Negative and non-negative numbers

A negative number is a number that is less than zero, such as −3. A positive number is a number that is greater than zero, such as 3. Zero itself is neither positive nor negative. The non-negative numbers are the real numbers that are not negative (positive or zero). The non-positive numbers are the real numbers that are not positive (negative or zero).

In the context of complex numbers, positive implies real, but for clarity one may say "positive real number".

Negative numbers

Negative integers can be regarded as an extension of the natural numbers, such that the equation x − y = z has a meaningful solution for all values of x and y. The other sets of numbers are then derived as progressively more elaborate extensions and generalizations from the integers.

Negative numbers are useful to describe values on a scale that goes below zero, such as temperature, and also in bookkeeping where they can be used to represent debts. In bookkeeping, debts are often represented by red numbers, or a number in parentheses.

Non-negative numbers

A number is nonnegative if and only if it is greater than or equal to zero, i.e. positive or zero. Thus the nonnegative integers are all the integers from zero on upwards, and the nonnegative reals are all the real numbers from zero on upwards.

A real matrix A is called nonnegative if every entry of A is nonnegative.

A real matrix A is called totally nonnegative by matrix theorists or totally positive by computer scientists if the determinant of every square submatrix of A is nonnegative.

Sign function

It is possible to define a function sgn(x) on the real numbers which is 1 for positive numbers, −1 for negative numbers and 0 for zero (sometimes called the signum function):

\sgn(x)=\left\{\begin{matrix} -1 & : x < 0 \\ \;0 & : x = 0 \\ \;1 & : x > 0 \end{matrix}\right.

We then have (except for x=0):

\sgn(x) = \frac{x}{|x|} = \frac{|x|}{x} = \frac{d{|x|}}{d{x}} = 2H(x)-1.

where |x| is the absolute value of x and H(x) is the Heaviside step function. See also derivative.

Arithmetic involving signed numbers

Addition and subtraction

For purposes of addition and subtraction, one can think of negative numbers as debts.

Adding a negative number is the same as subtracting the corresponding positive number:

5 + (-3) = 5 - 3 = 2 \,

(if you have $5 and acquire a debt of $3, then you have a net worth of $2)

-2 + (-5) = -2 - 5 = -7 \,

Subtracting a positive number from a smaller positive number yields a negative result:

4 - 6 = -2 \,

(if you have $4 and spend $6 then you have a debt of $2).

Subtracting a positive number from any negative number yields a negative result:

-3 - 6 = -9 \,

(if you have a debt of $3 and spend another $6, you have a debt of $9).

Subtracting a negative is equivalent to adding the corresponding positive:

5 - (-2) = 5 + 2 = 7 \,

(if you have a net worth of $5 and you get rid of a debt of $2, then your new net worth is $7).

Also:

(-8) - (-3) = -5 \,

(if you have a debt of $8 and get rid of a debt of $3, then you still have a debt of $5).

Multiplication

Multiplication of a negative number by a positive number yields a negative result: (−2) × 3 = −6. Multiplication of two negative numbers yields a positive result: (−4) × (−3) = 12.

One way of understanding this is to regard multiplication by a positive number as repeated addition. Thus, 2 × 3 = 2 + 2 + 2 = 6 and so naturally (−2) × 3 = (−2) + (−2) + (−2) = −6.

Multiplication by a negative number can be regarded as repeated subtraction. For instance, 3 × (−2) = − 3 − 3 = −6. Notice that this keeps multiplication commutative: 3 × (−2) = (−2) × 3 = −6. Applying the same interpretation of "multiplication by a negative number" for a value that is also negative, we have:

(−4) × (−3)	= − (−4) − (−4) − (−4)
	= 4 + 4 + 4
	= 12

Division

Division is similar to multiplication. If both the dividend and the divisor have different signs, the result is negative:

\; 8 \;/\; (-2) = (-4) \,

(-10) \;/\; 2 = (-5) \,

If both numbers are of the same sign, the result is positive (even if they are both negative):

(-12) \;/\; (-3) = 4 \,

Formal construction of negative and non-negative integers

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:

( a , b ) + ( c , d ) = ( a + c , b + d ) \,

( a , b ) \times ( c , d ) = ( a \times c  + b \times d ,  a \times d + b \times c ) \,

We define an equivalence relation ~ upon these pairs with the following rule:

(a, b)\sim(c, d)

if and only if

a + d = b + c . \,

This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z to be the quotient set N²/~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense.

We can also define a total order on Z by writing

( a , b ) \leq ( c , d ) \,

if and only if

a + d \leq b + c . \,

This will lead to an additive zero of the form (a, a), an additive inverse of (a, b) of the form (b, a), a multiplicative unit of the form (a+1, a), and a definition of subtraction

( a , b ) - ( c , d ) = ( a + d , b + c ). \,

First usage of negative numbers

For a long time, negative solutions to problems were considered "false" because they couldn't be found in the real world (in the sense that one cannot have a negative number of, for example, seeds). The abstract concept was recognised as early as 100 BC - 50 BC. The Chinese "Nine Chapters on the Mathematical Art" (Jiu-zhang Suanshu) contains methods for finding the areas of figures; red rods were used to denote positive coefficients, black for negative. They were able to solve simultaneous equations involving negative numbers. At around the same time in ancient India, the Bakhshali Manuscript written sometime between 200 BC and 200 CE carried out calculations with negative numbers, using a "+" as a negative sign. These are the earliest known uses of negative numbers.

In Hellenistic Egypt, Diophantus in the 3rd century CE referred to the equation equivalent to $4x + 20 = 0$ (the solution would be negative) in Arithmetica, saying that the equation was absurd, indicating that no concept of negative numbers existed in the ancient Mediterranean.

During the 7th century, negative numbers were in use in India to represent debts. The Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta (written in 628) discusses the use of negative numbers to produce the general form quadratic formula that remains in use today. He also finds negative solutions to quadratic equations and gives 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 a "debt". In the 12th century in India, Bhaskara also gives negative roots for quadratic equations but rejects the negative roots since they were inappropriate in the context of the problem, stating that the negative values "is in this case not to be taken, for it is inadequate; people do not approve of negative roots."

From the 8th century, the Islamic world learnt about negative numbers from Arabic translations of Brahmagupta's works, and by about 1000 AD, Arab mathematicians had realized the use of negative numbers for debt.

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

European mathematicians however, 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, 1202) and later as losses (in Flos). At the same time, the Chinese were indicating negative numbers by drawing a diagonal stroke through the right-most non-zero digit. The first use of negative numbers in a European work was by Chuquet during the 15th century. He used them as exponents, but referred to them as “absurd numbers”.

The English mathematician Francis Maseres ^* wrote in 1759 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 did not exist.

Negative numbers were not well-understood until modern times. As recently as the 18th century, the Swiss mathematician Leonhard Euler believed that negative numbers were greater than infinity, and it was common practice to ignore any negative results returned by equations on the assumption that they were meaningless.

Footnotes

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."
Colva Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews, stated this on the BBC Radio 4 "In Our Time", on Negative Numbers, 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". ^*
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.

External links

BBC Radio 4 series "In Our Time", on Negative Numbers, March 9, 2006

Elementary arithmetic | Integers

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