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Negative is a term in mathematics that usually means "opposite." An electron's charge is called negative not because it is "below" but because it is opposite that of a proton. A surface with negative curvature bulges in from the point of view of someone on one side of the surface but bulges out from the point of view of someone on the other side. A line with negative slope is downhill for someone moving to the right but uphill for someone moving to the left.

The term negative is most commonly applied to numbers. When negative is an adjective applied to a number or integer, the reference is to the opposite of a positive number. As a noun, negative is the opposite of any given number. Thus, -4, -3/5, and - √ 2 are all negative numbers, but the negative of -4 is +4. The integers, for example, are often defined as the natural numbers plus their negatives plus zero. Sometimes the word opposite is used to mean the same as the noun negative.

Technically, negative numbers are the opposites with respect to addition. If a is a positive number then -a is a negative number because: a + (-a) = 0.

Allowing numbers and other mathematical elements to be negative as well as positive greatly expands the generality and usefulness of the mathematical systems of which they are a part. For example, if one owes a credit card company $150 and mistakenly sends $160 in payment, the company automatically subtracts the payment from the balance due, leaving -$10 as the balance due. It does not have to set up a separate column in its ledger or on its statements. A balance due of -$10 is mathematically equivalent to a credit of $10.

When the Fahrenheit temperature scale was developed, the starting point was chosen to be the coldest temperature which, at that time, could be achieved in the laboratory. This was the temperature of a mixture of equal weights of ice and salt. Because the scale could be extended downward through the use of negative numbers, it could be used to measure temperatures all the way down to absolute zero.

The idea of negative numbers is readily grasped, even by young children. They usually do not raise objection to extending a number line beyond zero. They play games that can leave a player "in the hole." Nevertheless, for centuries European mathematicians resisted using negative numbers. If solving an equation led to a negative root, it would be dismissed as without meaning.

In other parts of the world, however, negative numbers were used. The Chinese used two abaci, a black one for positive numbers and a red one for negative numbers, as early as two thousand years ago. Brahmagupta, the Indian mathematician who lived in the seventh century, not only acknowledged negative roots of quadratic equations, he gave rules for multiplying various combinations of positive and negative numbers. It was several centuries before Euopean mathematicians became aware of the work of Brahmagupta and others, and began to treat negative numbers as meaninful.

Negative numbers can be symbolized in several ways. The most common is to use a minus sign in front of the number. Occasionally the minus sign is placed behind the number, or the number is enclosed in parentheses. Children, playing a game, will often draw a circle around a number which is "in the hole." When a minus sign appears in front of a letter representing a number, as in -x, the number may be positive or negative depending on the value of x itself. To guarantee that a number is positive, one can put absolute value signs around it, for example |-x|. The absolute value sign can also guarantee a negative value, which is -|x|.

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