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## Equality

In English, for a thing to be equal'' to another is to assert that two different things are in fact the same - a rather oxymoronic thing if one thinks about it. In mathematics it means that two (possibly) different symbols in fact stand for the same thing - a rather more sensible quality.

The simplest statement of equality, one that is always correct for any possible symbol that stands anything at all is the tautology:

 (3.1)

That is, whatever stands for is the same as itself. It is what it is. As a symbol for a number, it has whatever value it has. This law is so fundamental that it is difficult to imagine any sort of logical system where it is not true.

Simple or not, obvious or not, there are (amazingly enough) times we will want to start with this in a derivation. Again observe that in algebra the statement:

 (3.2)

doesn't mean that and are different things that are the same. It means that and are two symbols that stand for the same thing - whereever the symbol is used in any expression one can indifferently use and end up with exactly the same thing.

While we are discussing basic equality, we will add to these fairly obvious statements the less obvious rule of transitivity. If

 (3.3)

and
 (3.4)

then
 (3.5)

Again, this seems trivial, but it really is not. In mathematics, it means that all three symbols basically stand for the same thing. We'll use this rule implicitly all the time, since we will often change just one side of an equation with an algebraic rule applied to that side only, and the preservation of equality with the other side basically follows from transitivity. In any event it can't hurt to formally state the rule and name it at least one time.

Next: Addition Rules Up: Algebraic Transformations of Equations Previous: Algebraic Transformations of Equations   Contents
Robert G. Brown 2009-07-27