About a Line and a Triangle

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Given ∆ ABC, extend the side AB beyond the vertices. Now, rotate the line AB around the vertex A until it falls on the side AC. Next rotate it (from its new position) around C until it falls on the side BC.

Lastly, rotate it around B till it takes up its erstwhile position.

It is virtually obvious that although the line now occupies exactly the same position as before, something has changed. After three rotations, the line turned around 180°. So, for example, the point A will now lie on a different side from B than before. We say that turning the line around the triangle changed its orientation.

It appears that the line occupies the same position but not quite: points on the line did not preserve their locations. However, since there are just two possible orientations of the line, we come up with an interesting question: what happens to the line after it turns around the triangle twice? Will it occupy its original position exactly (point-for-point)?

The answer is easily obtained from the following observation.

After the first rotation the line occupies the same position but with a different orientation. Let's turn the line into coordinate axis. In other words, let's choose the origin – point O, the unit of measurements, and the positive direction. If, after the rotation, the point originally at the distance x from O will be now located at the position b-x. Therefore, there exists one point on the line that does not move even after a single rotation. This is the fixed point of the transformation. The fixed point solves the equation x = b-x. The rotation of the line around the triangle is simply equivalent to the rotation of the line around that point through 180°.

Text 7. Make the written translation into Russian (time 90 minutes)

Computer Algebra

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Symbols as well as numbers can be manipulated by a computer.

New, general-purpose algorithms can undertake a wide variety of routine mathematical work and solve intractable problems by Richard Pavelle, Michael Rothstein and John Fitch

Of all the tasks to which the computer can be applied none is more daunting than the manipulation of complex mathematical expressions. For numerical calculations the digital computer is now thoroughly established as a device that can greatly ease the human burden of work. It is less generally appreciated that there are computer programs equally well adapted to the manipulation of algebraic expressions. In other words, the computer can work not only with numbers themselves but also with more abstract symbols that represent numerical quantities.

In order to understand the need for automatic systems of algebraic manipulation it must be appreciated that many concepts in science are embodied in mathematical statements where there is little point to numerical evaluation. Consider the simple expression 3π²/π.

As any student of algebra knows, the fraction can be reduced by cancelling π from both the numerator and the denominator to obtain the simplified form 3π. The numerical value of 3π may be of interest, but it may also be sufficient, and perhaps of greater utility, to leave the expression in the symbolic, nonnumeric form. With a computer programmed to do only arithmetic, the expression 3π²/π must be evaluated; when the calculation is done with a precision of 10 significant figures, the value obtained is 9,424777958. The number, besides being a rather uninformative string of digits, is not the same as the number obtained from the numerical evaluation (to 10 significant figured) of 3π.

The latter number is 9, 424777962; the discrepancy in the last two decimal places results from round-off errors introduced by the computer. The equivalence of 3 π²/π and 3π would probably not be recognized by a computer programmed in this way.

Text 8. Make the written translation into Russian (time 90 minutes)