Negative number

The minus sign "−" signifies the operator for both the binary (two-operand) operation of subtraction (as in y − z) and the unary (one-operand) operation of negation (as in −x, or twice in −(−x)). A special case of unary negation occurs when it operates on a positive number, in which case the result is a negative number (as in −5).

The ambiguity of the "−" symbol does not generally lead to ambiguity in arithmetical expressions, because the order of operations makes only one interpretation or the other possible for each "−". However, it can lead to confusion and be difficult for a person to understand an expression when operator symbols appear adjacent to one another. A solution can be to parenthesize the unary "−" along with its operand.

For example, the expression 7 + −5 may be clearer if written 7 + (−5) (even though they mean exactly the same thing formally). The subtraction expression 7 – 5 is a different expression that doesn't represent the same operations, but it evaluates to the same result.

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[25]

Addition

A visual representation of the addition of positive and negative numbers. Larger balls represent numbers with greater magnitude.

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 and negative numbers, one can think of the negative numbers as positive quantities being subtracted. For example:

8 + (−3)  =  8 − 3  =  5  and (−2) + 7  =  7 − 2  =  5.

In the first 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 2 + (−7)  =  2 − 7  =  −5.

Here the credit is less than the debt, so the net result is a debt.

Multiplication

A multiplication by a negative number can be seen as a change of direction of the vector of magnitude equal to the absolute value of the product of the factors.

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) × 3  =  −6

and

(−2) × (−3)  =  6.

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

(−2) × 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 ) × (−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) × (−3)  +  2 × (−3)  =  (−2 + 2) × (−3)  =  0 × (−3)  =  0.

Since 2 × (−3) = −6, the product (−2) × (−3) must equal 6.

These rules lead to another (equivalent) rule—the sign of any product a × b depends on the sign of a as follows:

• if a is positive, then the sign of a × b is the same as the sign of b, and
• if a is negative, then the sign of a × b is the opposite of the sign of b.

The justification for why the product of two negative numbers is a positive number can be observed in the analysis of complex numbers.

The negative version of a positive number is referred to as its negation. For example, −3 is the negation of the positive number 3. 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. Specifically:

• The negation of 0 is 0, and
• The negation of a negative number is the corresponding positive number.

For example, the negation of −3 is +3. In general,

The absolute value of a number is the non-negative number with the same magnitude. For example, the absolute value of −3 and the absolute value of 3 are both equal to 3, and the absolute value of 0 is 0.

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:

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

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. 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 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

This construction is a special case of the Grothendieck construction.

Uniqueness

The additive inverse of a number is unique, as is shown by the following proof. As mentioned above, an additive inverse of a number is defined as a value which when added to the number yields zero.

Let x be a number and let y be its additive inverse. Suppose y′ is another additive inverse of x. By definition,

$${\displaystyle x+y'=0,\quad {\text{and}}\quad x+y=0.}$$

And so, x + y′ = x + y. Using the law of cancellation for addition, it is seen that y′ = y. Thus y is equal to any other additive inverse of x. That is, y is the unique additive inverse of x.

For a long time, understanding of negative numbers was delayed by the impossibility of having a negative-number amount of a physical object, for example "minus-three apples", and negative solutions to problems were considered "false".

In Hellenistic Egypt, the Greek mathematician Diophantus in the 3rd century AD referred to an equation that was equivalent to ${\displaystyle 4x+20=4}$ (which has a negative solution) in Arithmetica, saying that the equation was absurd.[26] For this reason Greek geometers were able to solve geometrically all forms of the quadratic equation which give positive roots; while they could take no account of others.[27]

Negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art (九章算術, Jiǔ zhāng suàn-shù), which in its present form dates from the Han period, but may well contain much older material.[3] The mathematician Liu Hui (c. 3rd century) established rules for the addition and subtraction of negative numbers. 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.[4] The Chinese were able to solve simultaneous equations involving negative numbers. The Nine Chapters used red counting rods to denote positive coefficients and black rods for negative.[4][28] 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. Liu Hui writes:

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.[4]

The ancient Indian Bakhshali Manuscript carried out calculations with negative numbers, using "+" as a negative sign.[29] The date of the manuscript is uncertain. LV Gurjar dates it no later than the 4th century,[30] Hoernle dates it between the third and fourth centuries, Ayyangar and Pingree dates it to the 8th or 9th centuries,[31] and George Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century,[32]

During the 7th century AD, negative numbers were used in India to represent debts. The Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta (written c. AD 630), discussed the use of negative numbers to produce the general form quadratic formula that remains in use today.[26] 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".[33][34]

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.[5] Al-Khwarizmi in his Al-jabr wa'l-muqabala (from which the word "algebra" derives) did not use negative numbers or negative coefficients.[5] But within fifty years, Abu Kamil illustrated the rules of signs for expanding the multiplication ${\displaystyle (a\pm b)(c\pm d)}$,[35] and al-Karaji wrote in his al-Fakhrī that "negative quantities must be counted as terms".[5] In the 10th century, Abū al-Wafā' al-Būzjānī considered debts as negative numbers in A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen.[35]

By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve polynomial divisions.[5] As al-Samaw'al writes:

the product of a negative number—al-nāqiṣ (loss)—by a positive number—al-zāʾid (gain)—is negative, and by a negative number is positive. 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. If we subtract a positive number from an empty power (martaba khāliyya), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number.[5]

In the 12th century in India, Bhāskara II 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."

Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of Liber Abaci, 1202 AD) and later as losses (in Fibonacci's work Flos).

In the 15th century, Nicolas Chuquet, a Frenchman, used negative numbers as exponents[36] but referred to them as "absurd numbers".[37]

Michael Stifel dealt with negative numbers in his 1544 AD Arithmetica Integra, where he also called them numeri absurdi (absurd numbers).

In 1545, Gerolamo Cardano, in his Ars Magna, provided the first satisfactory treatment of negative numbers in Europe.[26] He did not allow negative numbers in his consideration of cubic equations, so he had to treat, for example, ${\displaystyle x^{3}+ax=b}$ separately from ${\displaystyle x^{3}=ax+b}$ (with ${\displaystyle a,b>0}$ in both cases). In all, Cardano was driven to the study of thirteen types of cubic equations, each with all negative terms moved to the other side of the = sign to make them positive. (Cardano also dealt with complex numbers, but understandably liked them even less.)

In 1748 Leonhard Euler, by formally manipulating complex power series while using the square root of ${\displaystyle -1,}$ obtained Euler's formula of complex analysis:[38]

$${\displaystyle \cos \theta +i\sin \theta =e^{i\theta }}$$

where ${\displaystyle i={\sqrt {-1}}.}$

In 1797 AD, Carl Friedrich Gauss published a proof of the fundamental theorem of algebra but expressed his doubts at the time about "the true metaphysics of the square root of −1".[39]

However, European mathematicians, for the most part, resisted the concept of negative numbers until the middle of the 19th century.[40] In the 18th century it was common practice to ignore any negative results derived from equations, on the assumption that they were meaningless.[41] In 1759 AD, the English mathematician Francis Maseres 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.[42]

Wikiquote has quotations related to Negative number.

• Maseres' biographical information
• BBC Radio 4 series In Our Time, on "Negative Numbers", 9 March 2006
• Endless Examples & Exercises: Operations with Signed Integers
• Math Forum: Ask Dr. Math FAQ: Negative Times a Negative