Complex numbers

The simplest way to define a complex or imaginary number is a number that is a multiple of i in which i is defined by the property of i squared equals -1. This is puzzling to most people, because it is hard to imagine any number having a negative square. The result: it is tempting to believe that i doesn't really exist, but is just a convenient mathematical fiction. This isn't the case. Imaginary numbers …

…does there exist a number between 1 and 2?" has the answer no in the first two contexts (you cannot go to the beach and pick up more than one but fewer than two pebbles), but yes in the last two contexts (you could eat three cookie halves, which is in between one whole cookie and two whole cookies). Therefore a complex number does exist because it is still a number, but in a different number system