Hence, multiplication by a negative quantity indicates that we are to proceed just as if the multiplier were positive, and then change the sign of the product.
44. Note on Arithmetical and Symbolical Algebra
Arithmetical Algebra is that part of the science which deals solely with symbols and operations arithmetically intelligible. Starting from purely arithmetical definitions, we are enabled to prove certain fundamental laws.
Symbolical Algebra assumes these laws to be true in every case, and thence finds what meaning must be attached to symbols and operations which under unrestricted conditions no longer bear an arithmetical meaning. Thus the results of Arts. 41 and 42 were proved from arithmetical definitions which require the symbols to be positive whole numbers, such that a is greater than b and c is greater than d. By the principles of Symbolical Algebra we assume these results to be universally true when all restrictions are removed, and accept the interpretation to which we are led thereby.
Henceforth we are able to apply the Law of Distribution and the Rule of Signs without any restriction as to the symbols used.
45. To familiarize the beginner with the principles we have just explained we add a few examples in substitutions where some of the symbols denote negative quantities.
Ex.1. If a= — 4, find the value of a^3.
Here a^3 =(— 4)^3 =(— 4)x(—4)x(—)=— 64.
Ex. 2. If a=—1, b=3, c=— 2, find the value of — 3 a^4bc^3.
Here — 3a^4bc^3 =— 3 x (— 1)^4x 3 x (— 2)^3
=—3x1x3x(—8)=72.
EXAMPLES IV. b.
iea=—2, b=3, c=—1, x =-5, y=4, find the value of
1. 3a^2b. 6. 3a^2c. 11. —4a^2c^4. 16. 4c^5x^3, 2. 8abc^2. 7. — b^2c^2. 12. 3 c^3x^3. 17. — 5a^2b^2 c^2. 3. — 5c^3. 8. 3a^3c^. 13. 5 a^2x^2. 18. — 7 a^3c^3. 4. 6a^2c^2. 9. — 7 a^4bc. 14. -7c^4xy. 19. 8c^4x^3. 5. 4c^3y. 10. — 2abx. 15. — 8 ax^3. 20. 7 a^4c^4.
