CHAMBERS'S INFORMATION FOR THE PEOPLE. 



means that 5 and 3 are added together. The sign 

 , which is called minus, means that the two 

 quantities between which it is placed are to be 

 subtracted; thus 5 3 is read, 5 minus 3, and 

 means that from 5 3 is to be subtracted. The 

 symbol X is the sign for multiplication, and -=- for 

 division, while = is the symbol for equality ; thus 

 5 + 3 = 8, is read, 5 plus 3 equals 8. V \/> V 

 denote respectively the square root, cube root, 

 fourth root of the quantity before which they stand, 

 thus A/5, \/5> t/5 mean respectively square root 

 5, cube root 5, fourth root 5. The letters a, b, c, 

 at the beginning of the alphabet, are usually taken 

 to denote known quantities ; while those at the end, 

 r,y, z, are taken to denote unknown quantities, or 

 quantities to be found. 



ADDITION. 



Two quantities in algebra are added by simply 

 putting the sign of addition between them ; thus, if 

 we wish to add x and y together, this operation is 

 indicated by x + y. Again, if we have to add 

 zx; yc, 4* together, we have as before for 

 their sum 2x -f "$x + $x ; but we can go a step 

 farther here, since 2x + yc + 4^ is the same as 9*-, 

 and write the sum as 9-r. 



Again, let it be required to add 7^ 3^ + 4^ 

 -\-$x -\-6x 2x. Here the plus part of this ex- 

 pression is jx + $x -\- $x -\- 6x, or 22r > an d tne 

 minus part of it is yc 2x, or $x, so that the 

 expression is on the whole positive, and equal to 

 I7.r. When like and unlike quantities are mixed 

 together, as in the following example, the like 

 quantities must first be collected together, and the 

 unlike quantities annexed in order : 



4*yy+ y 



2y -\- $x 2xy 



a + 3^ + 



SUBTRACTION. 



One quantity is subtracted from another by 

 simply writing the sign of subtraction between 

 them ; thus, to subtract x from y, we merely 

 write y x. Again, to subtract jrfrom^ + ^, we 

 write y + z x. When the quantity to be sub- 

 tracted, however, consists of several parts with 

 different signs, some care is necessary, as the 

 process is a little complicated. To explain this, 

 we will do as we do in arithmetic, and write the 

 quantity to be subtracted below the other given 

 quantity. We will first suppose that each quan- 

 tity consists only of one part. Thus, let it be 

 required to subtract + b from + a. Writ- 

 ing these as in the margin ; now a is the i a 

 same as a b + b, then from a we have to i 

 take + b ; instead, however, of taking + b 

 from a, we will take it from a b + b, 

 there therefore remains a b. 



Again, let it be required to subtract from a the 

 quantity b. As before, a is the same as 

 a b -f- b, then if from this we take away 

 bj we have a + b left. From these two 

 examples it appears that the sign of the 

 quantity below is changed, and then the 

 two quantities are supposed to be added. 



ab 



i a 



As another example, let it be required to su 

 tract a + b c from a b c. 

 Arranging the quantities as here a I 

 shewn, we see that the result will be a -\-b c 



-\-2a- 2b, the c below becoming 



+ c, and therefore destroying the + 2a 26 

 c above. 



Sometimes the subtraction is merely indicated, 

 the part to be subtracted being inclosed within 

 brackets ; thus in the expression 



2a + b - (a b + c}, 



the part a b + c, which is within the brackets, 

 is understood to be subtracted from 2a + b. In 

 that case, if the subtraction were performed, 

 a b + c would be made the under line, and, as 

 before explained, its signs would be changed : 

 bearing this in mind, we may save ourselves the 

 trouble of writing it down in two lines if we take 

 away the brackets, and change the signs of the 

 part inclosed ; when we do so, the expression 

 will then become 



2a -f- b a-\- b c. 

 Or a -f- 2b e. 



MULTIPLICATION. 



The multiplication of two different quantities is 

 performed by merely indicating the process. 

 Thus a multiplied by b is written a X b, or a-b, or 

 simply ab. Similarly, a, b, c multiplied together 

 is abc. Again, 2x by 3j is 6xy. So also a multi- 

 plied by a that is, by itself is written aa or a 2 . 

 Similarly, a X a X a = a 3 , and a X a X a X a 

 = a 4 ; and these expressions, <z a , a 8 , a 4 , &c. are 

 read, a square, a cube, a fourth power. The 

 numbers 2, 3, and 4 here introduced are called 

 indices. Again, a* multiplied by a 8 is a 8 . Since 

 a 2 means aa, and a 3 means aaa, then, when # 2 and 

 a 3 are multiplied, their product will be aaaaa or 

 a s . Hence multiplication is performed by adding 

 the indices ; thus, a 3 X a 3 = a 6 , a 3 X a* = a 1 . 

 If the quantities which are multiplied together 

 have the same sign, the sign of their product is + 

 If they have different signs, the sign of their pro- 

 duct is . 



Example I. Multiply 2x $y + $z by yc 



2*- 37 

 yc 2y 



-4- S 

 +2 



cur 3 <ycy + i ycz 



zxz 



\ycy 



17x2 



Example 2. Multiply a + b by a + b. 

 a + b 

 a +b 

 a* + ab 



a* -f- 2ab -f- b* 

 Example 3. Multiply a + b by a b. 



a + 

 a b 



ab-b* 



