By Leo Zippin
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21, you will have a proof, by the method of Eudoxus-Archimedes, of the famous formula for the sum of all the terms of an infinite geometric progression whose ratio r is rational and smaller than j. However, there are many proofs of this formula valid for any r numerically less than 1 . F R O M N A T U R A L N UM B E R S T O vZ 31 Fi nally, convert your answer to 1 1 -r' and observe that a 1 -r· 1 or can r exceed 1 i n the > n? (The si gn " > " i s read a + ar + ar2 + . . Can a = 0 i n thi s formula?
It is interesting to see what the first few approximations to V2 come out to be. We start with 1. Next we get l(1 + 2) = J. ) = H¥-) H. 414215 . . and is reasonably close. 4142140. It is clear that we get an infinite sequence of estimates in this way, but it is at least conceivable that after a certain number of steps we would find a fraction whose square is 2. If that could indeed happen, then the process would continue but it would give the same answer over and over. ' preceding section, V2 is not a rational number (not the ratio of two integers).
What happens when you "add" these expressions as if they were numbers? What do you expect to get? This non-terminating decimal form of the number 1 is one of the minor problems that has to be cleared up before one accepts non-terminating decimals as representing numbers. 11 . 4. 6 Infinities in Geometry Euclidian geometry abounds in infinities. Some elementary theo rems of plane geometry may be interpreted as theorems about infinite sets. 5 THE ORE M 1 . Let AB denote a line segment. The locus of all points in the plane equidistant from A and B is a line perpendicular to the segment AB through its midpoint M.