The object of this address is to pass in review various theories of complex numbers and to scrutinise them from the standpoint of the teacher engaged in initiating his pupils into this subject. The material of the address is therefore very elementary and well known and roughly a century old, but it is arranged and combined in a way which may perhaps throw some new light on an old problem. It is important to realise at the outset the fundamental issues which are involved; and, although it is undesirable to intrude explicit philosophy into a first lesson on complex numbers, it is well for the teacher to have these metamathematical considerations clear in his own mind.