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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?

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Loves Probability
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    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 05:33
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    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 05:35
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    @fleablood : you mean algebraic closure vs metric completion, hard to understand though – reuns Jun 24 '16 at 05:38
  • @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 05:40
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    I do mean algebraic closure vs. metric completion. There are two issues. The real rationals are not closed. Neither are the real algebraics but we usually note it with the rationals. The rationals do not have the least upper bound property. Take the set of all rationals whose square is less than 2. There is no value that is the precise least upper bound of the set. (There is is the algebraics. root(2) is such a least upper bound. But the set of all algebraic numbers less than pi doesn't.) So we close the rationals to get the reals (which incidentally but unitentionally) ...tbc.. – fleablood Jun 24 '16 at 06:14
  • ... also include all the real algebraics. To extend the reals to include $x^2 + 1 = 0$ solution gives to complex. But had we never closed the rationals to begin with we would only have acheived the algebraic colmplex. .. tbc.. – fleablood Jun 24 '16 at 06:22
  • Basically history is: Integers aren't enough; we need to solve nx + b = 0. But that has holes. Analyst invented the reals that close the rationals. Algebrists evaluate the reals and distinguish those that solve polynomials (algebraic numbers) and those that don't (trancedental) and like that except for root(neg numbers) the rationals became completed as well. Both seem be aware that root(-1) is undefined but both thing that is weird. Until it isn't. So complex numbers are born. But this is a completion of the already closed reals. – fleablood Jun 24 '16 at 06:22
  • Okay, those last comments were probable confusing. Including solutions is to "complete" and yes we go from Natural numbers, to integers, to rationals, to real algebraic, to complete. But in the mean time non-algebraist analyst faced another issue altogether with the rationals. For sets A where all x in A; x < M, then sup A doesn't always exist in the rationals. So analists extended the rationals to the reals so sup A always does exist.This isn't completing. This is "closing". Then with all the new irrational reals, algebrists complete the reals to get the complex numbers. There was a detour. – fleablood Jun 24 '16 at 06:30
  • @fleablood that's a nice argument. Maybe, you can write one single comprehensible answer combining them all together?? – Loves Probability Jun 24 '16 at 06:36
  • Maybe I will but not tonight... – fleablood Jun 24 '16 at 06:43
  • @fleablood thank you. :) – Loves Probability Jun 24 '16 at 06:43
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    Historically, this story is completely false! Positive rational and irrational numbers were used long before negative and complex numbers were accepted (which happened roughly at the same time), and complex numbers were accepted not because they were needed for solving the quadratic equation $x^2+1=0$, but because they were needed for finding real roots of cubic equations. – Hans Lundmark Jun 24 '16 at 17:18
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    By the way, this is maybe of interest: http://math.stackexchange.com/questions/259584/why-dont-we-define-imaginary-numbers-for-every-impossibility – Hans Lundmark Jun 24 '16 at 17:19

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