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I am trying to grasp the intuition behind the historical construction of negative integers. Can subtraction be defined as follows...?

$\forall a,b\in \mathbb{N}: \exists c\in \mathbb{N}: a+b=c\iff b=c-a$

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    You may extend it to real numbers also ( also here you are assuming that addition operation is already defined) – Lalit Tolani Aug 04 '21 at 15:25
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    It's unclear if you either want a historical or formal definition of subtraction. If you want a historical construction I guess it highly depend on the civilisation. – EtienneBfx Aug 04 '21 at 15:27
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    No, you can't define subtraction like that. What is $1-2$? You can't define subtraction until you have defined the negative integers. – TonyK Aug 04 '21 at 15:28
  • @LalitTolani I'm confused, does it work as a definition? I'm indeed assuming addition is already defined. –  Aug 04 '21 at 15:30
  • @EtienneBfx I understand. Thank you. –  Aug 04 '21 at 15:31
  • @TonyK May you help me understand why? –  Aug 04 '21 at 15:37
  • @@HannyBoy as @TonyK suggests, you need negative numbers to define subtraction , you can't define negative numbers from subtraction – Lalit Tolani Aug 04 '21 at 15:59
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    @LalitTolani Let me get this straight. The problem with this definition is that if $a=4, c=5$ for instance then here's no $b\in \mathbb{N}$ such that $4+c=3$. Meaning this definition does not work because it does not tell me what to do when $b\notin \mathbb{N}$? –  Aug 04 '21 at 16:09
  • @HannyBoy Yes , you need negative numbers for such a definition to exist – Lalit Tolani Aug 04 '21 at 16:12
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    Negative numbers are far older than modern mathematical formalism. They were first discovered in ancient China, as far as I know before mathematical proofs themselves were introduced to the region. In some sense, the integers are the smallest extension of the natural numbers in which your identity holds. Formally, they're the smallest group containing the monoid of the natural numbers. – Thomas Anton Aug 04 '21 at 16:18

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