Negative Numbers

The Greeks rejected them. Diophantus, the most algebraically inclined ancient Greek mathematician, called negative solutions absurd and threw them out. The Indian mathematician Brahmagupta in the seventh century CE was the first to systematically accept negative numbers as legitimate mathematical objects, in his treatise Brāhmasphuṭasiddhānta. He gave them a financial interpretation: positive numbers are assets, negative numbers are debts. The interpretation worked, the operations made sense, and over the next millennium the rest of the mathematical world reluctantly came around. The full acceptance of negative numbers in Western European mathematics did not come until the eighteenth century, and even then mathematicians like Augustus De Morgan in the nineteenth century continued to call them "impossible."

A negative number is a number less than zero; equivalently, the additive inverse of a positive number: a + (−a) = 0. The set of integers ℤ extends the natural numbers to include negatives: ℤ = {…, −3, −2, −1, 0, 1, 2, 3, …}. Arithmetic with negatives extends naturally if you commit to the sign rules: positive plus negative is whichever has greater absolute value (in sign); positive times negative is negative; negative times negative is positive. The last rule was controversial for centuries — it is not obvious from any physical interpretation why two debts should multiply to an asset — and is best motivated by requiring the distributive law to keep working. If we want a × (b + (−b)) = a × 0 = 0 and also = a × b + a × (−b), we are forced to set a × (−b) = −(a × b); if we want both factors to be negative and the product to satisfy the same algebra, we are forced to set (−a) × (−b) = +ab. The geometric interpretation as movement on a number line — positives go right, negatives go left, multiplication scales and possibly flips — finally rationalized the rules. Absolute value |x| measures distance from zero, ignoring sign. Order on ℤ extends from ℕ: a < b iff b − a is a positive integer.