Negative numbers/Talk - Philosophical Investigations

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Here's what wikipedia has to say...

Negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art (Jiu zhang suan-shu), which in its present form dates from the period of the Han Dynasty (202 BC – 220 AD), but may well contain much older material.[1] The Nine Chapters used red counting rods to denote positive coefficients and black rods for negative.[2] (This system is the exact opposite of contemporary printing of positive and negative numbers in the fields of banking, accounting, and commerce, wherein red numbers denote negative values and black numbers signify positive values). The Chinese were also able to solve simultaneous equations involving negative numbers.

For a long time, negative solutions to problems were considered "false". In Hellenistic Egypt, Diophantus in the third century A.D. referred to an equation that was equivalent to 4x + 20 = 0 (which has a negative solution) in Arithmetica, saying that the equation was absurd.

The use of negative numbers was known in early India, and their role in situations like mathematical problems of debt was understood.[3] Consistent and correct rules for working with these numbers were formulated.[4] The diffusion of this concept led the Arab intermediaries to pass it to Europe.[3]

The ancient Indian Bakhshali Manuscript, which Pearce Ian claimed was written some time between 200 B.C. and A.D. 300,[5] while George Gheverghese Joseph dates it to about 400 AD and Takao Hayashi to no later than the early 7th century,[6] carried out calculations with negative numbers, using "+" as a negative sign.[7]

During the 7th century A.D., negative numbers were used in India to represent debts. The Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta (written in A.D. 628), discussed the use of negative numbers to produce the general form quadratic formula that remains in use today. He also found negative solutions of quadratic equations and gave rules regarding operations involving negative numbers and zero, such as "A debt cut off from nothingness becomes a credit; a credit cut off from nothingness becomes a debt. " He called positive numbers "fortunes," zero "a cipher," and negative numbers "debts." [8][9]

During the 8th century A.D., the Islamic world learned about negative numbers from Arabic translations of Brahmagupta's works, and by A.D. 1000 Arab mathematicians were using negative numbers for debts.

In the 12th century A.D. in India, Bhaskara also gave negative roots for quadratic equations but rejected them because they were inappropriate in the context of the problem. He stated that a negative value is "in this case not to be taken, for it is inadequate; people do not approve of negative roots."

Knowledge of negative numbers eventually reached Europe through Latin translations of Arabic and Indian works.

European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of Liber Abaci, A.D. 1202) and later as losses (in Flos).

In the 15th century, Nicolas Chuquet, a Frenchman, used negative numbers as exponents and referred to them as “absurd numbers.”[citation needed]

In A.D. 1759, Francis Maseres, an English mathematician, wrote that negative numbers "darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple". He came to the conclusion that negative numbers were nonsensical. [10]

In the 18th century it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless.[11]

WP also notes:

Brahmagupta stated in Brahmasputhasiddhanta "positive times positive is positive and negative times negative is positive". Diophantus had earlier stated the rule but only as a route towards getting an eventual positive result. However due to a distrust of negative numbers even as late as the 18th century this rule was challenged by Lazare Carnot. He asked how the square of a smaller number could be larger than the square of a larger number, for example, how could the square of −3 be larger than the square of −2, as −3 is smaller than −2?

2010-09-04 04:59:37 A related topic (also see our PI article on Fermat) In The Eye Of The Beholder: Art, Justin Bieber And The Best Equation Ever
http://www.npr.org/templates/story/story.php?storyId=129610905&sc=fb&cc=fp

The best equation ever - the incomparable and glorious Euler's Identity. Here is it:

ei(π)+1 =0

Why are so many mathematically inclined folks sent into paroxysms of delight over this string of symbols which seem like gibberish to others. Let me quote from Paul Nahin's wonderful Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills:

I think [Euler's Identity] is beautiful because it is true even in the face of enormous constraint. The equability is precise: the left hand is not "almost" or "pretty near" or "just about" zero, but exactly zero. That five numbers each with vastly different origins, and each with roles in mathematics that can not be exaggerated, should be connected by such a simple relationship, is just stunning. It is beautiful. And unlike the physics or chemistry or engineering of today which will surely appear archaic to technicians of the future, Euler's equation will still appear, to the arbitrarily advanced mathematicians ten thousand years hence, to be beautiful and stunning and untarnished by time.

So to begin with, its the appearance of those five magic numbers that starts things off. Let's go down the list.

Zero (0) which can be added to any number and the number is unchanged
One (1) which can be multiplied to any number and the number is unchanged
Pi which appears everywhere from trigonometry on to the most abstract domains of mathematics
e which is the base of natural logarithms. It is also a number which "just shows up" again and again in the exploration of mathematics
i which is the fundamental imaginary number. There is no number which when multiplied by itself gives a negative result i.e. 1*1 = 1; (-1)*(-1)=1. So someone just invented a new number: i = Square Root(-1). They called it imaginary (cause it is) and so much mathematical magic suddenly appeared as to almost defy imagination. How can an imaginary number be so central to mathematics?

Each of these numbers plays so deep a role in mathematics and therefore physics, engineering etc. that in and of themselves they draw out a sense of wonder. The power of 0 and 1 is obvious. But the other numbers appear again and again in the places you least expect it. That ubiquity makes the mathematical explorer draw back in awe. But to find all these numbers united in the single relationship that is Euler's Identity speaks to something even more powerful. —PerigGouanvic

2011-02-24 03:52:39

about complex numbers

Fromstanford encyclopedia... an excerpt for later use...
Style, according to Granger, is a way of imposing structure to an experience. Experience must be taken here to go beyond empirical experience. In general the kind of experience the mathematician appeals to is not empirical. From this experience come the “intuitive” components that are structured in mathematical activity. But one should not think that there is an “intuition” to which, as it were externally, one then applies a form. The mathematical activity gives rise at the same time to form and content within the background of a certain experience.

Style appears to us here on the one hand as a way of introducing the concepts of a theory, of connecting them, of unifying them; and on the other hand, as a way of delimiting the what intuition contributes to the determination of these concepts. (Granger 1968, 20)
As an example Granger gives three ways of introducing the complex numbers; all three ways account for the structural properties which characterize the algebraic structure in question. The first way introduces the complex numbers by trigonometric representation using angles and directions. The second introduces them as operators applied to vectors. In the first case, one defines a complex number as a pair of real numbers and the additive properties are then immediate. By contrast, in the second case, it is the multiplicative properties that are immediately seized. But, and this is the third way, one can also introduce complex numbers by regular square matrices. This leads to seeing the complex numbers as a system of polynomials in x modulo x2+1. —PerigGouanvic