Introduction. i i 



and since we know what is meant by —3 multiplied by 5, the 

 equation ab=ba gives a meajiing to 5 multiplied by —3. 

 .•.5 multiplied by —3= — 15. 

 From this we are led to say that when the multiplier is neg- 

 ative, the product is just the opposite from what it would be if 

 the multiplier were positive. 



Therefore, if a and b are any two positive quantities, we may 

 wTite the following four equations : 



a.b=ab (i) 



{—a).b=—ab (2) 



a.{—b)^—ab (3) 



{^-a).{-b) = ab. (4) 



From the ist and 4th we conclude that the product of two posi- 

 tive quantities or two iiegative quayitities is positive, and from the 2d 

 and 3d, the product of 07ie positive and o?ie negative quantity is neg- 

 ative. 



24. The four equations just written are true whether a and 

 b are positive nor not. 



Consider, for example, the second equation under the supposi- 

 tion that a is negative and b positive ; then {—a).b becomes the 

 product of two positive quantities and is therefore positive, but 

 —ab is also positive in this case, as it should be, rendering the 

 equation still true. And so of the other equations, whether a and 

 b are positive or not. Therefore, directing our attention to the 

 signs, we may say that the product of two quantities preceded b}' 

 like signs is a quantity preceded by the -j- sign, and the product 

 of two quantities preceded by tmlike signs is a quantity preceded 

 by a — sign. 



This statement is usually shortened into the following — 



In multiplication like signs give plus and unlike sights give 

 minus. 



This is often confused with the statement in italics in the pre- 

 ceding article. They are not identical, but both are true. 



25. As division is the inverse of multiplication, it easily fol- 

 lows that the quotient of tivo positive or two 7icgative quantities is 

 positive, and that the quotient of a positive by a negative quantity, 



