Symbol Manipulation
For this section, you must be familar with the Basic Rules of Arithmetic which was stated in the Basic Rules of Algebra section.
 NOTE:
 We will use the notational convention that b+(a) is the same as ba and 1/a is the same as a^{1}

01. If a+b = a+c then b = c
(Proof)
02. (a) = a
03. If a+b = 0 then b = a
04. If ab = ac and a 0 then b = c
05. a0 = 0
06. If ab = 1 then b = 1/a
07. If ab = 0 then a = 0 or b = 0
08. (a)b = (ab)
09. (a)(b) = ab (Proof)
10. (1)b = b
11. 1/(1/a) = a if a 0
12. 1/(ab) = (1/a)(1/b) if a, b 0 (Proof)
13. ac/(bc) = a/b if b, c 0
14. a/b + c/d = (ad + bc)/(bd) if b, d 0
15. (a/b)(c/d) = (ac)/(bd) if b, d 0
16. (a/b)/(c/d) = (ad)/(bc) if b, c, d 0
17. If b 0 and d 0, then a/b = c/d implies ad = bc
18. If b 0 and d 0, then ad = bc implies a/b = c/d (Proof)
An expression which is true for all values of x for which the expression is meaningful is known as an IDENTITY. See #5 in previous set of examples.
When an expression can only take on certain values, then it is called an EQUATION. See #6 in previous set of examples.