ALGEBRAICAL SIGNS. 



-f- Resolution, or Division, denotes that 

 the number before it is to be divided 

 by the number after it : as 15-^-3 = 5. 

 When the number after the sign is 

 greater than that before it, the quo- 

 tient, or result of the division, cannot 

 be expressed in a common number, 

 because it is less than 1, which is the 

 least common number. In these cases 

 the quotient is indicated by placing 

 the number to be divided above a line, 

 and the divisor below. Thus the quo- 

 tient of 3 -j- 4 is expressed by |. When 

 we require only to express the divi- 

 sion and not perform it, the fraction 

 is suflScient : as ^ is the same as 

 15 -j- 3. Arithmetical operations can- 

 not be performed with letters, and 

 thus the fraction is the only form in 

 which we can point out the dividing 

 of one letter by another : as | is the 

 only way in which we can express the 

 quotient of a-r-b. 

 ', Ratio, denotes that the numbers or 

 quantities between which it is placed 

 have some relation or proportion to 

 each other. In expressing ratios that 

 are equal, instead of =, the usual sign 

 of equality, ; ; is used. Thus the 

 expression a \ h ', ', c ', d, means 

 that as o is to 6 so is c to d; and 

 2 : 4 :: 6 : 12, as 2 is to 4 so is 6 

 to 12. Any one relation of the mag- 

 nitude or value of one thing, or quality, 

 is called a ratio. 



> Majority, denotes that the number or 

 quantity which is placed before it is 

 greater than that which follows: as 

 a > 6, that the quantity expressed 

 by a is greater than that represented 

 by 6. 



< Minority is the reverse of majority, 

 as c ■< d expresses that the quantity 

 c is less than that of d. 



= , > and <, are used to denote the re- 

 lations of ratios, or proportions, as 

 well as of single numbers and quan- 

 tities : thus a \ b = c \ d, means that 

 a is the same part or portion of b that 

 c is of d ; a '. b> c \ d, means that 

 a is a greater part of b than c is of ti ; 

 and a '. b <. c \ d, means that a is a 

 less part of h than c is of d. The 

 same may be expressed by making 

 the first, or antecedent term of each 

 ratio, numerator of a fraction, and the 

 last or consequent term, denominator. 

 Thus J =^, ^>J, and ^<-^, are 

 respectively the same as, a '. h = c \ d, 

 16 



a ! 6 > c ; rf, and a ', b -^ c '. d. 

 When ratios vary, the signs are con- 

 veniently written = , >, <. 



Connexion {vinculum, or tie) drawn 



over numbers or quantities, connected 

 by signs, or the enclosing of such be- 

 tween parenthetical characters, de- 

 notes that they are to be taken as 

 one, that is, as the single number or 

 quantity that would result after all 

 the operations were performed. Thus, 

 8+6-^7, or (8-l-fi)-^7, denotes that 

 the sum of 8 and 6 is to be divided 

 by 7, and is the same as ',*, or 2 ; but 

 8+6 -^ 7, without the sign of con- 

 nexion, is 8l^. Again, 24 — 3 x 8, or 

 (24— 3)x8, is the same as 21x8, that 

 is, 168; but if the sign of connexion 

 be taken away, it becomes 24 — 24, 

 or 0. 



« a Power. A number or letter written 

 over the right of another in a smaller 

 character is called an exponent, and 

 denotes that the number or letter 

 over which it is written is understood 

 to be used as a factor in multiplica- 

 tion as often as is expressed by the 

 exponent. Thus 4^ is the same as 

 4x4x4, or 64. An expression of 

 this kind is called a power of the 

 number or quantity to which the ex- 

 ponent is affixed, and that number or 

 quantity is called the root. The per- 

 forming of the multiplications is 

 called involution; and the number of 

 multiplications is always one less than 

 the number expressed by the expo- 

 nent, because two factors are required 

 for the first multiplication, and one 

 additional factor for every succeeding 

 one. If the root consist of several 

 numbers or letters, they must be in- 

 closed in parentheses, or placed under 

 a vinculum. 



^ a Root. When a number or letter is 

 considered as a power, the root of it 

 is denoted by placing the sign t^ 

 before it, and writing the exponent 

 over the sign, in the place of the small 

 letter n : thus ^64 is the third root 

 of 64, that is, it is 4. If the power 

 consist of several numbers or let- 

 ters, they must be connected. Thus, 

 \/4 4-12, or V(4 + 12), is 4; but 

 ^4+12, without the connexion, is 

 14. For the second root, or, as it is 

 called, the square root, the sign is 

 used without the exponent : as 

 ^9=3. Roots are also expressed by 



