Numbers

Question 1

ℕ

Answer 1

1. Natural Numbers (ℕ)
   Definition: Counting numbers (with or without 0).

Traditional Math: ℕ = {1, 2, 3, …}

Discrete Math/CS: ℕ₀ = {0, 1, 2, …}

Trick: “Natural means no negatives, no fractions!”

Example: 5, 12, 100 (but not −3 or 0.5).

Question 2

𝕎

Answer 2

Whole Numbers (𝕎)
Definition: Natural numbers + zero.

𝕎 = {0, 1, 2, 3, …}

Trick: “Whole = Natural + Whole 0!”

Example: 0, 7, 42 (but not −1 or ½).

Question 3

ℤ

Answer 3

Definition: Whole numbers + negatives.

ℤ = {…, −2, −1, 0, 1, 2, …}

Trick: “ℤ = Zero + Zigzag (pos/neg)!”

Example: −3, 0, 42 (but not 1.5 or √2).

Question 4

(𝕀)

Answer 4

Irrational Numbers (𝕀)
Definition: Non-repeating, non-terminating decimals.

𝕀 = {√2, π, e, …}

Trick: “Irrational = Insane decimals (never end, no pattern)!”

Example: π = 3.14159…, √3 = 1.732…

Question 4

(ℚ)

Answer 4

Rational Numbers (ℚ)
Definition: Fractions of integers (a/b where b ≠ 0).

ℚ = {½, 0.75, −2, 0.333…, 5 (since 5 = 5/1)}

Trick: “ℚ = Quotient (because fractions = division)!”

Example: ¼, −2, 0.3̄ (repeating), but not π or √2.

Question 5

(ℝ)

Answer 5

Real Numbers (ℝ)
Definition: All rational + irrational numbers.

ℝ = {−∞, …, −1.5, 0, ½, √2, π, … ∞}

Trick: “ℝ = Rational + Root (irrational)!”

Example: −5, 0.3, √9 (=3), π.

Question 6

ℤ⁺

Answer 6

ℤ⁺ (Positive Integers)
Definition: The set of strictly positive integers.

ℤ⁺ = {1, 2, 3, …}

Key Point:

Same as traditional ℕ (natural numbers), but explicitly excludes zero.

Trick: “ℤ⁺ = Zero is +out!”

Example: 42 ∈ ℤ⁺, but 0 ∉ ℤ⁺, −5 ∉ ℤ⁺.