ROOT. The mathematical use of this term has gradually been extended, until it may be defined as follows : every value of an unknown quantity which satisfies a given equation is called a root of that equation. Thus, 2, 1, 1 + 3) aud 1 \/( 3) are the roots, and all the roots, of the equation since they are the only algebraical formulas and arithmetical numbers which satisfy it. On this general use of the term root, see THEORY or EQUATIONS and INVOLUTION.
The more common use of the term root is as follows : the seventh root of 8 is the incommensurable fraction whose seventh power is 8, or the solution of the equation There are altogether seven such solutions, one only arithmetical, the others of the form a -14 I) ; the method of obtaining the arithmetical solution has already been discussed in the article INVOLUTION; the importance of the SQUARE HOOT will justify its consideration in an article apart. We reserve for the present article the method of finding and using any root (in the common sense) of any algebraical quantity.
Every algebraical result is of the form a + bs/( 1) at widest, or may be reduced to that form. Hero a and b are meant to be real algebraical quantities, that is, reducible to positive or negative whole numbers or fractious, commensurable or incommensurable. Thus, if b= 0, we have the simple real quantity a; if a= 0, we have the simple impossible quantity 1).. V(l). It is indifferent, as to the present article, in what the impossible quantity s/(-1) is considered ; whether upon that extended system of definitions which makes it explicab e and rational, or upon the more common system in which it is used without such explanation : for we are now merely considering all algebraic formulie as results, subject to certain laws by which their nee is to be regulated, and without any reference to inter pretation. When we desire to consider only the arithmetical root of an arithmetical quantity, we shall use the symbols a/, -V, :V, etc., but the exponential A, 4, &e., will denote any one of the alge braical roots of a formula. Thus a/16 means simply 4 ; but (16)1 is
an ambiguous symbol standing for either + 4 or 4. And when we have an equation which presents an ambiguous formula equated to an finambiguons one, we mean that the unambiguous side of the equation is one of the values of the ambiguous one : in this sense (1)1 = &(-1 + ). When we use the simple arithmetical symbol a/ before an algebraical quantity, as in a/(-3), we merely mean to signify that the two values of ( 3)1 are distinguished into + s/( 3) and 3).
Let us now take a quantity of the form a +6 V(-1). Assume r cos 0=a, r sin 0 =b, which gives r= tan 0= ;.
Let us choose for r, which is called the mndultu of the expression, the positive value 'We can then always make the angle 0 give the equation a+b V(-1)=r cos 0+r sin 0 V(-1) Identically true. If a and b be both positive, 8 must lie between 0 and a right angle, or between 0 and iv [ANGLE) : if a be positive and negative, 0 must lie between ir and 2w: if b be positive and a negative, 0 must lie between it and or : and if both be negative, B must lie between w and jw. Thus reducing angles to degrees and minutes, 2 + 3 V(-1)= V131cos 56' 39' +sin 19'. A 1) } -2+ 3V(-1)= V13{cos 41'+sin 41',/(-1)} 2-3 V(-1)= V131coa 303' 41'+ sin 303' 4I's/(-1)} -2-3 V(-1)= s/13{cos 236' sin 236' 19' V( -1)} Generally, if a andb be positive, and if, returning to the arcual mode of measnring angles, 0 be that angle which lies between 0 and and has :a for its tangent, we must use 0 for a + -1), for - a + 6V( - 1), fur a -6V(-1), and w + 0 for -a-6 V( -1).
Again, since 0 + 2kr has the same sine and cosine as 0, when k is any whole number, positive or negative, if we take 0 so as to satisfy (1), we find that the following is also satisfied : a + ./( -1)=ricoe (0+ 2kr)+ sin (0+ 217r) . V(-1)} .... (2) for all integer values of k positive or negative, but for no frac tional value of k whatsoever. This and various other results of common trigonometry should be familiar to every student who attempts the present subject.