By Isaac Asimov
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Extra info for Adding a Dimension
If, when you finish, you are disappointed to find that your answer isn't 44 A PIECE OF PI ADDING A DIMENSION as close to TT as the value of |ff, don't give up. Just add more terms. Add -^ to your answer, then subtract ^ , then add -^ and subtract -^, and so on. You can go on as long as you want to, and if any of you find out how many terms it takes to improve on -f^f, drop me a line and tell me that, too. Of course, all this may disappoint you. To be sure, the endless series is a mathematical representation of the true and exact value of TT.
Then, in 1873, the French mathematician Charles Hermite worked out a method of analysis that showed that e could not be the root of J ,. • and the other is -B-y/B2-4AC 7r-— ~• Solutions get progressively more complicated and eventually, for equations of the fifth degree and higher, no general solution can be given, although specific solutions can still be worked out. The principle remains, however, that in all polynomial equations, the value of x can be expressed by use of a finite number of integers involved in a finite number of operations, these operations consisting of addition, subtraction, multiplication, division, raising to a power ("involution"), and extracting roots ("evolution").
This holds true for forces, accelerations, and so on. For such vector quantities, complex numbers are necessary to the mathematical treatment, since complex numbers include both magnitude and direction (which was my reason for making the analogy between the four types of numbers and the compass points). Now, when my sociology professor demanded "the square root of minus one pieces of chalk," he was speaking of a scalar phenomenon for which the real numbers were sufficient. " Now, you see, he would have been dealing with a vector quantity for which the real numbers are insufficient.
Adding a Dimension by Isaac Asimov