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M4005 Geometry |
February 20, 2006 |
Quiz (10 minutes)
After the quiz, there was an extended discussin of Homework set 4. Students got together in pairs to read and comment on one another's work.
Regarding problem 1.5.1. The problem as posed was: Prove that (Sqrt[2]+1)/1 = 1/(Sqrt[2]-1)and hence conclude that the rectangle with sides of length Sqrt[2]+1 and 1 is similar to the rectangle with sides 1 and Sqrt[2]-1
It is not enough to "cross multiply", observe that you get a true statement and say, "Done." The problem is, why does doing that prove anything?
Cross-multiplying is an operation on mathematical sentences of a particuar kind. Cross multiplying turns a statement of proportionality into a statement of equality between two products. The important thing is, the statement of proportionality is true exactly when the statement into which it is transformed by cross-multiplying is true.
We say that two statements are EQUIVALENT if (under the same assumptions) they are both true or both false. We write a double arrow between statements to make the assertion that they are equivalent. The double arrow stands for the phrase, "is equivalent to", or "if and only if". (Another commonly used abbreviation for equivalence is the word "iff"---a handy shorthand for "if and only if".)
Examples:
x = 1+1 <=> x = 2
x x = c c <=> {x = c OR x = -c}
A statement such as
(Sqrt[2]+1)/1 = 1/(Sqrt[2]-1)
is called an identity. One method for proving an identity is to show that the statement-to-be-proved is equivalent to another statement, which in turn is equivalent to another, etc., which is fanally equivalent to a statement that is known to be true.
Here is a proof of the identity in 1.5.1 presented in this fashion:
(Sqrt[2]+1)/1 = 1/(Sqrt[2]-1) <=> (Sqrt[2]+1)(Sqrt[2]-1) = 1 (multiplying both sides by same quantity) <=> 2-1 = 1 (arithmetic) <=> TRUE (arithmetic)
Another way of approaching 1.5.1.
There is another method available for proving identities. Rather than working with sentences and proving that the sentence to be proved is equivalent to a true sentence, one might present the following string of equalities:
1/(Sqrt[2]-1) = 1/(Sqrt[2]-1)*((Sqrt[2]+1)/(Sqrt[2]+1)) (multiplying by 1) = (Sqrt[2]+1)/((Sqrt[2]-1)(Sqrt[2]+1)) (arithmetic) = (Sqrt[2]+1)/(2-1) (arithmetic) = Sqrt[2]+1 (arithmetic) In this case, we are not transforming sentences. We are transforming terms. To explain the underlying rationale of this method of proof, we need to realize that in mathematics the things we write down are of two kinds:
- Terms (or expressions), which become meaningful by referring to a mathematical object. E.g., the numbers 1, 2,...; P (a specific point) or l (a specific line) or x (a specific but unknown number). In addition to such simple or "atomic" terms, there are complex terms that are built up from the atomic ones using operations such as addition or multiplication. For example, (x+1)*(x+2) and x*x + 3*x + 2 are complex terms.
- Sentences (e.g., equalities, inequalities, statements of congruence, similarity, etc.) which are either true or false. Sentences usually are built up from terms using the symbols for equality, inequality, congruence or whatever.
There are numerous rules that tell us that terms of some specific kind can be transformed in certain ways and that the transformation does not alter what the term refers to. For example, the distributive law of arithmetic tells us that a term of the form a*(b+c) always refers to the same thing as the term we get by making the transformation to a*b + a*c.
Now, an identity is a statement asserting that two terms, that have a different appearnce, refer to the same thing. Thus, one way to prove an identity is to show that one of the terms can be transformed into the other by using operations on terms that do not change what the terms refer to. This is the style of argument seen above. The transformations used are suggested by the comments in the right hand column.
Summary
Some proofs are made by means of linguistic transformations. One style of proof uses transformations of sentences. A proof in this style is complete when one succeeds in showing how the sentence to be proved arises from a sentence known to be true by means of transformations that preserve truth. Another style of proof---used in proving identities---proceeds by transforming terms. The goal is to show that a sequence of transformations that preserve reference leads from one side of the identity to the other.