# Are complex numbers complete in every way?

I was told many times a story. Indeed a fascinating one to me as a student learning mathematics.

First there were natural numbers. People started adding things and finding solutions to finding the unknowns when the results of the addition are known.

The intriguing "non-existent" solutions to certain additive equations involving natural numbers, lead to finding negative numbers and zero. Complementing the set such that there is a solution to every problem of simple addition.

Then came the extensive use of multiplication to ease the laborious addition operations. Leading to problems asking to find the unknowns when the results of multiplication are known.

Extending the story, what lead to the discovery of rationals is to solve any equations involving simple multiplication.

And what lead to the discovery of irrationals is the solutions to equations involving simple exponents, and even more. (such as?)

Finally, the exciting polynomials gave birth to complex numbers in a way that every polynomial equation has all solutions within complex numbers.

My question is simply this.

### Is it the end of the story?

Can we expect anything more?

Is there a set of numbers that is sufficient for every operation that we can imagine?

Or, is it a never ending story?

• The complex numbers are algebraically closed, but depending on context, we may be interested in other number systems. For example, we could be interested in using the quaternions, which contain the complex numbers as a proper subset. Unlike the complex numbers, we have three distinct "imaginary units" $i,j$ and $k$, satisfying $i^2=j^2=k^2=ijk=-1$. In doing so, we lose the commutativity property of multiplication however, we can no longer say that $xy = yx$. – JMoravitz Jun 24 '16 at 5:33
• No. This only accounts for algebraic numbers (real and complex ) but not transcendental numbers such as pi or e. Complex Algebraic numbers, which do solve all algebraic polynomials are closed but not complete. Completing it (by declaring all cauchy sequences converge) is the and of the story.... if we are using the euclidean distance metric. If we use the p-adic metric each closure brings incompleteness and each completing lead to non-closure and the story never ends. – fleablood Jun 24 '16 at 5:35
• @fleablood : you mean algebraic closure vs metric completion, hard to understand though – reuns Jun 24 '16 at 5:38
• @fleablood It now seems incorrect of me to state that, complex numbers are to encompass all solutions of all possible polynomials. Whereas that seems to require just the "algebraic numbers", leading to say complex numbers being much more. Any alternative constructive definition to complex numbers this way? – Loves Probability Jun 24 '16 at 5:40
• By the way, this is maybe of interest: math.stackexchange.com/questions/259584/… – Hans Lundmark Jun 24 '16 at 17:19