18 MATHEMATICS 



desired to indicate that all the quantities included by them 

 are to be subjected to the same operation. Thus, suppose 

 that the sum of 5 and 8 is to be multiplied by 7, and that 

 the addition is to precede the multiplication. Any one 

 of the four signs of aggregation maybe employed to indicate 

 the operation. Thus, 5 + 8X7, (5 + 8) X 7, [5+8] X 7, {5 + 8} 

 X7 The vinculum is placed above the quantities that are 

 to be treated as one quantity and subjected to the same 

 operations. 



While any one of the four signs may be used as shown 

 above, custom has restricted their use somewhat. The vincu- 

 lum is rarely used except in connection with the radical sign. 

 Thus, instead of writing -^(5 + 8), >/ [5 + 8], or $ {s + s} 

 for the cube root of 5 plus 8, all of which would be correct, 

 the vinculum is nearly always used, ^/5 + 8. 



In cases where only one sign of aggregation is needed 

 (except, of course, when a root is to be indicated), the 

 parenthesis is always used. Hence, (5 + 8)X7 would be the 

 usual way of expressing the product of 5 plus 8 by 7. 



If two signs of aggregation are needed, the brackets and 

 parenthesis are used, so as to avoid having a parenthesis 

 within a parenthesis, the brackets being placed outside. For 

 example, [(20 5)-^-3] X9 means that the difference between 

 20 and 5 is to be divided by 3, and this result multiplied by 9. 



If three signs of aggregation are required, the brace, 

 brackets, and parenthesis are used, the brace being placed 

 outside, the brackets next, and the parenthesis inside. For 

 example, {[(20-5) * 3] X 9 21 } -4-8 means that the quo- 

 tient obtained by dividing the difference between 20 and 5 

 by 3 is to be multiplied by 9, and that 21 is to be subtracted 

 from the product thus obtained, and the result divided by 8. 



Should it be necessary to use all four signs of aggrega- 

 tion, the brace would be put outside, the brackets next, 

 the parenthesis next, and the vinculum inside. For example, 

 {l(20-5-i-3)X9-21]H-8}xi2. The reason for using the 

 brace in this last instance will be explained, as it is not 

 generally understood. 



