EQUATION. 163 EQUATION. of certain letters representing the unknown quantities is called an equation. These particu lar values are called the roots of the equation, and the determination of these roots is called the solution of the equation. Thus 2x + 3 = 9 is an equation, because the equality is true only for a particular value of the 1 unknown quantity .<-, viz., for x := 3. The expression 2 + 5=7 ex- presses an equality, but it is not an equation as the word is technically used in mathematics. Expressions like (a + x) 2 = a 2 + 2a.c + x- are true for all values of the letters and are called identities to distinguish them from equations. If an algebraic function f{x) equals zero, and is arranged according to the descending, integral, positive powers of x. and in its relation to (I ex- pressed as an equation, it has the form /'(.<■) — u„x« + o,a> „_, + + o„..i <" + «„ = 0. Such an equation is called a complete equation of the Kth degree with one unknown quantity; e.g. a x- + a,a> + 03 =: is a complete equation, while OoX" + o 2 = is an incomplete equation, both of the second degree. The letters «,„ o,, a» , , o n stand for known quantities, and in the theory of equations, so called, they stand for real quantities. They are all coefficients of powers of X, except the absolute term, o n , which might, however, be considered the coefficient of x°. In case <i„- a, o„ arc all expressed as numbers, the equation is said to be numerical ' ; otherwise it is known as literal. Equations may be classified as to the number of their unknown quantities. Those already mentioned involve a single unknown, but x' + xy + y" = and xy = 1 involve two unknowns. There is no theoretical limit to the number of unknown quantities. Equations may also be classified as to degree, this being determined by the value of n in the complete equation already given. Thus, a a x-- a x = a x 2 +o x x + a 2 = «„ X 3 + O x X 2 4 a 2 x + « 3 = a x i +- a 1 I s + o 2 x 1 + a 3 x -j- a 4 = are equations, respectively, of the first degree (linear equation), of the second degree (quad- ratic equation), of the third degree (cubic equa- tion), and of the fourth degree (quartic or biquadratic equation). If two or more equations are satisfied by the same value of the unknown quantities they are said to be simultaneous, as in the case of x' + y = 7, x -- y 2 — l, where x = 2, i/ = 3; but w 2 + y = 7 and 3x 2 +3!/=:9 are not simulta- neous; they are inconsistent, there being no val- ues of x and y that will satisfy both ; and x" + y = 7 and Zx* + 3j/ = 21 are said to be identical, each being derivable from the other. In case sufficient relations are not given to en- able the roots of an equation to be determined, exactly or approximately, the equation is said to be indeterminate; e.g. in the equation x + 2y = 10, anv of the following pairs of values satisfies the equation: (0. 5), (1. 4.5), (2. 4), (3, 3.5), (10, 0). (11, —0.5), In general, n linear equations, each containing n + 1 or more unknown quantities, are indeterminate. Thus Zx + Zy 4 z = 10, 3x + 2y + z — S. give rise to the simple equation x + y = 2. which is indeterminate. Equations may also be clas- sified as rational, irrational, integral, or fraction- al, according as the two members, when like terms are united, are composed of expressions which are rational, irrational (oi partly so), in-
- ional i or pari i. bo), respectively,
with respect to thi unknown quantities; e.g.: Sx + j/' 5 =0 is a rational integral equation, 6 + 3 xl = is an irrational integral equation, 2 — -+l/]4: = is a rational fractional equation, £ , (x + .'l)"- = 5 is an irrational fractional equation. Algebra is chiefly concerned with the solution of equation,, and definite methods have been de- vised for determining the roots of algebraic equa tions of the first, second, third, and fourth de grees. Equations of the first degree are solved bj applying the common axioms: if equals are added to equals, the results are equal; if equals are subtracted from equals, the results are equal: and the corresponding ones of multiplica- tion and division. Equation- of the second de- gree may be solved by reducing the quadr function to the product of two linear factors, tnus making the solution of the quadratic equa tion depend upon that of two linear equations. Tnus x" + px + q = reduces to P Gs+§+jyi»*-*a) (s+§-i l /» a -4 P =o ■whence x + £ ± ^ ^/p'-iq = 0. Similarly, the solution of the cubic equation is made to depend upon that of the quadratic equa- tion, and that of the biquadratic equation upon that of the cubic equation. These formulas, however, when applied to numerical equations often involve operations upon complex numbers not readily performed, and hence are of little value in such cases; e.g. in applying the general formula for the roots of the cubic equation, the cube root of a complex number is often required, in which case the methods of trigonometry are employed. The real roots of numerical equations of any degree may be calculated approximately by the methods of Newton, Lagrange, and Horner, the last being the most recent and gen- erally preferred of the three. Equations of the first degree were familiar to the Egyptians in the time of Ahme"s (q.v. ), since a papyrus transcribed by him contains an equa- tion in the following form: Heap (hau), its - :1 . its V", its i, its whole, gives 37 ; that is, 'x+-x-- lx + x = 37. The ancient Greeks knew little of linear equa- tions except through proportion, but they treated in geometric form many quadratic and cubic equations. (See Cube.) Diophantus (c.300a.d.), however, distinguished the coefficients ( iri;Sos) of the unknown quantity, gave the equation a symbolic form, classified equations, and gave definite rules for reducing them to their simplest forms. His work was chiefly concerned with in- determinate systems of equations, and his method of treatment is known as Diophantine analysis (q.V.). The Chinese likewise solved quadratic equa- tions geometrically, and Sun Tse (third century ) . like Diophantus. 'developed a method of solving linear indeterminate equations. The Hindus ad- vanced the knowledge of the Greeks. Bhaskara (twelfth century) used only one type of quel ratic equation, a.r- + ox + c =0. considered both signs of the square root, and distinguished the surd values, while the Greeks accepted only posi- tive integers. The Arabs improved the methods of their predecessors. They developed quite an
