Diophantus called negative answers absurd, yet Han China already used them
A negative number sits below zero—the opposite of a positive quantity—handy for debts, losses, or opposite charges. Han-era Nine Chapters on the Mathematical Art already used them; Liu Hui set addition and subtraction rules by the third century, and seventh-century Indian mathematicians continued the thread.
Notation usually prefixes a minus, so −3 reads minus three or negative three, and a plus sign can stress that a number is positive. Every real number except zero is one or the other, and zero is usually treated as having no sign. Integers gather the negative whole numbers, zero and the positive ones, while the natural numbers are the non-negative whole numbers, though some definitions leave zero out. Bookkeepers often mark amounts owed in red or in parentheses instead of with a minus.
Negatives express losses, shortfalls and debts, which behave like negative assets. When a quantity comes in two opposite senses, such as electric charge, one sense can be labelled positive, even arbitrarily. Temperature scales in Celsius and Fahrenheit run below zero. The rules of arithmetic encode the idea of opposites, which is why −(−3) equals 3.
On the number line, rightward means greater and leftward lesser, with zero in the middle. So a negative with bigger magnitude is smaller: 8 beats 5, yet −8 sits below −5. Size and order become separate ideas once signs appear.
The Nine Chapters on the Mathematical Art, in its present form from the Han dynasty (202 BC to AD 220) but possibly holding far older material, already used negatives, and Liu Hui gave rules for adding and subtracting them around the third century. By the seventh century Brahmagupta in India was describing them, and Islamic mathematicians extended the rules to subtraction and multiplication and tackled problems with negative coefficients. Earlier, Diophantus had dismissed negative solutions as false, and even Leibniz considered negatives invalid while still calculating with them.
Source: Negative number
