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 b-a and 1/a is the same as a^-1

There are many algebraic manipulations which are listed below that result from the basic properties. Try to prove each of these statements on your own. The proofs are given for some of them if you run into difficulties.

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)

From now on, these Basic rules and manipulations will be used without reference. Examples of uses of these rules.

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.

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