Problem Sheet 8 MT1100 From Euclid to Mandelbrot Not to be handed in 1. Let A = {(x, y) : 0 < < < 1 and - 2 < y < 2}, and let > = 3. Draw a picture of AA. Which, if any, of the following points (x, y) are in AA? (a) (0,0) (b) (3,3) (c) (2, -7). 2. This problem sheet is printed on A4 paper. This is one of a sequence of paper sizes (A0, A1, A2, A3, A4, A5 and so on) with the following properties. If you fold a piece of An paper in half, you get the A(n + 1) size. All the sizes are similar rectangles. The area of A0 is one square metre. (i) What is the area (in square metres) of an A4 sheet of paper? (ii) What is the ratio between the lengths of the sides of an A4 sheet of paper? [You need to justify your answers.] 3. Evaluate (if possible) the following sums: (a) j=1 ? (b) =1 2(c) 700 0.01 j=1 0.025. 4. Derive a formula for (0.abc)3 (in other words for O.abcabcabc ... ) where a, b, c E {0, 1, 2}, and apply this formula to write (0.110)3 in the form n/m where n and m are integers (written in decimal). 1