Fundamentals of Number Sets and Operations

1. 📌 Essentials

Number sets: ℕ = {0,1,2,...}, ℤ = {..., -2, -1, 0, 1, 2, ...}, ⅅ = finite decimals, ℚ = rationals, ℝ = reals.

• Set inclusion: ℕ ⊂ ℤ ⊂ ⅅ ⊂ ℚ ⊂ ℝ.
• Intervals notation: [a; b], ]a; b[, [a; +∞[, ]-∞; b[.
• Interval types: open ( ]a; b[ ), closed ( [a; b] ), infinite ( [a; +∞[ ).
• Operations: Intersection ( ∩ ), Union ( ∪ ).
• Absolute value: |x| = x if x ≥ 0; = -x if x < 0.
• Distance between points: |a - b|.
• Equation with absolute value: |x - c| = d → x = c ± d.
• Inequality with absolute value: |x - c| ≤ d → x ∈ [c - d; c + d].
• Scientific notation: number = a × 10^n, with 1 ≤ a < 10.
• Bounding (Encadrement): find a < x < b for approximation.
• Inclusion relationships: ℕ ⊂ ℤ ⊂ ⅅ ⊂ ℚ ⊂ ℝ.

2. 🧩 Key Structures & Components

• Natural numbers (ℕ): Counting numbers, including zero.
• Integers (ℤ): All positive and negative whole numbers.
• Finite decimals (ⅅ): Numbers with a finite decimal expansion.
• Rational numbers (ℚ): Numbers expressible as a/b, with a, b ∈ ℤ, b ≠ 0.
• Real numbers (ℝ): All numbers on the number line, including irrationals like π, √2.
• Intervals: Defined by bounds, open or closed, finite or infinite.
• Absolute value: Measures magnitude, regardless of sign.
• Equations/inequalities: Involving |x - c|, solved via ± d or interval notation.
• Scientific notation: Compact form for large/small numbers.

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