NEGATIVE QUANTITY. The inverse op erations of mathematics. such as subtraction, division, and evolution, often lead to results which cannot be expressed in terms of the sante unit as the numbers entering the operation. The interpretation of these results loads to the so called artificial numbers, and in the particular ease of subtraction to the notion of negative num ber. For example. $2 $3 is impossible if the result is to be expressed in terms of the positive unit $1, but, since the result of subtraction is the number which added to the subtrahend will produce the minuend. it is easy to see that the number whhdi added to $3 will make $2 must be equivalent to the number Which subtracted from $3 will make $2. In other words. instead of subtract ing $1 front $3 to reduce it to f2, a number must be 'added which will produce the same result. Such a number is called a negative number and is desig nated by the sign placed before it. Hence $2 --$3 = $1. This notion of negative num ber as the opposite of positive number, :Ind ap parently growing out of an arbitrary interpre tation of a mathematical process. has its counter part in concrete magnitudes opposed in function or extent. For example. in the above ease, if a man's assets are $2 anti his debts $3, the number expressing his financial status is $1 of indebted ness. which may be expressed by $1. Similarly, time A.D. is often expressed by positive numbers. and time B.C. by negative numbers. In astronomy north latitude is expressed by positive numbers and south latitude by negative numbers; west longitude is designated as positive and east longi tude as negative. Such extensions of the mean ing of signs and modes of operation are the nat ural outgrowths of a constantly progressive science. The introduction of the negative num ber doubles the number space of arithmetic by adding an series of numbers opposite in meaning and having a 1 to 1 correspondence (see ConnEsIaNDENcE) with the series of positive numbers.
The negative quantity enters geometry through the phases of motion and direction. For ex ample, the segments AB, BC, and Cl), of a hori zontal straight line Al) thought of as extending to the right are considered positive. but the seg ments DC. CB, and BA thought of as extending to the left are considered negative. Similarly, many writers regard all angles generated by a line revolving counter-clockwise about a point as positive and those generated by a clockwise mo tion as negative. Flue introduction of negative quantities into geometry, especially in connection with the theory of continuity (see CoxricUrsy), has greatly increased the power and scope of the subject.
The meaning of negative quantities as eln ployed in the physical sciences may be illustrated from elementary mechanics. A material point
confined to a horizontal straight line may move to the ripbt, remain stationary. or move to the left. The first condition may be expressed by a positive velocity toward the right. the second by a zero velocity, and the third by a negative velocity.
By analogy to mathematical usage. the positive and negative notation is sometimes applied to quantities measured by scales like those of the ordinary thermometers, on which an arbitrary point is denoted as the zero-point and all degrees zero are denoted by negative numbers. Such conventional notations are convenient, but not always well founded. Thus. the temperature 1° C. is not the physical opposite of ± 1 0 the two temperatures would he the physical oppo sites of each other only if 0° C. represented a state in which bodies would have no heat at all. and if it were possible that a body should have less than no heat. (hn the other hand. in the ease of physical magnitudes whose character, like that of electricity, may be dual, the positive and negative notation has again a definite natural meaning, Negative quantities have been thoroughly un derstood only within recent times. Although Dem of Alexandria in his St ercomel rico eon sideml the expression I $1 144 as possible, the I'eShlt i5 recorded as 8 I/I6, which shows that negative quantities were understood by the Greeks. The Hindus were more suc cessful, for Aryabhatta, c.530, distinguished between (Maim (assets), positive quantities, and negative numbers. Bhaskara, v.11.50, was aware that a square root can be both positive and negative, and that t a not exist for the ordinary num ber-system. Al-Khowariztni, c.s30, a celebrated mathematician under the Arab supremacy, ob tained two roots for the quadratic- equation. but the negative roots were rejected as not valid. Among the early European mathematicians, Fi bonavei (1202) went no further than the Arabs. Paceioli (1494) definitely stated the rule, limes always y ices plus; hut this fact was known to the Arabs and Tlindus, Ilhaskara, for example, having stated that the square of a negative number is always positive. Cardan ( 1545 )recognized negative roots, but called them oast i ma t lanes fa Isw :?ti fel ( 1544 ) called negative numbers n meri absurdi. and Ilarriot (1631) was the first to consider such a number capable of forming a member of an equation. Pieta (1591) distinguished between positive and negative numbers, and Descartes (1637) in his geometry used the same letter for both positive and negative quantities.