# Imaginary into real but not vice versa?

1. Jun 3, 2009

### aaryan0077

Its kinda weird (atleast to me) that we can convert imaginary numbers into real, there is just a simple procedure/function called multiplication. You multiply two no. and you get a real ,though negative. But not to worry we can still multiply it with 2 $$i's$$ or say $$i^2$$. To make it positive.
There is no such operation. Yeah, we can surely say that
$$2 =\frac{\ 2i}{i}$$ but this is not a satisfactory answer.
So why is there a way to imaginary to rela but not vice versa?

2. Jun 4, 2009

### arildno

Quite simply because fewer laws/restrictions constitute the definition of the complex numbers than the reals, and that all the laws valid for complex numbers are valid for the reals as well.
That is, the reals can be regarded as a subset of the complex numbers, and "producible" from other complex numbers through an operation like multiplication.

Fully analogous is the case with rationals viz. integers with respect to the operation of multiplication:

It is impossible to multiply two integers and get a non-integer fraction, but 3/2*2/3, for example, equals the integer 2.

3. Jun 4, 2009

### HallsofIvy

Staff Emeritus
The real numbers are "closed" under the usual arithmetic operations. The complex numbers are also but include the real numbers as a subset.