# The fundamental sets of numbers

𝔸 𝔹 ℂ 𝔻 𝔼 𝔽 𝔾 ℍ 𝕀 𝕁 𝕂 𝕃 𝕄 ℕ 𝕆 ℙ ℚ ℝ 𝕊 𝕋 𝕌 𝕍 𝕎 𝕏 𝕐 ℤ​ 𝕒 𝕓 𝕔 𝕕 𝕖 𝕗 𝕘 𝕙 𝕚 𝕛 𝕜 𝕝 𝕞 𝕟 𝕠 𝕡 𝕢 𝕣 𝕤 𝕥 𝕦 𝕧 𝕨 𝕩 𝕪 𝕫 𝟘 𝟙 𝟚 𝟛 𝟜 𝟝 𝟞 𝟟 𝟠 𝟡

The fundamental sets of numbers, AKA Types of Numbers

* Natural numbers, ℕ
  * Whole numbers, 𝕎, ℕᐩ
  * Odd numbers, 𝕆
  * Even numbers, 𝔼
  * Prime numbers, ℙ
* Integer numbers, ℤ
  * positive integers, ℤᐩ
  * negative integers, ℤ⁻
  * nonnegative integers, {0} ∪ ℕ \~ ℕ₀
  * nonpositive integers, ℤ  ℕ ∪ {0}
* Rational numbers, ℚ
  * fractional nmbers, 𝔽
* Real numbers, ℝ
  * Irrationa numbers, 𝕁
* Complex numbers, ℂ
  * Imagenary numbers, 𝕀
* Hamiltonian numbers, ℍ
* Octionian numbers, 𝕆
* Transfinite numbers, 𝕋
* Algebraic numbers, 𝔸

## Integers

Integer numbers include the natural numbers, zero and negative whole numbers. The reason for using ℤ to denote integers came from German word for whole number, viz. **Zahlen**.

Leopold Kronecker criticised Georg Cantor for his work on set theory with the quip: _"Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk"_ ("God made the integers, and all the rest is man's work"), implying that the other numbers are artificial. However, Cantor's work on _transfinite numbers_ proved to be far from artificial.

## Rational numbers

A number that equals the quotient of one integer divided by another (non-zero) integer is a rational number.

p/q for ∀p,q ∈ ℤ ∧ q ≠ 0
