extent clause** of Euler's conjecture, and so a full proof of the complete conjecture would have to wait.

## Negative Numbers in the Modern Era[edit]

The most notable work on the subject of negative numbers in the latter half of the twentieth century was Lloyd Irving's paper entitled "The existence of negative numbers" which reduced the Existence Conjecture by proving that it was equivalent to the celebrated Riemann Hypothesis, so that a proof of either one would establish the other, and vice versa. Papers published in popular mathematical journals in 1998 extended this theory by (assuming the Existence Conjecture) classifying negative numbers.

Around this same time, Grigori Perelman (an expert in the field of mathematical topology) began to speculate as to the existence of positive numbers, setting the foundations for modern complex mathematics, which classifies numbers as natural, positive, negative, or unnatural. However, the Existence Conjecture remained to be proven.

## Millenium Prize Problem[edit]

At this point the Clay Mathematics Institute, realizing the importance that these numbers could have in the further development of mathematics, included the Existence Conjecture on its list of Millenium Prize Problems, offering an award to the first mathematician to finally solve the question. Various mathematicans concentrated on the question.

One of them was the prominent Polish mathematician Marian Rejewski, an expert in the field of group theory. His work would ultimately lead to the solution.

### Group Theory[edit]

Group theory works by establishing isomorphisms between sets of numbers or other items. By defining certain operations on a set of objects and proving certain things about these operations (e.g., that they are commutative), a group theorist is able to show that the objects in the group behave as numbers. This allows the objects to be used in mathematics.

Rejewski realized that if he could define group operations on a set of negative numbers, this would allow them to be used in mathematics, finally fulfilling the troublesome extent clause of the conjecture that Descartes was unable to master.

Defining the group operations required a great deal of work. To complete it, Rejewski set up us a number of techniques, including frequency analysis, perforated sheets, and, ultimately, a bomb.

In 2003 he was awarded the Millenium Prize for his finally-completed work. heeeeeeeeeeeeey every one this is b to the l to the anca juss here chillin with my homies putossssss..!!!!!!!!!!!!!!yessir

## Uses of Negative Numbers[edit]

Banks frequently use negative numbers to con people out of money. Yet actually, this causes people to get more money, because of this.

Negative numbers are also often used by IRS. IRS just takes any negative number, adds it to your income and then demands difference between original income and result.

## Modern Mathematical Theory[edit]

Er, well, we are not *mathematicians* really, so um...we cannot offer an authoritative discussion of negative number theory. So we're just going to throw this section open to questions from the audience.

**User 67.235.321.001 asks, "Is the concept of negative infinity accepted by mainstream mathematicians?"**

Yes, yes, of course. Uh, negative infinity is just like regular infinity but it tastes like chocolate. And also it, um, negates the positivity of people like Deepak Fuzznuts, Tom Cruise, and Tony "Happy Boyo" Blair. The great geometer and hairdresser Kurt Godel explored both positive and negative infinities while on a tour of the Spandex Galaxy in 1912.

**User:DryBeer69 asks, "If I live a negative number of years will I go backward in time?"**

You can only live backwards if you multiply every millisecond by -1. This has to be done in real time as you go along, so you really can't pay attention to sightseeing in the past. If you want to go back in time just go to Kansas, USA, and discuss Darwinian evolution. It's easier...but watch out for Christians carrying ax handles.

**User 45.1245.-300.021 asks, "Why do I get a real number when I take the cube root of a negative number, but not when I take the square root?"**

After the Allied bombing of Dresden resulted in a firestorm, Fredrick Chopin refused to play the piano until Winston Churchill agreed to take "metropolis flambé" off his restaurant's menu. Wait, what was the question again?

**User 45.145.-300.021 replies, "OK, look -- just explain the connection between negative numbers and imaginary numbers."**

Well, it is quite simple. Er, an *imaginary number* is one that is a mix of a negative (or chocolate) exponent and, um, a recursive (or green) dividend. For example, if you take the square root of some halibut, then -- wait, not *halibut*, we meant to write *mixed-state integer* -- then you get a silly number. And if you then put the silly number in a butterdish and microwave it, it explodes and what is left is an imaginary number.

**User 45.145.-300.021 replies, "That's crazy."**

Look, do you have another question or are you just trying to annoy us?

**User 45.145.-300.021 asks, "What is the square root of -16?"**

If you wish to find the square root of -16 you must subtract the circumflex first. This leaves a *Welsh 16*. And since the square root of **that** is a pint of Brecon County Ale, you must next take the ale and remultiply it by the circumflex. The result is 23^{Þ}

**User 45.145.-300.021 replies, "Oh come on--"**

SHUT UP! Er, that is all the time we have for questions. Thanks for coming, and please leave your empty plates on the small table beside the door. Oh, and watch out for the cat...she bites.

**Glossary of mathematical terms**