Guide One Plus Two Minus One: Lust, Love, and Number Theory

In this case, losing two debts of three each is the same as gaining a credit of six:. The convention that a product of two negative numbers is positive is also necessary for multiplication to follow the distributive law.

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In this case, we know that. The justification for why the product of two negative numbers is a positive number can be observed in the analysis of complex numbers. The sign rules for division are the same as for multiplication.

If dividend and divisor have the same sign, the result is positive, if they have different signs the result is negative. The negative version of a positive number is referred to as its negation. The sum of a number and its negation is equal to zero:. That is, the negation of a positive number is the additive inverse of the number. Using algebra , we may write this principle as an algebraic identity :. This identity holds for any positive number x.

It can be made to hold for all real numbers by extending the definition of negation to include zero and negative numbers. In general,. The absolute value of a number is the non-negative number with the same magnitude. In a similar manner to rational numbers , we can extend the natural numbers N to the integers Z by defining integers as an ordered pair of natural numbers a , b. We can extend addition and multiplication to these pairs with the following rules:.

How To Count Past Infinity

Note that Z , equipped with these operations of addition and multiplication, is a ring , and is in fact, the prototypical example of a ring. We can also define a total order on Z by writing. This construction is a special case of the Grothendieck construction. Let x be a number and let y be its negative.

By an axiom of the real number system. Thus y is equal to any other negative of x. That is, y is the unique negative of x. For a long time, negative solutions to problems were considered "false". 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 BC — AD , but may well contain much older material. The historian Jean-Claude Martzloff theorized that the importance of duality in Chinese natural philosophy made it easier for the Chinese to accept the idea of negative numbers.

The Nine Chapters used red counting rods to denote positive coefficients and black rods for negative. Liu Hui writes:.

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Now there are two opposite kinds of counting rods for gains and losses, let them be called positive and negative. Red counting rods are positive, black counting rods are negative. LV Gurjar dates it no later than the 4th century,  Hoernle dates it between the third and fourth centuries, Ayyangar and Pingree dates it to the 8th or 9th centuries,  and George Gheverghese Joseph dates it to about AD and no later than the early 7th century, . During the 7th century AD, negative numbers were used in India to represent debts.

AD , discussed the use of negative numbers to produce the general form quadratic formula that remains in use today. In the 9th century, Islamic mathematicians were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid. By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve polynomial divisions. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. 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. 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 , AD and later as losses in Flos.

In , Gerolamo Cardano , in his Ars Magna , provided the first satisfactory treatment of negative numbers in Europe. In all, Cardano was driven to the study of thirteen different types of cubic equations, each expressed purely in terms of positive numbers. He came to the conclusion that negative numbers were nonsensical. In the 18th century it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless. Gottfried Wilhelm Leibniz was the first mathematician to systematically employ negative numbers as part of a coherent mathematical system, the infinitesimal calculus.

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Calculus made negative numbers necessary and their dismissal as "absurd numbers" slowly faded. From Wikipedia, the free encyclopedia. Main article: Number line. Main article: Sign mathematics. Main article: Additive inverse.

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Signed zero Additive inverse History of zero Integers Positive and negative parts Rational numbers Real numbers Sign function Sign mathematics Signed number representations. For example, in the French convention, zero is considered to be both positive and negative. Oxford University Press.

Retrieved 5 December Geological Society of America. Retrieved 6 January — via Google Books. Academic Press. Retrieved 5 December — via www. The Independent.

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Think again". Wiggins; Jay McTighe Understanding by design. ACSD Publications. Cambridge: Cambridge University Press. Page The MacTutor History of Mathematics archive. Retrieved 24 July Page 65— Reidel Publishing Co. The western Ghubar Arabs' main contribution was to make the longer line diagonal rather than straight, though they showed some tendencies to making the character more rectilinear.

The eastern Arabs developed the character from a 6 lookalike into an uppercase V lookalike. Both modern Arab forms influenced the European form, a two-stroke character consisting of a horizontal upper line joined at its right to a line going down to the bottom left corner, a line that is slightly curved in some font variants. As is the case with the European glyph, the Cham and Khmer glyph for 7 also evolved to look like their glyph for 1, though in a different way, so they were also concerned with making their 7 more different.

For the Khmer this often involved adding a horizontal line above the glyph. This horizontal stroke is, however, important to distinguish the glyph for seven from the glyph for one in writings that use a long upstroke in the glyph for 1. On the seven-segment displays of pocket calculators and digital watches, 7 is the number with the most common glyph variation 1, 6 and 9 also have variant glyphs. While the shape of the 7 character has an ascender in most modern typefaces , in typefaces with text figures the character usually has a descender , as, for example, in.

Most people in Continental Europe,  and some in Britain and Ireland as well as Latin America, write 7 with a line in the middle " 7 " , sometimes with the top line crooked. The line through the middle is useful to clearly differentiate the character from the number one , as the two can appear similar when written in certain styles of handwriting.