Consider the sequence of squares (S), with Si, S2, and S as illustrated in Figur
ID: 3282196 • Letter: C
Question
Consider the sequence of squares (S), with Si, S2, and S as illustrated in Figure 9.7.1. In Si, the sum of the lengths of the thicker lines, that is, I +1, approximates the length 5. S2 Figure 9.7.1 of the diagonal, that is, 2. The diagonal is a curve with an arc length of 2. In S2, the sum of the lengths of the thicker lines, that is, +, approximates V2, and in S3. +1++approximates 2. Observe that the shaded area between the diagonal and the starlike curves tends to 0 as k ? 00, The diagonal has an arc length of 2, whereas the length of the stairlike curve in each S, that is to approximate the length of the diagonal is consistently 2. But the value of 2 is nowhere near that of v2. How can this be?Explanation / Answer
This happens because the number of 'steps' of the 'stair' is always at most countable; in fact, it is always finite. The diagonal on the other hand consists of an uncountable number of points.
Hence, the stair, even in a countably infinite number of steps limit, never approximates the diagonal, as there can exist no bijection between an at-most countable set and an uncountable set. The stair is hence not a limiting figure to the diagonal [though the area sandwiched between the two tends to 0].
The metric used to determine the length of the diagonal is our usual Euclidean metric. But the value of 2 that we get for the length of the staircase can, in proper co-ordinates (axes parallel to the sides of the stairs), be both the Euclidean and the Taxicab metric. In the infinite limit, we never 'make the switch' from the stair to the diagonal, and we never switch from a possible Taxicab length to what can only be purely Euclidean length.