+ | - | × | ÷ | a^{b} | |

natural numbers | yes | only larger number minus smaller or equal number | yes | only if it divides evenly; can't divide by zero | yes |

integers | yes | yes | yes | only if it divides evenly; can't divide by zero | only positive and zero powers |

rational numbers | yes | yes | yes | can't divide by zero | integer powers; some fractional powers |

real numbers | yes | yes | yes | can't divide by zero | not allow some fractional powers of negative numbers e.q. (-1)^{(1/2)} |

complex numbers | yes | yes | yes | can't divide by zero | yes |

These increasingly larger groups of numbers can be seen as attempts to make subtraction, division and exponentiation work without restrictions.

The natural numbers allow exponentiation without restriction, but restrict subtraction and division. We can introduce negative numbers to allow subtraction (giving us the integers), but this forces restrictions on exponentiation.

We can then introduce fractions (giving us the rational numbers) to allow almost all divisions. Then adding irrational numbers (giving us the reals) allows fractional powers of all positive numbers and some powers of negative numbers.

To finally get back to having no restrictions on exponentiation, we need to include imaginary numbers leaving us with the complex numbers.

Beyond the complex numbers are the quaternions, octonians and sedenions.

+ | - | × | ÷ | a^{b} | |

complex numbers | yes | yes | yes | can't divide by zero | yes |

quaternions | yes | yes | not commutative | can't divide by zero | yes |

octonions | yes | yes | not commutative, not associative | can't divide by zero | yes |

sedenions | yes | yes | not commutative, not associative, not alternative | can't divide by zero | yes |

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