

ALGEBRA. 



The results of the last two examples ought to be 

 remembered, as they are very useful. 



DIVISION. 



The division of two different quantities is per- 

 formed by merely indicating the process. Thus, 



terms, in algebra as in arithmetic, by dividing the 

 numerator and denominator by any quantity cap- 

 able of dividing them both without leaving a re- 

 mainder. Thus, in the fraction Ioa3 + 2 ^Jg! 



35<z 2 



it is evident that the coefficient of every term can 

 be divided by 5 ; and as the letter a enters into 



divided by * is written . + * or simply 4. ^S^^^^^^^S^^SSi 



/ 







but as 



Again, 6> divided by "$y is written 



*>y 



the division of 6 by 3 can actually be performed, 

 this result becomes *. Again, a* divided by a 2 



will be a 2 , since a 4 means aaaa, and a* means aa. 

 Hence division is performed by subtracting the 

 indices, and the signs in division will be the same 

 as in multiplication that is, like signs in division 

 give + ; unlike, give - . Thus, 



+ abx* -; ax. 



Or 



Again, 



The following example will shew how to divide 

 when the divisor consists of more than one 

 quantity : 



a* + 4a*b + eofF + 4a^ + b* (a* + 



N 



ad 3 + P 



measure of this fraction, because it can divide 

 both the numerator and the denominator. The 

 numerator (ioa 3 + 2oab -f- 5<2 2 ) -j- $a = aa 2 -|- 46 

 + a ; and the denominator, 35 2 -j- $a = "ja ; hence 

 the fraction, in its lowest terms, is 2a + 4^ + ^ 



When two or more fractions are to be added or 

 subtracted, they must, as in arithmetic, be first 

 brought to others having a common denominator. 



For example, let it be required to add and . 



We have for their sum + . Taking i2yz for 



4- y "\% 

 the denominator of each, the two fractions will be 



and . and therefore their sum will be 



FRACTIONS. 



The rules regulating the management of frac- 

 tions in algebra are similar to those in arithmetic. 



A mixed quantity is reduced to a fraction by \ 

 multiplying the whole or integral part by the 

 denominator of the fraction, and annexing the 

 numerator with its proper sign to the product ; | 

 the former denominator, if placed under this sum, 

 will give the required fraction. Thus the mixed 



ie thus reduced to a 



quantity ix + ^~ 



O 



fraction: 2x X 6e = i iex, and as $ab must be 

 added to form the numerator, and the former 

 denominator be retained, the required fraction is 



the following : - .An operation exactly 



the reverse of this would of course be requisite 

 were it proposed to reduce a fraction to a mixed 



quantity. Thus the fraction - ex "7" ma y be 



reduced to a mixed number by dividing the 

 numerator by the denominator ; the numerator of 

 the fractional part must be formed by that term 

 which is not divisible without a remainder ; the 

 following is therefore the required mixed quantity : 



zx + 5|r. A fraction is reduced to its lowest 



i . Again, to multiply one fraction by 



another, we do so by multiplying their numerators 

 together for the numerator of the product, and 

 their denominators together for the denominator 



of their product. Thus, multiplying - by - we 



ac b a" 



have for the result y-y Then, again, to divide one 



fraction by another, as in arithmetic we invert the 

 divisor, and proceed as in multiplication. 



Involution and Evolution. 



The raising of a quantity to any required power 

 is called involution, and is performed by suc- 

 cessive multiplications ; thus (a -f- b) multiplied by 

 itself gives the square or second power of a -f- b ; 

 this is equal to a 3 + 2ab -f- P, and is thus ex- 

 pressed : 



(a + b) 2 = a 3 + 2ab + b 2 . 

 Similarly (a + b) 3 = a 3 + y?b + lab 2 + b 9 ; 

 and (a + b)* = a* + ^b + 6a 2 ^ + 40?+ b 4 . 



A theorem discovered by Sir Isaac Newton, named 

 the Binomial Theorem, enables us to write down 

 at once such expressions as the above. The 

 operations of evolution are the inverse of those of 

 involution, and are designed to enable us to dis- 

 cover the square root, cube root, &c. of any given 

 quantity. The extraction of the square root of a 

 compound quantity is precisely similar to the 

 method employed for the extraction of the arith- 

 metical square root. 



Irrational Quantities, or Surds. 



Some numbers have no exact root ; for instance, 

 no number multiplied into itself can produce five. 

 The roots of such quantities are expressed by 

 fractional indices, or by the sign ^, which is 

 called the radical sign, from the Latin radix, a 

 root ; thus the square root of 5, and the cube 

 root of (a + b)', may be expressed either by VS) 



The approximate value of such quantities can 

 be ascertained to any required degree of exact- 

 ness by the common rules for extracting roots ; 



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