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