742 



THE IRKIGATION AGE. 



The value of the number 4 is greater by 3 units than 

 the number 7. Also the sum of a positive and negative 

 number of equal units is equal to zero, thus : 

 +77=0 



4. Multiplication. In a sum like : 5+5+5 where the sum- 

 mand 5 appears three times the operation is indicated by 

 writing the equal summand but once, and the number which 

 shows how often that equal summand is present is written 

 along with the equal summand with the symbol of multipli- 

 cation, thus : 5X3 or 3X5 and the result being 15, we have : 

 3X5=15 



This operation is called multiplication ; the numbers 3 

 and 5 are called factors and the number 15 is called the 

 product. 



In a similar manner if we have the sum: A+A+A+A 

 'we could say 4Xa or 4a; thus, when numbers are expressed 

 by letters no multiplication sign need to be placed between, 

 hence, 7X means seven times X, ab means A times B, 

 abc means A times B times C, etc. When a sum or a differ- 

 ence is to be multiplied by a number, each term of the sum 

 and difference must be multiplied, and the sum or differ- 

 ence placed in brackets, for instance: Multiply the sum 4+5 

 by 7 ; this should be written thus : 7 (4+5) ; this would be 

 equal to 7X4+7X5 or 28+35 which is equal to 63; this is 

 obvious from the fact that 4+5=9 and 7X9=63; hence, if 

 instead of given numbers we have algebraic symbols the same 

 principles obtain, thus : 



A i(B+C)=AB+AC 



Likewise: A (B C)=AB AC 



This principle also governs if sums and differences are 

 multiplied by sums and differences, for example: Multiply 

 9 9 by 4+7; (9 2) X (44-7) multiply each term of one 

 with each term of the other, thus : 



.9X4+9X72X42X7= 



36+63 814= 



9922=77 



That this is correct is seen if, instead of 9 2 we write 7, 

 and instead of 4+7 we write 11, then 7X11=77. 



Example: Multiply 115 with 83; this should be 

 written: (115) (83) and the result should be 30, since 

 11 5=6 and 8 3=5 ; multiply as above : 



11X811X35X8+5X3=883340+15=30 



Observe that the product of two positive or two negative 

 numbers is positive and that the product of a positive with 

 a negative number is negative. These principles are neces- 

 sary to understand when working with algebraic terms or 

 formulas. 



Thus, multiply: (A+B) with (C+D) = (A+B) (C+D) 

 =AC+AD+BC+BD. 



Also (A+B) (C D'=AC AD+BC BD ; also ( A B) 

 (C D) =AC AD BC+BD. 



When expressions occur in which like numbers appear 

 after multiplication they are united thus: 



(A+B) (A+B)=AA+AB +AB+BB 



Where the same number is multiplied by itself, like 

 AA and BB, it is called a square and is written A 2 or B 2 ; 

 also as the number AB appears twice instead of AB+AB 

 we say 2AB, thus instead of AA+AB+AB+BB write A 2 

 +2AB+B 2 . 



This expression also illustrates an important principle 

 that the square of the sum of two numbers is equal to the 

 sum of the squares of the two numbers plus twice their 

 product. To illustrate an application of this multiply 

 203 by 203. 



Solution: Instead of 203 set (200+3) (200+3) this is 

 equal according to the above principle : 



200X200+3X3+2X3X 200=40000 +9+1200=41209 



Another principle is found by multiplying a difference by 

 itself, thus: (A B) (A B)=A J AB-AB+B 2 =A 2 2AB 

 +B 2 . This principle says: The square of the difference of 

 two numbers is equal to the sum of the squares of these two 

 numbers minus twice their product. Application, multiply 

 98 by 98. 



Solution: Set 100 2 for 98, then we have: (100 2) 

 (100 2)=:100X100 2X100 2X100+4=10000+4 400=9604. 



Another principle is illustrated as follows : 



Multiply the sum of two numbers by their difference, as 

 (A+B) (A B)=A 2 +AB AB B 2 . 



In this answer +AB and AB cancel, hence the answer 

 is A* B J . This principle, expressed in words, says : The 

 product of the sum of two numbers into their difference is 

 equal to the difference of their squares. This is also true 

 in the reverse way, for instance: Subtract the square of 13 

 from the square of 17, which operation would ordinarily re- 

 quire, first, to multiply 17 by 17, next multiply 13 by 13, and 

 then subtract the latter product from the former, thus: 

 17X17=289, 13X13=169; 289 169=120. By the aid of the 

 above principle this problem is solved much easier as the 

 answer will be found by multiplying the sum by the difference, 

 thus: 17 2 13*= (17+13) (17 13) =30X4=120. 



5. Division. If in a product one factor is unknown it can 

 be found when all the other quantities are known, for instance : 

 3X=21, which means to find a number which, multiplied by 3, 

 gives 21 ; this number is evidently 7, because 3 times 7 equal 

 21, and is found by dividing 3 into 21; hence, the operation 

 of finding a factor is called division ; the operation is indi- 

 cated thus : 



X=21 or X=2lH-3 



In this expression the number 21 is called the dividend or 

 numerator, the number 3 is called the divisor or denominator 

 and' the number X (or 7) is called the quotient or fraction. 



21 



As =7 the relation between the different terms may 



B 



be expressed that the dividend is equal to the quotient multi- 

 plied by the divisor, and that the divisor is equal to the divi- 

 dend divided by the quotient. 



The rules for the signs in division are the same as in 

 multiplication; thus, like signs produce positive, and unlike 

 signs produce negative numbers, for instance : 



^= - ; and -A-, -B=. 



This general law regarding the signs may be shortly ex- 

 pressed thus : Like signs produce positive and unlike signs 

 produce negative results. 



6. Involution. When in a product all the factors are equal 

 for instance, 5X5X5X5, then, instead of writing the factor 

 "> four times, we write it but once, and the number 4' which 

 indicates how many times said factor is to be multiplied by 

 itself on the upper right hand of the number 5, making the 

 number 4 about half the size of the 5, thus: 

 5'=625 



Which is read: 5 to the 4th power equals 625. The num- 

 ber 5 is called the base, 4 is called the exponent and 625 

 is called the power. Thus, the power is found by multiply- 

 ing the base as often with itself as the exponent indicates. 

 Thus, 2 5 means that the number 2 is multiplied 5 times by 

 itself, or 2X2X2X2X2=32. 



Also AAAAAA should be written A', XXXXXXX=X 7 . 



