User:Jim.belk/Draft:Negative number

A negative number is any number that is less than zero. Such numbers are often used to represent the amount of a loss or absence. For example, a debt that is owed may be thought of as a negative asset, or a decrease in some quantity may be thought of as a negative increase. Negative numbers are also used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature.

Negative numbers are usually written with a minus sign in front. For example, $-3$ would represent a negative quantity with a magnitude of three, and would be pronounced “negative three”. In the context of negative numbers, a number that is not negative is called positive, though zero is usually thought of as neither positive nor negative. The positivity of a number may be emphasized by placing a plus sign before it, e.g. $+3$ . In general, the negativity or positivity of a number is called referred to as its sign.

In mathematics, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative whole numbers (together with zero) are referred to as integers.

The number line

The relationship between negative numbers, positive numbers, and zero is often expressed in the form of a number line:

The number line

Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are less. Thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left.

Note that a negative number with greater magnitude is considered less. For example even though (positive) $8$ is greater than (positive) $5$ , written

8 > 5

negative 8 is considered to be less than negative 5:

-8 < -5.

In addition, any negative number is less than any positive number, so

-8 < 5

and

-5 < 8.

As the result of subtraction

Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero:

0 - 3 = -3.

In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers. For example,

5 - 8 = -3

since $8 - 5 = 3$ .

Arithmetic with negative numbers

The minus sign "−" is used for both the operation of subtraction and to signify that a number is negative. The ambiguity does not generally cause problems in arithmetic, as the result of adding a negative number to another is the same as subtracting the number. A negative number may be parenthesised with its sign, e.g. an addition is clearer if written $7 + (-5)$ rather than $7 + -5$ , and gives the same result as the subtraction $7 - 5$ .

Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus sign to explicitly distinguish negative and positive numbers as in^[1]

- 2 + - 5

gives

- 7

.

Addition

Addition of two negative numbers is very similar to addition of two positive numbers. For example,

(-3) + (-5) = -8

.

The idea is that two debts can be combined into a single debt of greater magnitude.

When adding together a mixture of positive of negative numbers, one can think of the negative numbers as being subtracted. As with any addition, the order of the two numbers does not matter:

8 + (-3) = 8 - 3 = 5

and

(-3) + 8 = 8 - 3 = 5

.

In this example, a credit of $8$ is combined with a debt of $3$ , which yields a total credit of $5$ . If the negative number has greater magnitude, then the result is negative:

(-8) + 3 = 3 - 8 = -5

and

3 + (-8) = 3 - 8 = -5

.

Here the credit is $3$ but the debt is $8$ , so the net result is a debt of $5$ .

Subtraction

As discussed above, it is possible for the subtraction of two non-negative numbers to yield a negative answer:

5 - 8 = -3

In general, subtraction of a positive number is the same thing as addition of a negative. Thus

5 - 8 = 5 + (-8) = -3

and

(-3) - 5 = (-3) + (-5) = -8

On the other hand, subtracting a negative number is the same as adding a positive. (The idea is that losing a debt is the same thing as gaining a credit.) Thus

3 - (-5) = 3 + 5 = 8

and

(-5) - (-8) = (-5) + 8 = 3

.

Multiplication

When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The sign of the product is determined by the following rules:

The product of one positive number and one negative number is negative.
The product of two negative numbers is positive.

Thus

(-2) \times 3 = -6

and

(-2) \times (-3) = 6

.

The reason behind the first example is simple: adding three $-2$ 's together yields $-6$ :

(-2) \times 3 = (-2) + (-2) + (-2) = 6

.

The reasoning behind the second example is more complicated. The idea again is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six:

(-2

debts

) \times (-3

each

) = +6

credit.

The convention that a product of two negative numbers is positive is also necessary for multiplication to follow the distributive law. In this case, we know that

(-2) \times (-3) + 2 \times (-3) = (-2 + 2) \times (-3) = 0 \times (-3) = 0

.

Since $2 \times (-3) = -6$ , the product $(-2) \times (-3)$ must equal $6$ .

Division

The sign rules for division are the same as for multiplication. For example,

8 \div (-2) = -4

,

(-8) \div 2 = -4

,

and

(-8) \div (-2) = 4

.

If dividend and divisor have the same sign, the result is always positive.

Formal construction

The integers (including positive integer, negative integers, and zero) can be formally constructed from the natural numbers as certain equivalence classes of ordered pairs. See construction of the integers for details on this construction.

History

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 B.C. – A.D. 220), but may well contain much older material.^[2] The Nine Chapters used red counting rods to denote positive coefficients and black rods for negative.^[3] (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.^[4] Consistent and correct rules for working with these numbers were formulated.^[5] The diffusion of this concept led the Arab intermediaries to pass it to Europe.^[4]

The ancient Indian Bakhshali Manuscript, which Pearce Ian claimed was written some time between 200 B.C. and A.D. 300,^[6] while George Gheverghese Joseph dates it to about A.D. 400 and no later than the early 7th century,^[7] carried out calculations with negative numbers, using "+" as a negative sign.^[8]

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." ^[9]^[10]

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. ^[11]

In the 18th century it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless.^[12]

Notes

^ Grant P. Wiggins; Jay McTighe (2005). Understanding by design. ACSD Publications. p. 210. ISBN 1416600353.
^ Struik, page 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."
^ Temple, Robert. (1986). The Genius of China: 3,000 Years of Science, Discovery, and Invention. With a forward by Joseph Needham. New York: Simon and Schuster, Inc. ISBN 0671620282. Page 141.
^ ^a ^b Bourbaki, page 49
^ Britannica Concise Encyclopedia (2007). algebra
^ Pearce, Ian (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. Retrieved 2007-07-24.{{cite web}}: CS1 maint: date and year (link)
^ Teresi, Dick. (2002). Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas. New York: Simon and Schuster. ISBN 0684837188. Page 65–66.
^ Teresi, Dick. (2002). Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas. New York: Simon and Schuster. ISBN 0684837188. Page 65.
^ Colva M. Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews, stated this on the BBC Radio 4 programme "In Our Time," on 9 March 2006.
^ Knowledge Transfer and Perceptions of the Passage of Time, ICEE-2002 Keynote Address by Colin Adamson-Macedo. "Referring again to Brahmagupta's great work, all the necessary rules for algebra, including the 'rule of signs', were stipulated, but in a form which used the language and imagery of commerce and the market place. Thus 'dhana' (= fortunes) is used to represent positive numbers, whereas 'rina' (= debts) were negative".
^ Maseres, Francis (1731–1824). A dissertation on the use of the negative sign in algebra: containing a demonstration of the rules usually given concerning it; and shewing how quadratic and cubic equations may be explained, without the consideration of negative roots. To which is added, as an appendix, Mr. Machin's Quadrature of the Circle, 1758. Quoting from Maseres' work, "If any single quantity is marked either with the sign + or the sign − without affecting some other quantity, the mark will have no meaning or significance, thus if it be said that the square of −5, or the product of −5 into −5, is equal to +25, such an assertion must either signify no more than 5 times 5 is equal to 25 without any regard for the signs, or it must be mere nonsense or unintelligible jargon."
^ Alberto A. Martinez, Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton University Press, 2006; a history of controversies on negative numbers, mainly from the 1600s until the early 1900s.

References

Bourbaki, Nicolas (1998). Elements of the History of Mathematics. Berlin, Heidelberg, and New York: Springer-Verlag. ISBN 3540647678.
Struik, Dirk J. (1987). A Concise History of Mathematics. New York: Dover Publications.

External links

[[Category:Elementary arithmetic]] [[Category:Integers]] [[Category:Numbers]]

[1] Grant P. Wiggins; Jay McTighe (2005). Understanding by design. ACSD Publications. p. 210. ISBN 1416600353.

[struik33-2] Struik, page 32–33. "In these matrices we find negative numbers, which appear here for the first time in history."

[3] Temple, Robert. (1986). The Genius of China: 3,000 Years of Science, Discovery, and Invention. With a forward by Joseph Needham. New York: Simon and Schuster, Inc. ISBN 0671620282. Page 141.

[bourbaki49-4] Bourbaki, page 49

[ebcal-5] Britannica Concise Encyclopedia (2007). algebra

[6] Pearce, Ian (May 2002). "The Bakhshali manuscript". The MacTutor History of Mathematics archive. Retrieved 2007-07-24.{{cite web}}: CS1 maint: date and year (link)

[7] Teresi, Dick. (2002). Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas. New York: Simon and Schuster. ISBN 0684837188. Page 65–66.

[8] Teresi, Dick. (2002). Lost Discoveries: The Ancient Roots of Modern Science–from the Babylonians to the Mayas. New York: Simon and Schuster. ISBN 0684837188. Page 65.

[9] Colva M. Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews, stated this on the BBC Radio 4 programme "In Our Time," on 9 March 2006.

[10] Knowledge Transfer and Perceptions of the Passage of Time, ICEE-2002 Keynote Address by Colin Adamson-Macedo. "Referring again to Brahmagupta's great work, all the necessary rules for algebra, including the 'rule of signs', were stipulated, but in a form which used the language and imagery of commerce and the market place. Thus 'dhana' (= fortunes) is used to represent positive numbers, whereas 'rina' (= debts) were negative".

[11] Maseres, Francis (1731–1824). A dissertation on the use of the negative sign in algebra: containing a demonstration of the rules usually given concerning it; and shewing how quadratic and cubic equations may be explained, without the consideration of negative roots. To which is added, as an appendix, Mr. Machin's Quadrature of the Circle, 1758. Quoting from Maseres' work, "If any single quantity is marked either with the sign + or the sign − without affecting some other quantity, the mark will have no meaning or significance, thus if it be said that the square of −5, or the product of −5 into −5, is equal to +25, such an assertion must either signify no more than 5 times 5 is equal to 25 without any regard for the signs, or it must be mere nonsense or unintelligible jargon."

[12] Alberto A. Martinez, Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton University Press, 2006; a history of controversies on negative numbers, mainly from the 1600s until the early 1900s.

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v t e Number systems
Sets of definable numbers	Natural numbers ( $\mathbb {N}$ ) Integers ( $\mathbb {Z}$ ) Rational numbers ( $\mathbb {Q}$ ) Constructible numbers Algebraic numbers ( $\mathbb {A}$ ) Closed-form numbers Periods ( ${\mathcal {P}}$ ) Computable numbers Arithmetical numbers Set-theoretically definable numbers Gaussian integers Gaussian rationals Eisenstein integers
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Classification List

The number line

As the result of subtraction

Arithmetic with negative numbers

Addition

Subtraction

Multiplication

Division

Formal construction

History

See also

Notes

References

External links