x = 1 + i maps onto y = (1 + i)2 = 2i x = i maps onto y = (–1 i)2 = 2i –1 – – x = 2 + i maps onto y = (2 + i)2 = 3 + 4i x = i maps onto y = (–2 i)2 = 3 + 4i –2 – – x = 1 + 2i maps onto y = (1 + 2i)2 = + 4i –3 x = 2i maps onto y = (–1 2i)2 = + 4i –1 – – –3 x = 1 i maps onto y = (1 i)2 = – – –2i x = + i maps onto y = (–1 + i)2 = –1 –2i x = 1 2i maps onto y = (1 2i)2 = 4i – – –3 – x = + 2i maps onto y = (–1 + 2i)2 = 4i –1 –3 – x = 2 i maps onto y = (2 i)2 = 3 4i – – – x = + i maps onto y = (–2 + i)2 = 3 4i –2 – I have put these points on as GREEN DOTS AND LINES. To make this a little bit clearer I will take off the red and purple lines and dots. Notice that the 12 points chosen on the complex x plane only mapped onto 6 points on the complex y plane. At this stage there does not seem to be a clear pattern forming from these complex y points. I decided to add just a few more: x = ½ + ½ maps onto y = (½ + ½ = 0 + i i)2 i x = ½ maps onto y = (–½ ½ = 0 + i i)2 i –½ – – x = + ½ maps onto y = (–½ + ½ = 0 i i)2 i –½ – x = +½ ½ maps onto y = (+½ ½ = 0 i i)2 i – – – darker green. The above 4 point are coloured a
x = 1 + i maps onto y = (1 + i)2 = 2i x = –1 – i maps onto y = (–1 – i)2 = 2i x = 2 + i maps onto y = (2 + i)2 = 3 + 4i x = –2 – i maps onto y = (–2 – i)2 = 3 + 4i x = 1 + 2i maps onto y = (1 + 2i)2 = –3 + 4i x = –1 – 2i maps onto y = (–1 – 2i)2 = –3 + 4i x = 1 – i maps onto y = (1 – i)2 = –2i x = –1 + i maps onto y = (–1 + i)2 = –2i x = 1 – 2i maps onto y = (1 – 2i)2 = –3 – 4i x = –1 + 2i maps onto y = (–1 + 2i)2 = –3 – 4i x = 2 – i maps onto y = (2 – i)2 = 3 – 4i x = –2 + i maps onto y = (–2 + i)2 = 3 – 4i I have put these points on as GREEN DOTS AND LINES. To make this a little bit clearer I will take off the red and purple lines and dots. Notice that the 12 points chosen on the complex x plane only mapped onto 6 points on the complex y plane. At this stage there does not seem to be a clear pattern forming from these complex y points. I decided to add just a few more: x = ½ + ½ i maps onto y = (½ + ½ i)2 = 0 + i x = –½ – ½ i maps onto y = (–½ – ½ i)2 = 0 + i x = –½ + ½ i maps onto y = (–½ + ½ i)2 = 0 – i x = +½ – ½ i maps onto y = (+½ – ½ i)2 = 0 – i The above 4 point are coloured a darker green.