Also, from (2)
(a-b)(c-d)=a(c - d )+b(c-d) =(c-d)a-(c-d)b [Art. 37.] =ac- ad - bc + bd.
If we consider each term on the right-hand side of this last result, and the way in which it arises, we find that
(+a) x(+c )=+ ac, (— b) x(—d)=+ bd, (—b)+( +c )=—bc, (+a) x(—d)=— ad.
These results enable us to state what is known as the Rule of Signs in multiplication.
Rule of Signs. The product of two terms with like signs is positive; the product of two terms with unlike signs is negative.
43. The rule of signs, and especially the use of the negative multiplier, will probably present some difficulty to the beginner. Perhaps the following numerical instances may be useful in illustrating the interpretation that may be given to multiplication by a negative quantity.
To multiply 3 by —4 we must do to 3 what is done to unity to obtain —4. Now —4 means that unity is taken 4 times and the result made negative; therefore 3 x (— 4) implies that 3 is to be taken 4 times and the product made negative.
But 3 taken 4 times gives + 12.
3 x(—1)=- 12. Similarly, —3 x —4 indicates that — 3 is to be taken 4 times, and the sign changed; the first operation gives — 12, and the second + 12.
Thus (—3)x(-4)=+ 12.
