= area(BCDE) + 4xarea(ABC) = a2+ 2bc . in .NET

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= area(BCDE) + 4xarea(ABC) = a2+ 2bc .
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(b+c)2 =a2 + 2bc . Whence,
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Figure 3.1 Proof of Pythagoras's theorem.
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statements is particularly great. The theorem proved is deep and interesting if a map is placed on the ground anywhere in the area covered by the map, then there will be a point on the map that is directly above the point on the ground that it
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3.1 The Nature of Proof
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Suppose a map of London is placed on the ground in the middle of Trafalgar Square. (Readers unfamiliar with London should note that Trafalgar Square is in Central London.) Then, there is a point on the map that is directly above the point on the ground that it represents, Proof. The map (of London) is directly above a part of London. Thus the (entire) map is directly above a part of the area which it represents. Now, the (smaller) area of the map representing Central London is also above a part of the area which it represents. Within the area representing Central London, Trafalgar Square is marked, and this (yet smaller) part of the map is directly above a part of the area which it represents. Continuing in this way, we can find smaller and smaller areas of the map each of which is directly above a part of the area which it represents. In the limit we reduce the area on the map to a single point which is directly above a part of the area it represents. That is, a point has been found that is directly above the point on the ground that it represents.
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Figure 3.2 A semantic proof.
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represents but it is quite difficult to understand even the statement of the theorem, and yet more difficult to understand the proof. Without a good understanding of the semantics of the proof, the reader cannot check its validity. Many would feel uneasy about the validity of the proof and would demand a more detailed justification for some of the steps. The difficulty is compounded, of course, for those for whom English is not the mother tongue. What is meant, for example, by 'the area represented' by a section of the map, and the meaning of 'in the limit'. In comparison, the language of mathematics is much simpler; moreover, unlike natural language, it is universal! At the other end of the scale, a formal proof is conducted entirely in the language of mathematics. A formal proof is a sequence of steps, each of which is a well-established fact or which follows from earlier statements by a process so simple that it is deemed to be self-evident. Figure 3.3 is an example of a formal proof. It is a proof of the fact that V2 + V is greater than V3 + VS. The first three lines of the proof state well-known facts. Each subsequent line combines one or more previous lines in order to derive some new fact; next to each of these lines is a hint explaining how the line is derived. Checking each line of Figure 3.3 is straightforward. The lines where the hint mentions 'arithmetic' involve squaring and other simple arithmetic calculations. For example, to check line 7 it is necessary to check that (1 + 2vl4) 2 is 57 + 4VT4 and that (2VT5) 2 is 57+3. The remaining lines are checked by confirming that the line is an instance of either line 0 or line 1. That is, one has to check that
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3: Calculational Proof
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0. if a > 0 and b > c> 0 then a+b > a+c > 0 1. if a > b > 0 then ya > Vb > 0 2. 224 > 9 > 0 3. ^/224 > x/9 > 0 (1 and 2) 4. 4714 > 3 > 0 (3 and arithmetic) (0 and 4) 5. 57 + 4x/l4 > 57+3 > 0 6. V57 + 4V14 > V57+3 > 0 (1 and 5) 7. 1 + 2/14 > 2vT5 > 0 (6 and arithmetic) 8. 8 + l + 2vT4 > 8 + 2/15 > 0 (0 and 7) (1 and 8) 9. V8 + 1 + 2/14 > V8 + 2/T5 > 0 (9 and arithmetic) 10. y2 + 7 7 > V 3 + 7 5 > 0
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