IMAGINARY EXPRESSIONS or QUANTITIES, or impos sible quantities in Algebra, are such as have the symbol .1-1 in their analytic expression. They are so called, because the square root of a negative quantity can have no real existence ; for whether a quantity be positive or negative, its square is a positive quantity.
The origin and nature of imaginary quantities have been explained in OUT article ALGEBRA, § 189-193. They are there shewn to be of two distinct kinds, one which is al together impossible, and can denote no real quantity ; and another which denotes real quantities.
The firv, class, when reduced to their most simple expression have this form a + 8/ 62, or a + b 4 - 1, where a may in some cases be = o. These occur, when a problem is to be resolved which from its nature requires that the data be contained within certain limits in respect of magnitude, while at thd same time, in the particular ease proposed, they pass those limits.
For example, if it be required to construct a right an gled triangle, tfie hypothenuse of which shall be equal to a given line a, and one of the sides equal to another given line b; from the nature of the case, a must be greater than b; and if, in the particular state of the data, a be less than b, the thing required cannot be done.
The unknown side of the triangle expressed in symhols, algebraically is ; now, if b be greater than a, the quantity is negative, and the expression for the side of the triangle has the form v = n 1, which is imaginary. The impossibility of giving a signi ficant numerical value to this symbol corresponds, in this instance, to the impossibility of placing between a given point, and a straight line given by position, a line of a given length, that is shorter than the perpendicular from the point on the line, or, which is the same, of determining the intersection of a straight line, and a circle which lies wholly on one side of the line.
In GEOMETRICAL Problems, passing the first order, the unknown quantities are determined either by the intersec tion of a straight line with a curve, or else by the intersec tion of two curves : Now, although it may be possible that the conditions to be fulfilled in a problem may be all satis fied at once, yet in nany cases there will be limitations of the data ; for example, by one condition a straight lint may be required to be of a given length ; and by another, that its extremities be on the circumference of a given circle. These can only be satisfied at once, when the
straight line is less than the Ciameter. In like manner, one condition requiring that a straight line touch a circle, and another, that it pass through a certain point, can both be satisfied only when the point is without the circle. When the data of a problem are in this way limited, as often as they cannot be all satisfied at once, the incongruity is indi cated geometrically, by there being no intersection of the lines, which should meet and determine the unknown quan tities ; and algebraically, by the impossible symbol V 1 entering into their values, and in such a way as not to admit of its being eliminated. The presence of the symbol v 1, in the algebraic expression for a quantity, serves not only to chew the impossibility of finding that quantity in the par ticular state of the data, but it also indicates the boundary which separates the possible from the impossible cases, and thus determines the greatest and least values that can be given to the different quantities concerned in the problem.
For example, let it be required to find a fraction which, together with its reciprocal, shall be equal to a given num ber, and the limits within which the problem is pos sible.
Calling the fraction x and the given number 2 a, the con dition to be satisfied will be expressed by this equation = 2 a, which produces the quadratic equation and, this resolved, gives x= V (az 1).
From this expression, it appears that the problem is im possible if a be a fraction, positive or negative, between the limits of + 1 and 1, because then will he less than 1, and az 1 a negative quantity, and j (az-1) an impossible quantity. However, if a be a positive quantity not less than + 1. or a negative quantity not greater than 1, (here we reckon 2 to be less than I, and 3 less than 2, and so on,) the problem will always be possible, and, excepting the cases a=-I- 1, and a = 1, x will have two values, which will be reciprocals of each other, because their product is unity.