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As mathematics advanced ,mathematicians found out new type of numbers such as complex numbers and hypercomplex numbers . I had been really fascinated by this idea and the uses of these numbers. However I still have a doubt about them which had actually occured to me recently.When we define a number how do we know that their existence alone does not contradict the existing axioms and theories?Also ,I have heard that there are theories about real numbers whose proofs are based oncomplex numbers. How do we know that those are correct unless we are sure that complex numbers numbers do exist?Thus ,my question is ,IS THERE AN ACTUAL PROOF FOR EXISTENCE OF COMPLEX NUMBERS or is it that I am mistaken some where?Please help. Thanks in advance...

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marked as duplicate by Zev Chonoles, dxiv, Arthur, David K, Claude Leibovici Jun 1 '17 at 8:12

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    $\begingroup$ There is a set-theoretical construction for complex numbers in terms of real numbers. $\endgroup$ – Lord Shark the Unknown Jun 1 '17 at 4:38
  • $\begingroup$ Actually, numbers don't exist. Not complex numbers or any other numbers. $\endgroup$ – immibis May 30 at 0:32
  • $\begingroup$ @immibis Why though? $\endgroup$ – Deepakms Jun 1 at 13:32
  • $\begingroup$ @Deepakms Numbers are constructs of the human mind that are used to explain things in the real world. But the numbers themselves don't exist in the real world. Like the world contains red things, but not the word "red" (unless you write it down). $\endgroup$ – immibis Jun 8 at 17:49
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We can define $\mathbb C$ as the set $\mathbb R^2$ with binary operations $+_C$ and $\times_C$, where $(a,b)+_C(a',b')=(a+a',b+b')$ and $(a,b)\times_C(a'b')=(aa-bb', ab'+ba').$ The reals are then isomorphically embedded in $\mathbb C$ as $\mathbb R\times \{0\}.$

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