lo Algebra. 



^ repeated 4 times is ^ ^nultiplied by 4, yet, in arithmetic, 4 

 multiplied by ^ is a familiar operation. 



Let us inquire how this comes to have a meaning, and how it 

 happens that 4 multiplied by \ turns out to be ^ ^4. 



21. As long as a and b are positive whole numbers it is easy to 

 see \h2Xab=ba. 



Suppose, to fix the ideas, that ^=3 and ^=5, then we may 

 write down 5 rows of dots with three dots in each row, thus — 



and we have in all 5 times 3 dots. But we may look at vertical 

 rows instead of horizontal ones and we see three rows with 5 dots 

 in each row, and of course the number of dots is the same ; so we 

 may say 



5x3=3x5. 

 Any other positive whole numbers would do as well as 3 and 5, 

 and so if a and b are ariy positive whole numbers, 



ab=-ba, 

 i. e. , in the product of two numbers, it is iiidiffereyit which is the 

 multiplier ajid which the multiplica?id, so long as both ?iu?nbers are 

 integers. ^ 



22. Now, in arithmetic^ the operation of multiplication is so 

 extended that eve7i when one of the quantities is a fractio7i it shall 

 still be indifferent which of the two quantities is the multiplier and 

 which the multiplicand. 



This gives a 77ieaning to multiplication when the multiplier is a 

 fraction, and thus it happens that 4 multiplied by ^ is taken to 

 mean the same as \ multiplied by 4. 



23. In exactly the same way in algebra, the operation of mul- 

 tiplication is extended so that whatever numbers, positive or neg- 

 ative, integral or fractional, are represented by a and b we vShall 

 always have 



ab=ba. 



